Displaying 1-10 of 19 results found.
0, 12, 48, 108, 192, 300, 432, 588, 768, 972, 1200, 1452, 1728, 2028, 2352, 2700, 3072, 3468, 3888, 4332, 4800, 5292, 5808, 6348, 6912, 7500, 8112, 8748, 9408, 10092, 10800, 11532, 12288, 13068, 13872, 14700, 15552, 16428, 17328, 18252, 19200, 20172, 21168, 22188
COMMENTS
Areas of perfect 4:3 rectangles (for n > 0).
Sequence found by reading the line from 0, in the direction 0, 12, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. Semi-axis opposite to A069190 in the same spiral. - Omar E. Pol, Sep 16 2011 (x,y,z) = (-a(n), 1 + n*a(n), 1 - n*a(n)) are solutions of the Diophantine equation x^3 + 2*y^3 + 2*z^3 = 4. - XU Pingya, Apr 30 2022
FORMULA
Sum_{n>=1} 1/a(n) = Pi^2/72 ( A086729). Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/144.
Product_{n>=1} (1 + 1/a(n)) = 2*sqrt(3)*sinh(Pi/(2*sqrt(3)))/Pi.
Product_{n>=1} (1 - 1/a(n)) = 2*sqrt(3)*sin(Pi/(2*sqrt(3)))/Pi. (End)
G.f.: 12*x*(1 + x)/(1-x)^3.
E.g.f.: 12*x*(1 + x)*exp(x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)
EXAMPLE
192 is on the list since 16*12 is a 4:3 rectangle with integer sides and an area of 192.
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {0, 12, 48}, 50] (* Harvey P. Dale, Jan 19 2020 *)
Centered 20-gonal (or icosagonal) numbers.
+10
15
1, 21, 61, 121, 201, 301, 421, 561, 721, 901, 1101, 1321, 1561, 1821, 2101, 2401, 2721, 3061, 3421, 3801, 4201, 4621, 5061, 5521, 6001, 6501, 7021, 7561, 8121, 8701, 9301, 9921, 10561, 11221, 11901, 12601, 13321, 14061, 14821, 15601, 16401, 17221, 18061, 18921, 19801
COMMENTS
Equals binomial transform of [1, 20, 20, 0, 0, 0, ...]. - Gary W. Adamson, Jun 13 2008 Sequence found by reading the line from 1, in the direction 1, 21, ..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. Semi-axis opposite to A033583 in the same spiral. - Omar E. Pol, Sep 16 2011
FORMULA
a(n) = 10n^2 - 10n + 1.
G.f.: x*(1 + 18*x + x^2)/(1-x)^3. - R. J. Mathar, Feb 04 2011 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=1, a(1)=21, a(2)=61. - Harvey P. Dale, Apr 29 2011 Sum_{n>=1} 1/a(n) = Pi*tan(sqrt(3/5)*Pi/2)/(2*sqrt(15)).
Sum_{n>=1} a(n)/n! = 11*e - 1.
Sum_{n>=1} (-1)^n * a(n)/n! = 11/e - 1. (End)
EXAMPLE
a(5)=201 because 201 = 10*5^2 - 10*5 + 1 = 250 - 50 + 1.
MATHEMATICA
Table[10n^2-10n+1, {n, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {1, 21, 61}, 50] (* Harvey P. Dale, Apr 29 2011 *)
CROSSREFS
Cf. centered polygonal numbers listed in A069190.
0, 1, 26, 99, 244, 485, 846, 1351, 2024, 2889, 3970, 5291, 6876, 8749, 10934, 13455, 16336, 19601, 23274, 27379, 31940, 36981, 42526, 48599, 55224, 62425, 70226, 78651, 87724, 97469, 107910, 119071, 130976, 143649, 157114, 171395, 186516
COMMENTS
The general formula for alternating sums of powers of odd integers is in terms of the Swiss-Knife polynomials P(n,x) A153641 (P(n,0)-(-1)^k*P(n,2*k))/2. Here n=3, thus a(k) = |(P(3,0)-(-1)^k*P(3,2*k))/2|. - Peter Luschny, Jul 12 2009
FORMULA
a(n) = cosh(3*arccosh(n)) = cos(3*arccos(n)). - Artur Jasinski, Feb 14 2010 a(n) = Imaginary part of -(1/2)*(2*n*i-1)^3.
