Displaying 1-10 of 17 results found.
Triangle read by rows: T(n,k) = k for n >= 1, k = 1..n.
+10
465
1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14
COMMENTS
Old name: integers 1 to k followed by integers 1 to k+1 etc. (a fractal sequence).
Start counting again and again.
The PARI functions t1, t2 can be used to read a square array T(n,k) (n >= 1, k >= 1) by antidiagonals downwards: n -> T(t1(n), t2(n)). - Michael Somos, Aug 23 2002 Reading this sequence as the antidiagonals of a rectangular array, row n is (n,n,n,...); this is the weight array (Cf. A144112) of the array A127779 (rectangular). - Clark Kimberling, Sep 16 2008 The upper trim of an arbitrary fractal sequence s is s, but the lower trim of s, although a fractal sequence, need not be s itself. However, the lower trim of A002260 is A002260. (The upper trim of s is what remains after the first occurrence of each term is deleted; the lower trim of s is what remains after all 0's are deleted from the sequence s-1.) - Clark Kimberling, Nov 02 2009 The triangle sums, see A180662 for their definitions, link this triangle of natural numbers with twenty-three different sequences, see the crossrefs. The mirror image of this triangle is A004736. - Johannes W. Meijer, Sep 22 2010 A002260 is the self-fission of the polynomial sequence (q(n,x)), where q(n,x) = x^n + x^(n-1) + ... + x + 1. See A193842 for the definition of fission. - Clark Kimberling, Aug 07 2011
Sequence B is called a reluctant sequence of sequence A, if B is triangle array read by rows: row number k coincides with first k elements of the sequence A. Sequence A002260 is reluctant sequence of sequence 1,2,3,... ( A000027). - Boris Putievskiy, Dec 12 2012 This is the maximal sequence of positive integers, such that once an integer k has occurred, the number of k's always exceeds the number of (k+1)'s for the remainder of the sequence, with the first occurrence of the integers being in order. - Franklin T. Adams-Watters, Oct 23 2013 A002260 are the k antidiagonal numerators of rationals in Cantor's proof of 1-to-1 correspondence between rationals and naturals; the denominators are k-numerator+1. - Adriano Caroli, Mar 24 2015
T(n,k) gives the distance to the largest triangular number < n. - Ctibor O. Zizka, Apr 09 2020
REFERENCES
Clark Kimberling, "Fractal sequences and interspersions," Ars Combinatoria 45 (1997) 157-168. (Introduces upper trimming, lower trimming, and signature sequences.)
M. Myers, Smarandache Crescendo Subsequences, R. H. Wilde, An Anthology in Memoriam, Bristol Banner Books, Bristol, 1998, p. 19.
F. Smarandache, Sequences of Numbers Involved in Unsolved Problems, Hexis, Phoenix, 2006.
LINKS
Jerry Brown et al., Problem 4619, School Science and Mathematics (USA), Vol. 97(4), 1997, pp. 221-222. Glen Joyce C. Dulatre, Jamilah V. Alarcon, Vhenedict M. Florida, and Daisy Ann A. Disu, On Fractal Sequences, DMMMSU-CAS Science Monitor (2016-2017) Vol. 15 No. 2, 109-113.
FORMULA
n-th term is n - m*(m+1)/2 + 1, where m = floor((sqrt(8*n+1) - 1) / 2).
The above formula is for offset 0; for offset 1, use a(n) = n-m*(m+1)/2 where m = floor((-1+sqrt(8*n-7))/2). - Clark Kimberling, Jun 14 2011 a(k * (k + 1) / 2 + i) = i for k >= 0 and 0 < i <= k + 1. - Reinhard Zumkeller, Aug 14 2001 a(n) = (2*n + round(sqrt(2*n)) - round(sqrt(2*n))^2)/2. - Brian Tenneson, Oct 11 2003 a(n) = n - binomial(floor((1+sqrt(8*n))/2), 2). - Paul Barry, May 25 2004 T(n,k) = Sum_{i=1..k} i*binomial(k,i)*binomial(n-k,n-i) (regarded as triangle, see the example). - Mircea Merca, Apr 11 2012 T(n,k) = Sum_{i=max(0,n+1-2*k)..n-k+1} (i+k)*binomial(i+k-1,i)*binomial(k,n-i-k+1)*(-1)^(n-i-k+1). - Vladimir Kruchinin, Oct 18 2013 G.f.: x*y / ((1 - x) * (1 - x*y)^2) = Sum_{n,k>0} T(n,k) * x^n * y^k. - Michael Somos, Sep 17 2014
EXAMPLE
First six rows:
1
1 2
1 2 3
1 2 3 4
1 2 3 4 5
1 2 3 4 5 6
MAPLE
at:=0; for n from 1 to 150 do for i from 1 to n do at:=at+1; lprint(at, i); od: od: # N. J. A. Sloane, Nov 01 2006
PROG
(PARI) t1(n)=n-binomial(floor(1/2+sqrt(2*n)), 2) /* this sequence */
(Haskell)
a002260 n k = k
a002260_row n = [1..n]
a002260_tabl = iterate (\row -> map (+ 1) (0 : row)) [1]
(Maxima) T(n, k):=sum((i+k)*binomial(i+k-1, i)*binomial(k, n-i-k+1)*(-1)^(n-i-k+1), i, max(0, n+1-2*k), n-k+1); /* Vladimir Kruchinin, Oct 18 2013 */ (Python)
from math import isqrt, comb
def A002260(n): return n-comb((m:=isqrt(k:=n<<1))+(k>m*(m+1)), 2) # Chai Wah Wu, Nov 08 2024
AUTHOR
Angele Hamel (amh(AT)maths.soton.ac.uk)
Number of (w,x,y) such that w,x,y are all in {0,...,n} and |w-x| = |x-y|.
