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Search: a008726 -id:a008726
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A118729
Rectangular array where row r contains the 8 numbers 4*r^2 - 3*r, 4*r^2 - 2*r, ..., 4*r^2 + 4*r.
+10
14
0, 0, 0, 0, 0, 0, 0, 0, 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
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,10
COMMENTS
The numbers in row r span the interval ]8*A000217(r-1), 8*A000217(r)].
The first difference between the entries in row r is r.
Partial sums of floor(n/8). - Philippe Deléham, Mar 26 2013
Apart from the initial zeros, the same as A008726. - Philippe Deléham, Mar 28 2013
a(n+7) is the number of key presses required to type a word of n letters, all different, on a keypad with 8 keys where 1 press of a key is some letter, 2 presses is some other letter, etc., and under an optimal mapping of letters to keys and presses (answering LeetCode problem 3014). - Christopher J. Thomas, Feb 16 2024
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
LeetCode, 3014. Minimum Number of Pushes to Type Word I.
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,1,-2,1).
FORMULA
From Philippe Deléham, Mar 26 2013: (Start)
a(8k) = A001107(k).
a(8k+1) = A002939(k).
a(8k+2) = A033991(k).
a(8k+3) = A016742(k).
a(8k+4) = A007742(k).
a(8k+5) = A002943(k).
a(8k+6) = A033954(k).
a(8k+7) = A033996(k). (End)
G.f.: x^8/((1-x)^2*(1-x^8)). - Philippe Deléham, Mar 28 2013
a(n) = floor(n/8)*(n-3-4*floor(n/8)). - Ridouane Oudra, Jun 04 2019
a(n+7) = (1/2)*(n+(n mod 8))*(floor(n/8)+1). - Christopher J. Thomas, Feb 13 2024
EXAMPLE
The array starts, with row r=0, as
r=0: 0 0 0 0 0 0 0 0;
r=1: 1 2 3 4 5 6 7 8;
r=2: 10 12 14 16 18 20 22 24;
r=3: 27 30 33 36 39 42 45 48;
MATHEMATICA
Flatten[Table[4r^2+r(Range[-3, 4]), {r, 0, 6}]] (* or *) LinearRecurrence[ {2, -1, 0, 0, 0, 0, 0, 1, -2, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 1, 2}, 60] (* Harvey P. Dale, Nov 26 2015 *)
CROSSREFS
Cf. similar sequences: A000217, A002620, A130518, A130519, A130520, A174709, A174738, A218470, A131242.
KEYWORD
nonn,tabf,easy
AUTHOR
Stuart M. Ellerstein (ellerstein(AT)aol.com), May 21 2006
EXTENSIONS
Redefined as a rectangular tabf array and description simplified by R. J. Mathar, Oct 20 2010
STATUS
approved
A375981
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
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
Table of n, a(n) for n=0..42.
Michael A. Allen, Combinations without specified separations, Communications in Combinatorics and Optimization (in press).
Michael A. Allen, Connections between Combinations Without Specified Separations and Strongly Restricted Permutations, Compositions, and Bit Strings, arXiv:2409.00624 [math.CO], 2024.
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.
Cf. A376743.
KEYWORD
easy,nonn,less
AUTHOR
Michael A. Allen, Sep 04 2024
STATUS
approved
A008728
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
a(n) = A179052(n) for n < 100. - Reinhard Zumkeller, Jun 27 2010
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 193
Index entries for Molien series
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)).
From Mitch Harris, Sep 08 2008: (Start)
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
CROSSREFS
Cf. A001840, A001972, A008724, A008725, A008726, A008727, A008732.
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
EXTENSIONS
More terms from Vladimir Joseph Stephan Orlovsky, Mar 14 2010
STATUS
approved
A008727
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Number of partitions of n into two kinds of 1's and one kind of 9. - Joerg Arndt, Dec 27 2014
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 192
Index entries for Molien series
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)).
From Mitch Harris, Sep 08 2008: (Start)
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
Drop[Accumulate[Floor[Range[70]/9]], 8] (* Jean-François Alcover, Mar 27 2013 *)
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)
def A008727_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P(1/((1-x)^2*(1-x^9))).list()
A008727_list(70) # G. C. Greubel, Sep 09 2019
(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
CROSSREFS
Cf. A001840, A001972, A008724, A008725, A008726, A008732. - Vladimir Joseph Stephan Orlovsky, Mar 14 2010
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
STATUS
approved
A008730
Molien series 1/((1-x)^2*(1-x^12)) for 3-dimensional group [2,n] = *22n.
+10
3
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 125, 130, 135, 140, 145, 150, 155, 160, 165, 170, 175, 180, 186, 192, 198, 204
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 195
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 1).
FORMULA
G.f. 1/( (1-x)^3 * (1+x) *(1+x+x^2) *(1-x+x^2) * (1+x^2) *(1-x^2+x^4)). - R. J. Mathar, Aug 11 2021
From Mitch Harris, Sep 08 2008: (Start)
a(n) = Sum_{j=0..n+12} floor(j/12).
a(n-12) = (1/2)*floor(n/12)*(2*n - 10 - 12*floor(n/12)). (End)
a(n) = A221912(n+12). - Philippe Deléham, Apr 03 2013
EXAMPLE
..1....2....3....4....5....6....7....8....9...10...11...12
.14...16...18...20...22...24...26...28...30...32...34...36
.39...42...45...48...51...54...57...60...63...66...69...72
.76...80...84...88...92...96..100..104..108..112..116..120
125..130..135..140..145..150..155..160..165..170..175..180
186..192..198..204..210..216..222..228..234..240..246..252
259..266..273..280..287..294..301..308..315..322..329..336
344..352..360..368..376..384..392..400..408..416..424..432
441..450..459..468..477..486..495..504..513..522..531..540
550..560..570..580..590..600..610..620..630..640..650..660
...
The columns are: A051866, A139267, A094159, A033579, A049452, A033581, A049453, A033580, A195319, A202804, A211014, A049598
- Philippe Deléham, Apr 03 2013
MAPLE
seq(coeff(series(1/(1-x)^2/(1-x^12), x, n+1), x, n), n=0..80);
MATHEMATICA
CoefficientList[Series[1/((1-x)^2*(1-x^12)), {x, 0, 70}], x] (* Vincenzo Librandi, Jun 11 2013 *)
LinearRecurrence[{2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 1}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16}, 70] (* Harvey P. Dale, Jan 01 2024 *)
PROG
(PARI) my(x='x+O('x^70)); Vec(1/((1-x)^2*(1-x^12))) \\ G. C. Greubel, Jul 30 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 70); Coefficients(R!( 1/((1-x)^2*(1-x^12)) )); // G. C. Greubel, Jul 30 2019
(Sage) (1/((1-x)^2*(1-x^12))).series(x, 70).coefficients(x, sparse=False) # G. C. Greubel, Jul 30 2019
CROSSREFS
Cf. A001840, A001972, A008724, A008725, A008726, A008727, A008728, A008729, A008732. - Vladimir Joseph Stephan Orlovsky, Mar 14 2010
Cf. A221912
KEYWORD
nonn,tabf,easy
AUTHOR
N. J. A. Sloane
EXTENSIONS
More terms from Vladimir Joseph Stephan Orlovsky, Mar 14 2010
STATUS
approved
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