a(n) = -4*(1/4 + n^2)^(3/2)*sin(3*arctan(2*n)). (End)
MATHEMATICA
lst={}; Do[AppendTo[lst, ChebyshevT[3, n]], {n, 0, 10^2}]; lst
Round[Table[N[Cosh[3 ArcCosh[n]], 100], {n, 0, 20}]] (* Artur Jasinski, Feb 14 2010 *) CoefficientList[Series[x*(1+22*x+x^2)/(1-x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 30 2012 *) LinearRecurrence[{4, -6, 4, -1}, {0, 1, 26, 99}, 40] (* Harvey P. Dale, Apr 02 2015 *)
PROG
(Magma) I:=[0, 1, 26, 99]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Jun 30 2012
a(n) = n-th concentric 12-gonal number.
+10
15
0, 1, 12, 25, 48, 73, 108, 145, 192, 241, 300, 361, 432, 505, 588, 673, 768, 865, 972, 1081, 1200, 1321, 1452, 1585, 1728, 1873, 2028, 2185, 2352, 2521, 2700, 2881, 3072, 3265, 3468, 3673, 3888, 4105, 4332, 4561, 4800, 5041, 5292, 5545, 5808, 6073, 6348
COMMENTS
Concentric dodecagonal numbers. [corrected by Ivan Panchenko, Nov 09 2013] Sequence found by reading the line from 0, in the direction 0, 12,..., and the same line from 1, in the direction 1, 25,..., in the square spiral whose vertices are the generalized octagonal numbers A001082. Main axis, perpendicular to A028896 in the same spiral.
FORMULA
a(n) = 3*n^2+(-1)^n-1.
a(n) = -a(n-1) + 6*n^2 - 6*n + 1. (End)
G.f.: -x*(1+10*x+x^2) / ( (1+x)*(x-1)^3 ). - R. J. Mathar, Sep 18 2011 a(n) = Sum_{k=1..n} (2*(-1)^(n-k+1) + 3*(2*k-1)), n>0, a(0) = 0. - L. Edson Jeffery, Sep 14 2014 Sum_{n>=1} 1/a(n) = Pi^2/72 + tan(Pi/sqrt(6))*Pi/(4*sqrt(6)). - Amiram Eldar, Jan 16 2023
MATHEMATICA
Table[Sum[2*(-1)^(n - k + 1) + 6*k - 3, {k, n}], {n, 0, 47}] (* L. Edson Jeffery, Sep 14 2014 *)
PROG
(Haskell)
a195143 n = a195143_list !! n
a195143_list = scanl (+) 0 a091998_list
CROSSREFS
Cf. A001082, A032527, A032528, A077221, A195043, A195045, A195040, A195142, A195145, A195146, A195147, A195148, A195149.
a(0) = 1, a(n) = 24*n^2 + 2 for n>0.
+10
12
1, 26, 98, 218, 386, 602, 866, 1178, 1538, 1946, 2402, 2906, 3458, 4058, 4706, 5402, 6146, 6938, 7778, 8666, 9602, 10586, 11618, 12698, 13826, 15002, 16226, 17498, 18818, 20186, 21602, 23066, 24578, 26138, 27746, 29402, 31106, 32858, 34658, 36506, 38402, 40346
COMMENTS
Number of points of L_infinity norm n in the simple cubic lattice Z^3. - N. J. A. Sloane, Apr 15 2008 Numbers of cubes needed to completely "cover" another cube. - Xavier Acloque, Oct 20 2003
FORMULA
a(n) = (2*n+1)^3 - (2*n-1)^3 for n >= 1. - Xavier Acloque, Oct 20 2003
a(n) = (2*n-1)^2 + (2*n+1)^2 + (4*n)^2 for n>0. - Bruno Berselli, Feb 06 2012 Sum_{n>=0} 1/a(n) = 3/4 + sqrt(3)/24*Pi*coth(Pi*sqrt(3)/6) = 1.065052868574... - R. J. Mathar, May 07 2024
PROG
(PARI) a(n) = if (n==0, 1, 24*n^2 + 2);
EXTENSIONS
More terms from Xavier Acloque, Oct 20 2003
Centered 18-gonal numbers.