+10
77
1, 4, 11, 20, 33, 48, 67, 88, 113, 140, 171, 204, 241, 280, 323, 368, 417, 468, 523, 580, 641, 704, 771, 840, 913, 988, 1067, 1148, 1233, 1320, 1411, 1504, 1601, 1700, 1803, 1908, 2017, 2128, 2243, 2360, 2481, 2604, 2731, 2860, 2993, 3128, 3267
COMMENTS
In the following guide to related sequences: M=max(x,y,z), m=min(x,y,z), and R=range=M-m. In some cases, it is an offset of the listed sequence which fits the conditions shown for w,x,y. Each sequence satisfies a linear recurrence relation, some of which are identified in the list by the following code (signature):
A: 2, 0, -2, 1, i.e., a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4);
B: 3, -2, -2, 3, -1;
C: 4, -6, 4, -1;
D: 1, 2, -2, -1, 1;
E: 2, 1, -4, 1, 2, -1;
F: 2, -1, 1, -2, 1;
G: 2, -1, 0, 1, -2, 1;
H: 2, -1, 2, -4, 2, -1, 2, -1;
I: 3, -3, 2, -3, 3, -1;
J: 4, -7, 8, -7, 4, -1.
...
A212959 ... |w-x|=|x-y| ...... recurrence type A
A212960 ... |w-x| != |x-y| ................... B
A212683 ... |w-x| < |x-y| .................... B
A212684 ... |w-x| >= |x-y| ................... B
A212963 ... see entry for definition ......... B
A212964 ... |w-x| < |x-y| < |y-w| ............ B
A006331 ... |w-x| < y ........................ C
A005900 ... |w-x| <= y ....................... C
A212965 ... w = R ............................ D
A212967 ... w < R ............................ E
A212968 ... w >= R ........................... E
A077043 ... w = x > R ........................ A
A212969 ... w != x and x > R ................. E
A212970 ... w != x and x < R ................. E
A011934 ... w < floor((x+y)/2) ............... B
A182260 ... w > floor((x+y)/2) ............... B
A055232 ... w <= floor((x+y)/2) .............. B
A011934 ... w >= floor((x+y)/2) .............. B
A212971 ... w < floor((x+y)/3) ............... B
A212972 ... w >= floor((x+y)/3) .............. B
A212973 ... w <= floor((x+y)/3) .............. B
A212974 ... w > floor((x+y)/3) ............... B
A212975 ... R is even ........................ E
A212976 ... R is odd ......................... E
A212980 ... w < x + y and x < y .............. B
A212981 ... w <= x+y and x < y ............... B
A212982 ... w < x + y and x <= y ............. B
A212983 ... w <= x + y and x <= y ............ B
A002623 ... w >= x + y and x <= y ............ B
A087811 ... w = 2*x + y ...................... A
A008805 ... w = 2*x + 2*y .................... D
A000982 ... 2*w = x + y ...................... F
A001318 ... 2*w = 2*x + y .................... F
A008810 ... 3*x = 2*x + y .................... F
A001972 ... w = 4*x + y ...................... G
A212988 ... 4*w = x + y ...................... G
A016061 ... n < w + x + y <= 2*n ............. C
A000292 ... w + x + y <=n .................... C
A000292 ... 2*n < w + x + y <= 3*n ........... C
A143785 ... w < R < x ........................ E
A005996 ... w < R <= x ....................... E
A128624 ... w <= R <= x ...................... E
A213041 ... R = 2*|w - x| .................... A
A213045 ... R < 2*|w - x| .................... B
A087035 ... R >= 2*|w - x| ................... B
A213388 ... R <= 2*|w - x| ................... B
A171218 ... M < 2*m .......................... B
A213389 ... R < 2|w - x| ..................... E
A213390 ... M >= 2*m ......................... E
A213391 ... 2*M < 3*m ........................ H
A213392 ... 2*M >= 3*m ....................... H
A213393 ... 2*M > 3*m ........................ H
A213391 ... 2*M <= 3*m ....................... H
A047838 ... w = |x + y - w| .................. A
A213396 ... 2*w < |x + y - w| ................ I
A213397 ... 2*w >= |x + y - w| ............... I
A000384 ... max(|w-x|,|x-y|) = |w-y|
A213398 ... min(|w-x|,|x-y|) = x ............. A
A213399 ... max(|w-x|,|x-y|) = x ............. D
A213479 ... max(|w-x|,|x-y|) = w+x+y ......... D
A213480 ... max(|w-x|,|x-y|) != w+x+y ........ E
A006918 ... |w-x| + |x-y| > w+x+y ............ E
A213481 ... |w-x| + |x-y| <= w+x+y ........... E
A213482 ... |w-x| + |x-y| < w+x+y ............ E
A213483 ... |w-x| + |x-y| >= w+x+y ........... E
A213484 ... |w-x|+|x-y|+|y-w| = w+x+y
A213485 ... |w-x|+|x-y|+|y-w| != w+x+y ....... J
A213486 ... |w-x|+|x-y|+|y-w| > w+x+y ........ J
A213487 ... |w-x|+|x-y|+|y-w| >= w+x+y ....... J
A213488 ... |w-x|+|x-y|+|y-w| < w+x+y ........ J
A213489 ... |w-x|+|x-y|+|y-w| <= w+x+y ....... J
A213490 ... w,x,y,|w-x|,|x-y| distinct
A213491 ... w,x,y,|w-x|,|x-y| not distinct
A213493 ... w,x,y,|w-x|,|x-y|,|w-y| distinct
A213495 ... w = min(|w-x|,|x-y|,|w-y|)
A213492 ... w != min(|w-x|,|x-y|,|w-y|)
A213498 ... w != max(|w-x|,|x-y|,|w-y|)
...
A211795 includes a guide for sequences that count 4-tuples (w,x,y,z) having all terms in {0,...,n} and satisfying selected properties. Some of the sequences indexed at A211795 satisfy recurrences that are represented in the above list.
REFERENCES
A. Barvinok, Lattice Points and Lattice Polytopes, Chapter 7 in Handbook of Discrete and Computational Geometry, CRC Press, 1997, 133-152.
P. Gritzmann and J. M. Wills, Lattice Points, Chapter 3.2 in Handbook of Convex Geometry, vol. B, North-Holland, 1993, 765-797.
FORMULA
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
G.f.: (1+2*x+3*x^2)/((1+x)*(1-x)^3).
a(n) = (6*n^2 + 8*n + 3 + (-1)^n)/4. - Luce ETIENNE, Apr 05 2014
EXAMPLE
a(1)=4 counts these (x,y,z): (0,0,0), (1,1,1), (0,1,0), (1,0,1).
Numbers congruent to {1, 3} mod 6: 1, 3, 7, 9, 13, 15, 19, ...
a(0) = 1;
a(1) = 1 + 3 = 4;
a(2) = 1 + 3 + 7 = 11;
a(3) = 1 + 3 + 7 + 9 = 20;
a(4) = 1 + 3 + 7 + 9 + 13 = 33;
MATHEMATICA
t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[Abs[w - x] == Abs[x - y], s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 50]] (* A212959 *)
a(n) = Sum_{k=0..n} floor(k/4). (Partial sums of A002265.)
+10
25
0, 0, 0, 0, 1, 2, 3, 4, 6, 8, 10, 12, 15, 18, 21, 24, 28, 32, 36, 40, 45, 50, 55, 60, 66, 72, 78, 84, 91, 98, 105, 112, 120, 128, 136, 144, 153, 162, 171, 180, 190, 200, 210, 220, 231, 242, 253, 264, 276, 288, 300, 312, 325, 338, 351, 364, 378, 392, 406, 420, 435, 450
COMMENTS
Complementary to A130482 with respect to triangular numbers, in that A130482(n) + 4*a(n) = n(n+1)/2 = A000217(n). Disregarding the first three 0's the resulting sequence a'(n) is the sum of the positive integers <= n that have the same residue modulo 4 as n. This is the additive counterpart of the quadruple factorial numbers. - Peter Luschny, Jul 06 2011 Column sums of (shift of rows = 4):
1 2 3 4 5 6 7 8 9 10 11 12 13 14 ...
1 2 3 4 5 6 7 8 9 10 ...
1 2 3 4 5 6 ...
1 2 ...
.......................................
---------------------------------------
1 2 3 4 6 8 10 12 15 18 21 24 28 32 ...