+10
11
1, 19, 55, 109, 181, 271, 379, 505, 649, 811, 991, 1189, 1405, 1639, 1891, 2161, 2449, 2755, 3079, 3421, 3781, 4159, 4555, 4969, 5401, 5851, 6319, 6805, 7309, 7831, 8371, 8929, 9505, 10099, 10711, 11341, 11989, 12655, 13339, 14041, 14761, 15499, 16255, 17029, 17821
COMMENTS
Equals binomial transform of [1, 18, 18, 0, 0, 0, ...]. Example: a(3) = 55 = (1, 2, 1) dot (1, 18, 18) = (1 + 36 + 18). - Gary W. Adamson, Aug 24 2010 Sequence is a spoke of the hexagonal spiral built from the terms of A016777 (see illustration in links section). a(n) + 2 is a trisection of A002061. a(n) + 9 is the arithmetic mean of its neighbors.
4*a(n) + 5 is a square: A016945(n)^2. (End)
FORMULA
a(n) = 9*n^2 - 9*n + 1.
G.f.: ( x*(1+16*x+x^2) ) / ( (1-x)^3 ). - R. J. Mathar, Feb 04 2011 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(1)=1, a(2)=19, a(3)=55. - Harvey P. Dale, Jan 20 2014 Sum_{n>=1} 1/a(n) = Pi*tan(sqrt(5)*Pi/6)/(3*sqrt(5)).
Sum_{n>=1} a(n)/n! = 10*e - 1.
Sum_{n>=1} (-1)^n * a(n)/n! = 10/e - 1. (End)
a(n+1) - a(n) = 18*n = A008600(n). (End)
EXAMPLE
a(5) = 181 because 9*5^2 - 9*5 + 1 = 225 - 45 + 1 = 181.
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {1, 19, 55}, 50] (* Harvey P. Dale, Jan 20 2014 *)
CROSSREFS
Cf. centered polygonal numbers listed in A069190. Cf. A000217, A028387, A195042, A016945, A002378, A082040, A304163, A003215, A247792, A016777, A016778, A016790, A010008, A008600, A002061. Cf. A000290, A139278, A069129, A062786, A033996, A060544, A027468, A016754, A124080, A069099, A152740, A049598, A005891, A152741, A001844, A163756, A005448, A194715.
Centered 40-gonal numbers.
+10
11
1, 41, 121, 241, 401, 601, 841, 1121, 1441, 1801, 2201, 2641, 3121, 3641, 4201, 4801, 5441, 6121, 6841, 7601, 8401, 9241, 10121, 11041, 12001, 13001, 14041, 15121, 16241, 17401, 18601, 19841, 21121, 22441, 23801, 25201, 26641, 28121, 29641, 31201, 32801, 34441, 36121
COMMENTS
Also centered tetracontagonal numbers or centered tetrakaicontagonal numbers. Also sequence found by reading the line from 1, in the direction 1, 41, ..., in the square spiral whose vertices are the generalized dodecagonal numbers A195162. Semi-axis opposite to A195322 in the same spiral.
FORMULA
a(n) = 20*n^2 - 20*n + 1.
Sum_{n>=1} 1/a(n) = Pi*tan(Pi/sqrt(5))/(8*sqrt(5)). - Amiram Eldar, Feb 11 2022 E.g.f.: exp(x)*(20*x^2 + 1) - 1.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)
MATHEMATICA
Table[20*n^2 - 20*n + 1, {n, 1, 40}] (* Amiram Eldar, Feb 11 2022 *)
Centered 22-gonal numbers.
+10
8
1, 23, 67, 133, 221, 331, 463, 617, 793, 991, 1211, 1453, 1717, 2003, 2311, 2641, 2993, 3367, 3763, 4181, 4621, 5083, 5567, 6073, 6601, 7151, 7723, 8317, 8933, 9571, 10231, 10913, 11617, 12343, 13091, 13861, 14653, 15467, 16303, 17161, 18041, 18943, 19867, 20813
FORMULA
a(n) = 11*n^2 - 11*n + 1.
Sum_{n>=1} 1/a(n) = Pi*tan(sqrt(7/11)*Pi/2)/sqrt(77).