(End)
Conjecture: a(n+2) is the maximum effective weight of a numerical semigroup S of genus n (see Nathan Pflueger). - Stefano Spezia, Jan 04 2019
FORMULA
G.f.: x^4/((1-x^4)*(1-x)^2) = x^4/((1+x)*(1+x^2)*(1-x)^3).
a(n) = +2*a(n-1) -1*a(n-2) +1*a(n-4) -2*a(n-5) +1*a(n-6).
a(n) = floor(n/4)*(n - 1 - 2*floor(n/4)) = A002265(n)*(n - 1 - 2* A002265(n)). a(n) = round(n*(n-2)/8) = round((n^2-2*n-1)/8) = ceiling((n+1)*(n-3)/8). - Mircea Merca, Nov 28 2010 Euler transform of length 4 sequence [ 2, 0, 0, 1]. - Michael Somos, Oct 14 2011 a(n) = Sum_{i=1..ceiling(n/2)-1} (i mod 2) * (n - 2*i - 1). - Wesley Ivan Hurt, Jan 23 2014 a(n) = ( 2*n^2-4*n-1+(-1)^n+2*((-1)^((2*n-1+(-1)^n)/4)-(-1)^((6*n-1+(-1)^n)/4)) )/16 = ( 2*n*(n-2) - (1-(-1)^n)*(1-2*i^(n*(n-1))) )/16, where i=sqrt(-1). - Luce ETIENNE, Aug 29 2014 E.g.f.: (1/8)*((- 1 + x)*x*cosh(x) + 2*sin(x) + (- 1 - x + x^2)*sinh(x)). - Stefano Spezia, Jan 15 2019 Sum_{n>=4} 1/a(n) = Pi^2/12 + 5/2. - Amiram Eldar, Aug 13 2022
EXAMPLE
G.f. = x^4 + 2*x^5 + 3*x^6 + 4*x^7 + 6*x^8 + 8*x^9 + 10*x^10 + 12*x^11 + ...
[ n] a(n)
---------
[ 4] 1
[ 5] 2
[ 6] 3
[ 7] 4
[ 8] 1 + 5
[ 9] 2 + 6
[10] 3 + 7
[11] 4 + 8
MAPLE
quadsum := n -> add(k, k = select(k -> k mod 4 = n mod 4, [$1 .. n])):
MATHEMATICA
a[ n_] := Quotient[ (n - 1)^2, 8]; (* Michael Somos, Oct 14 2011 *)
PROG
(Maxima) makelist(floor((n-1)^2/8), n, 0, 70); /* Stefano Spezia, Jan 04 2019 */ (GAP) a:=List([0..65], n->Sum([0..n], k->Int(k/4)));; Print(a); # Muniru A Asiru, Jan 04 2019 (Python)
CROSSREFS
Cf. A000217, A001840, A002264, A002265, A002266, A002620, A004526, A010872, A010873, A010874, A130481, A130483, A130520.
Number of subsets of {1,2,...,n} such that no two elements differ by 1, 4, or 5.
+10
13
1, 2, 3, 5, 8, 11, 14, 19, 25, 34, 49, 70, 99, 141, 196, 270, 375, 520, 723, 1014, 1420, 1985, 2777, 3874, 5396, 7526, 10496, 14642, 20449, 28555, 39860, 55647, 77660, 108356, 151214, 211028, 294507, 411071, 573763, 800796, 1117679, 1559895, 2177002
COMMENTS
a(n) is the number of compositions of n+5 into parts 1, 6, 8, 9, 12, 15, 18, 21, ...
Other sequences related to restricted combinations along with the sets of disallowed differences between subset elements: A000045 {1}, A011973 {1}, A006498 {2}, A006500 {3}, A031923 {4}, A000930 {1,2}, A102547 {1,2}, A130137 {1,3}, A263710 {1,4}, A374737 {1,5}, A079972 {2,3}, A224809 {2,4}, A351873 {3,4}, A224810 {3,6}, A224815 {4,8}, A003269 {1,2,3}, A180184 {1,2,3}, A317669 {1,2,4}, A351874 {1,3,4}, A177485 {1,3,5}, A121832 {2,3,4}, A375982 {2,3,5}, A375983 {2,4,5}, A224808 {2,4,6}, A224814 {3,6,9}, A003520 {1,2,3,4}, A375185 {1,2,3,5}, A375186 {1,2,4,5}, A259278 {2,3,4,5}, A224811 {2,4,6,8}, A005708 {1,2,3,4,5}, A276106 {2,3,4,5,6}, A224812 {2,4,6,8,10}, A005709 {1,2,3,4,5,6}, A322405 {2,3,4,5,6,7}, A224813 {2,4,6,8,10,12}, A005710 {1,2,3,4,5,6,7}, A368244 {2,3,4,5,6,7,8}, A000027 {1,2,..}, A269445 {1,2,..}\{12,25,..}, A008730 {1,2,..}\{11,23,..}, A008729 {1,2,..}\{10,21,..}, A008728 {1,2,..}\{9,19,..}, A008727 {1,2,..}\{8,17,..}, A008726 {1,2,..}\{7,15,..}, A008725 {1,2,..}\{6,13,..}, A038718 {1,..,5,7,..}, A008724 {1,2,..}\{5,11,..}, A008732 {1,2,..}\{4,9,..}, A179999 {1,2,3,5,7,..}, A001972 {1,2,..}\{3,7,..}, A001840 {1,2,..}\{2,5,..}, A052955 {1,3,..}, A004277 {2,3,..}, A186384 {1,2,..}\{1,6,..}, A186347 {1,2,..}\{1,5,..}, A339573 {1,2,..}\{1,4,..}, A002620 {2,4,..}, A019442 {3,4,..}, A006501 {3,6,..}, A008233 {4,8,..}, A008382 {5,10,..}, A008881 {6,12,..}, A009641 {7,14,..}, A009694 {8,16,..}, A009714 {9,18,..}, A354600 {10,20,..}. [Keyword "less", because this comment should be moved to the Index to the OEIS, it is not appropriate here. - N. J. A. Sloane, Oct 25 2024]
LINKS
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1,0,1,0,1,0,0,-1).
FORMULA
a(n) = a(n-1) + a(n-3) - a(n-4) + a(n-6) + a(n-8) - a(n-11) for n >= 11.
G.f.: (1 + x + x^2 + x^3 + 2*x^4 + 2*x^5 - x^8 - x^9 - x^10)/(1 - x - x^3 + x^4 - x^6 - x^8 + x^11).
EXAMPLE
For n = 6, the 14 subsets are {}, {1}, {2}, {3}, {1,3}, {4}, {1,4}, {2,4}, {5}, {2,5}, {3,5}, {6}, {3,6}, {4,6}.
The a(4) = 8 compositions of 9 into parts 1, 6, 8, 9, ... are 1+1+1+1+1+1+1+1+1, 1+1+1+6, 1+1+6+1, 1+6+1+1, 6+1+1+1, 1+8, 8+1, 9.
MATHEMATICA
CoefficientList[Series[(1 + x + x^2 + x^3 + 2*x^4 + 2*x^5 - x^8 - x^9 - x^10)/(1 - x - x^3 + x^4 - x^6 - x^8 + x^11), {x, 0, 42}], x]
LinearRecurrence[{1, 0, 1, -1, 0, 1, 0, 1, 0, 0, -1}, {1, 2, 3, 5, 8, 11, 14, 19, 25, 34, 49}, 42]
CROSSREFS
See comments for other sequences related to restricted combinations.
Molien series for 3-dimensional group [2,n ] = *22n.
+10
9
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150, 156, 162, 168, 174, 180, 186, 192, 198, 204, 210, 217, 224, 231, 238
LINKS
Index entries for linear recurrences with constant coefficients, signature (2, -1, 0, 0, 0, 0, 0, 0, 0, 1, -2, 1).
FORMULA
G.f.: 1/((1-x)^2*(1-x^10)).
a(n) = Sum_{j=0..n+10} floor(j/10).
a(n-10) = (1/2)*floor(n/10)*(2*n - 8 - 10*floor(n/10)). (End)
MAPLE
g:= 1/((1-x)^2*(1-x^10)); gser:= series(g, x=0, 72); seq(coeff(gser, x, n), n=0..70); # modified by G. C. Greubel, Jul 30 2019
MATHEMATICA
CoefficientList[Series[1/((1-x)^2(1-x^10)), {x, 0, 70}], x] (* Vincenzo Librandi, Jun 11 2013 *)
PROG
(PARI) my(x='x+O('x^70)); Vec(1/((1-x)^2*(1-x^10))) \\ G. C. Greubel, Jul 30 2019 (Magma) R<x>:=PowerSeriesRing(Integers(), 70); Coefficients(R!( 1/((1-x)^2*(1-x^10)) )); // G. C. Greubel, Jul 30 2019 (Sage) (1/((1-x)^2*(1-x^10))).series(x, 70).coefficients(x, sparse=False) # G. C. Greubel, Jul 30 2019 (GAP) a:=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14];; for n in [13..70] do a[n]:=2*a[n-1]-a[n-2]+a[n-10]-2*a[n-11]+a[n-12]; od; a; # G. C. Greubel, Jul 30 2019
Partial sums of floor(n^2/8).
+10
8
0, 0, 0, 1, 3, 6, 10, 16, 24, 34, 46, 61, 79, 100, 124, 152, 184, 220, 260, 305, 355, 410, 470, 536, 608, 686, 770, 861, 959, 1064, 1176, 1296, 1424, 1560, 1704, 1857, 2019, 2190, 2370, 2560, 2760, 2970, 3190, 3421, 3663, 3916, 4180, 4456, 4744, 5044, 5356, 5681, 6019, 6370
COMMENTS
Degree of the polynomial P(n+1,x), defined by P(n,x) = [x^(n-1)*P(n-1,x)*P(n-4,x)+P(n-2,x)*P(n-3,x)]/P(n-5,x) with P(1,x)=P(0,x)=P(-1,x)=P(-2,x)=P(-3,x)=1.
Define the sequence b(n) = 1, 4, 10, 20, 36, 60,... for n>=0 with g.f. 1/((1+x)*(1+x^2)*(1-x)^5). Then a(n+3) = b(n)-b(n-1) and b(n)+b(n+1)+b(n+2)+b(n+3) = A052762(n+7)/24. - J. M. Bergot, Aug 21 2013 Maximum Wiener index of all maximal 4-degenerate graphs with n-1 vertices. (A maximal 4-degenerate graph can be constructed from a 4-clique by iteratively adding a new 4-leaf (vertex of degree 4) adjacent to four existing vertices.) The extremal graphs are 4th powers of paths, so the bound also applies to 4-trees. - Allan Bickle, Sep 15 2022
FORMULA
a(n) = Sum_{k=0..n} floor(k^2/8).
a(n) = round((2*n^3 + 3*n^2 - 8*n)/48) = round((4*n^3 + 6*n^2 - 16*n - 9)/96) = floor((2*n^3 + 3*n^2 - 8*n + 3)/48) = ceiling((2*n^3 + 3*n^2 - 8*n - 12)/48). - Mircea Mercaa(n+1) = cos((2*n+1)*Pi/4)/(4*sqrt(2)) + (2*n+3)*(2*n^2 + 6*n - 5)/96 + (-1)^n/32.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-4) - 3*a(n-5) + 3*a(n-6) - a(n-7). (End)
O.g.f.: x^3 / ( (1+x)*(x^2+1)*(x-1)^4 ). - R. J. Mathar, Jul 15 2008 a(n+3) = Sum_{k=0..6} min(6-k+1,k+1)* A190718(n+k-6). (End) a(n) = (4*n^3 + 6*n^2 - 16*n - 9 - 3*(-1)^n + 12*(-1)^((2*n - 1 + (-1)^n)/4))/96. - Luce ETIENNE, Mar 21 2014 E.g.f.: ((2*x^3 + 9*x^2 - 3*x - 6)*cosh(x) + 6*(cos(x) + sin(x)) + (2*x^3 + 9*x^2 - 3*x - 3)*sinh(x))/48. - Stefano Spezia, Apr 05 2023
EXAMPLE
a(6) = 10 = 0 + 0 + 0 + 1 + 2 + 3 + 4.
MATHEMATICA
p[n_] := p[n] = Cancel[Simplify[ (x^(n - 1)p[n - 1]p[n - 4] + p[n - 2]*p[n - 3])/p[n - 5]]]; p[ -5] = 1; p[ -4] = 1; p[ -3] = 1; p[ -2] = 1; p[ -1] = 1; Table[Exponent[p[n], x], {n, 0, 20}]
Accumulate[Floor[Range[0, 60]^2/8]] (* or *) LinearRecurrence[{3, -3, 1, 1, -3, 3, -1}, {0, 0, 0, 1, 3, 6, 10}, 60] (* Harvey P. Dale, Dec 23 2019 *)
PROG
(Magma) [Round((2*n^3+3*n^2-8*n)/48): n in [0..60]]; // Vincenzo Librandi, Jun 25 2011
EXTENSIONS
More formulas and better name from Mircea Merca, Nov 19 2010
Molien series 1/((1-x)^2*(1-x^8)) for 3-dimensional group [2,n] = *22n.
+10
6
1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 18, 20, 22, 24, 27, 30, 33, 36, 39, 42, 45, 48, 52, 56, 60, 64, 68, 72, 76, 80, 85, 90, 95, 100, 105, 110, 115, 120, 126, 132, 138, 144, 150, 156, 162, 168, 175, 182, 189, 196, 203, 210, 217, 224, 232, 240, 248, 256, 264, 272, 280
LINKS
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,1,-2,1).
FORMULA
G.f.: 1/((1-x)^2*(1-x^8)).
a(n) = Sum_{j=0..n+8} floor(j/8).
a(n-8) = (1/2)*floor(n/8)*(2*n-6-8*floor(n/8)). (End)
a(n) = 2*a(n-1) - a(n-2) + a(n-8) - 2*a(n-9) + a(n-10). - R. J. Mathar, Apr 20 2010
MAPLE
seq(coeff(series(1/(1-x)^2/(1-x^8), x, n+1), x, n), n=0..80);
MATHEMATICA
CoefficientList[Series[1/((1-x)^2*(1-x^8)), {x, 0, 80}], x] (* Vincenzo Librandi, Jun 11 2013 *) LinearRecurrence[{2, -1, 0, 0, 0, 0, 0, 1, -2, 1}, {1, 2, 3, 4, 5, 6, 7, 8, 10, 12}, 80] (* Harvey P. Dale, Jan 07 2015 *)
PROG
(PARI) my(x='x+O('x^80)); Vec(1/((1-x)^2*(1-x^8))) \\ G. C. Greubel, Sep 09 2019 (Magma) R<x>:=PowerSeriesRing(Integers(), 80); Coefficients(R!( 1/((1-x)^2*(1-x^8)) )); // G. C. Greubel, Sep 09 2019 (Sage)
P.<x> = PowerSeriesRing(ZZ, prec)
return P(1/((1-x)^2*(1-x^8))).list()
(GAP) a:=[1, 2, 3, 4, 5, 6, 7, 8, 10, 12];; for n in [11..80] do a[n]:=2*a[n-1] -a[n-2]+a[n-8]-2*a[n-9]+a[n-10]; od; a; # G. C. Greubel, Sep 09 2019
Expansion of 1/((1-2*x)*(1-x^4)).
+10
6
1, 2, 4, 8, 17, 34, 68, 136, 273, 546, 1092, 2184, 4369, 8738, 17476, 34952, 69905, 139810, 279620, 559240, 1118481, 2236962, 4473924, 8947848, 17895697, 35791394, 71582788, 143165576, 286331153, 572662306, 1145324612, 2290649224
COMMENTS
Here we let p = 4 to produce the above sequence, but p can be an arbitrary natural number. By letting p = 2, 3, 6, 7 we produce A000975, A033138, A195904 and A117302. We denote by U[p,n,m] the number of cases in which the first player gets killed in a Russian roulette game when p players use a gun with n chambers and m bullets. They never rotate the cylinder after the game starts. The chambers can be represented by the list {1,2,...,n}. We are going to calculate the following (0), (1), ..., (t) separately. (0) The first player gets killed when one bullet is in the first chamber and the remaining m-1 bullets are in {2,3,...,n}. We have binomial(n-1,m-1) cases for this. (1) The first gets killed when one bullet is in the (p+1)th chamber and the rest of the bullets are in {p+2,...,n}. We have binomial(n-p-1,m-1) cases for this. We continue to calculate and the last is (t), where t = floor((n-m)/p). (t) The first gets killed when one bullet is in the (pt+1)-st chamber and the remaining bullets are in {pt+2,...,n}. We have binomial(n-pt-1,m-1) cases for this. Therefore U[p,n,m] = Sum_{z=0..floor((n-m)/p)} binomial(n-pz-1,m-1). Let A[p,n] be the number of the cases in which the first player gets killed when p players use a gun with n chambers and the number of the bullets can be from 1 to n. Then A[p,n] = Sum_{m=1..n} U[p,n,m]. - Ryohei Miyadera, Tomohide Hashiba, Yuta Nakagawa, Hiroshi Matsui, Jun 04 2006 a(n) is the number of partitions of n into parts 1 and 4 where there are two colors of part 1 and the order of the colors of parts 1 matters. If the order of colors doesn't matter we get A001972. - Joerg Arndt, Jan 18 2024
FORMULA
a(n) = 2*a(n-1) + a(n-4) - 2*a(n-5).
If n is a multiple of 4, then a(n) = 2*a(n-1) + 1, otherwise a(n) = 2*a(n-1). - Gerald McGarvey, Oct 14 2008 a(n) = 2*a(n-1) + floor(((n-1) mod 4) /3), with a(0)=1. - Andres Cicuttin, Mar 29 2016 a(n) = 2*a(n-1) + 1 - ceiling((n mod 4)/4), with a(0)=1. - Andres Cicuttin, Mar 29 2016
MATHEMATICA
U[p_, n_, m_, v_]:=Block[{t}, t=Floor[(1+p-m+n-v)/p]; Sum[Binomial[n-v-p*z, m-1], {z, 0, t-1}]]; A[p_, n_, v_]:=Sum[U[p, n, k, v], {k, 1, n}]; (* Here we let p = 4 to produce the above sequence, but this code can produce A000975, A033138, A195904, A117302 for p=2, 3, 6, 7.*) Table[A[4, n, 1], {n, 1, 20}] (* Ryohei Miyadera, Tomohide Hashiba, Yuta Nakagawa, Hiroshi Matsui, Jun 04 2006 *) CoefficientList[Series[1/((1-2x)(1-x^4)), {x, 0, 40}], x] (* Vincenzo Librandi, Apr 04 2012 *) a[n_] := FromDigits[Table[(Mod[j, 4]/4) // Round, {j, 1, n + 3}], 2] (* Andres Cicuttin, Mar 25 2016 *) a[n_] := a[n] = 2 a[n - 1] + 1 - Ceiling[Mod[n, 4]/4]; a[0] = 1;
LinearRecurrence[{2, 0, 0, 1, -2}, {1, 2, 4, 8, 17}, 40] (* Harvey P. Dale, Apr 03 2018 *)
Molien series for 3-dimensional group [2,n] = *22n.
+10
5
1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 58, 62, 66, 70, 74, 78, 82, 86, 90, 95, 100, 105, 110, 115, 120, 125, 130, 135, 141, 147, 153, 159, 165, 171, 177, 183, 189, 196, 203, 210, 217, 224, 231, 238, 245, 252
COMMENTS
Number of partitions of n into two kinds of 1's and one kind of 9. - Joerg Arndt, Dec 27 2014
LINKS
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,0,1,-2,1).
FORMULA
G.f.: 1/((1-x)^2*(1-x^9)).
a(n) = Sum_{j=0..n+9} floor(j/9).
a(n-9) = (1/2)*floor(n/9)*(2*n - 7 - 9*floor(n/9)). (End)
MAPLE
seq(coeff(series(1/((1-x)^2*(1-x^9)), x, n+1), x, n), n = 0..70); # G. C. Greubel, Sep 09 2019
MATHEMATICA
CoefficientList[Series[1/(1-x)^2/(1-x^9), {x, 0, 70}], x] (* Vincenzo Librandi, Jun 11 2013 *) LinearRecurrence[{2, -1, 0, 0, 0, 0, 0, 0, 1, -2, 1}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13}, 120] (* Harvey P. Dale, Feb 13 2022 *)
PROG
(PARI) Vec(1/(1-x)^2/(1-x^9)+O(x^66)) /* Joerg Arndt, Mar 27 2013 */ (Magma) R<x>:=PowerSeriesRing(Integers(), 70); Coefficients(R!( 1/((1-x)^2*(1-x^9)) )); // G. C. Greubel, Sep 09 2019 (Sage)
P.<x> = PowerSeriesRing(ZZ, prec)
return P(1/((1-x)^2*(1-x^9))).list()
(GAP) a:=[1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13];; for n in [12..70] do a[n]:=2*a[n-1]-a[n-2]+a[n-9]-2*a[n-10]+a[n-11]; od; a; # G. C. Greubel, Sep 09 2019
Molien series for 3-dimensional group [2, n] = *22n.
+10
5
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 70, 74, 78, 82, 86, 90, 94, 98, 102, 106, 110, 115, 120, 125, 130, 135, 140, 145, 150, 155, 160, 165, 171, 177, 183, 189, 195, 201, 207, 213, 219
LINKS
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,0,0,0,1,-2,1).
FORMULA
a(n) = Sum_{j=0..n+11} floor(j/11).
a(n-11) = (1/2)*floor(n/11)*(2*n - 9 - 11*floor(n/11)). (End)
a(n) = 2*a(n-1) - a(n-2) + a(n-11) - 2*a(n-12) + a(n-13) for n > 12.
G.f.: 1/(1 - 2*x + x^2 - x^11 + 2*x^12 - x^13) = 1/((1-x)^3 *(1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^10)). (End)
EXAMPLE
..1....2....3....4....5....6....7....8....9...10...11
.13...15...17...19...21...23...25...27...29...31...33
.36...39...42...45...48...51...54...57...60...63...66
.70...74...78...82...86...90...94...98..102..106..110
115..120..125..130..135..140..145..150..155..160..165
171..177..183..189..195..201..207..213..219..225..231
238..245..252..259..266..273..280..287..294..301..308
316..324..332..340..348..356..364..372..380..388..396
405..414..423..432..441..450..459..468..477..486..495
505..515..525..535..545..555..565..575..585..595..605
...
MAPLE
g:= 1/((1-x)^2*(1-x^11)); gser:= series(g, x=0, 72); seq(coeff(gser, x, n), n=0..70); # modified by G. C. Greubel, Jul 30 2019
MATHEMATICA
CoefficientList[Series[1/((1-x)^2*(1-x^11)), {x, 0, 70}], x] (* Vincenzo Librandi, Jun 11 2013 *)
PROG
(PARI) my(x='x+O('x^70)); Vec(1/((1-x)^2*(1-x^11))) \\ G. C. Greubel, Jul 30 2019 (Magma) R<x>:=PowerSeriesRing(Integers(), 70); Coefficients(R!( 1/((1-x)^2*(1-x^11)) )); // G. C. Greubel, Jul 30 2019 (Sage) (1/((1-x)^2*(1-x^11))).series(x, 70).coefficients(x, sparse=False) # G. C. Greubel, Jul 30 2019 (GAP) a:=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 15];; for n in [14..70] do a[n]:=2*a[n-1]-a[n-2]+a[n-11]-2*a[n-12]+a[n-13]; od; a; # G. C. Greubel, Jul 30 2019
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