Sum_{n>=1} a(n)/n! = 12*e - 1.
Sum_{n>=1} (-1)^n * a(n)/n! = 12/e - 1. (End)
G.f.: x*(1 + 20*x + x^2)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)
EXAMPLE
a(5) = 221 because 11*5^2 - 11*5 + 1 = 275 - 55 + 1 = 221
For n=2, a(2)=22*2+1-22=23; n=3, a(3)=22*3+23-22=67; n=4, a(4)=22*4+67-22=133.
CROSSREFS
Cf. centered polygonal numbers listed in A069190.
a(n) = 25*n*(n + 1)/2 + 1.
+10
8
1, 26, 76, 151, 251, 376, 526, 701, 901, 1126, 1376, 1651, 1951, 2276, 2626, 3001, 3401, 3826, 4276, 4751, 5251, 5776, 6326, 6901, 7501, 8126, 8776, 9451, 10151, 10876, 11626, 12401, 13201, 14026, 14876, 15751, 16651, 17576, 18526, 19501, 20501, 21526, 22576, 23651
COMMENTS
Also centered 25-gonal (or icosipentagonal) numbers.
This is the case k=25 of the formula (k*n*(n+1) - (-1)^k + 1)/2. See table in Links section for similar sequences.
For k=2*n, the formula shown above gives A011379. Primes in sequence: 151, 251, 701, 1951, 3001, 4751, 10151, 12401, ...
REFERENCES
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 51 (23rd row of the table).
FORMULA
G.f.: (1 + 23*x + x^2)/(1 - x)^3.
a(n) = a(-n-1) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = A054254(5*n+2) with n>0, a(0)=1. Sum_{i>=0} 1/a(i) = 1.078209111... = 2*Pi*tan(Pi*sqrt(17)/10)/(5*sqrt(17)).
Sum_{n>=0} a(n)/n! = 77*e/2.
Sum_{n>=0} (-1)^(n+1) * a(n)/n! = 23/(2*e). (End)
MATHEMATICA
Table[25 n (n + 1)/2 + 1, {n, 0, 50}]
25*Accumulate[Range[0, 50]]+1 (* or *) LinearRecurrence[{3, -3, 1}, {1, 26, 76}, 50] (* Harvey P. Dale, Jan 29 2023 *)
PROG
(PARI) vector(50, n, n--; 25*n*(n+1)/2+1)
(Sage) [25*n*(n+1)/2+1 for n in (0..50)]
(Magma) [25*n*(n+1)/2+1: n in [0..50]];
CROSSREFS
Cf. centered polygonal numbers listed in A069190. Similar sequences of the form (k*n*(n+1) - (-1)^k + 1)/2 with -1 <= k <= 26: A000004, A000124, A002378, A005448, A005891, A028896, A033996, A035008, A046092, A049598, A060544, A064200, A069099, A069125, A069126, A069128, A069130, A069132, A069174, A069178, A080956, A124080, A163756, A163758, A163761, A164136, A173307.
Centered 17-gonal numbers: (17*n^2 - 17*n + 2)/2.
+10
7
1, 18, 52, 103, 171, 256, 358, 477, 613, 766, 936, 1123, 1327, 1548, 1786, 2041, 2313, 2602, 2908, 3231, 3571, 3928, 4302, 4693, 5101, 5526, 5968, 6427, 6903, 7396, 7906, 8433, 8977, 9538, 10116, 10711, 11323, 11952, 12598, 13261, 13941, 14638, 15352
COMMENTS
Equals binomial transform of [1, 17, 17, 0, 0, 0, ...]. - Gary W. Adamson, Mar 26 2010
FORMULA
a(n) = (17*n^2 - 17*n + 2)/2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=1, a(1)=18, a(2)=52. - Harvey P. Dale, Jun 05 2011 Sum_{n>=1} 1/a(n) = 2*Pi*tan(3*Pi/(2*sqrt(17)))/(3*sqrt(17)).
Sum_{n>=1} a(n)/n! = 19*e/2 - 1.
Sum_{n>=1} (-1)^n * a(n)/n! = 19/(2*e) - 1. (End)
EXAMPLE
a(5) = 171 because (17*5^2 - 17*5 + 2)/2 = (425 - 85 + 2)/2 = 342/2 = 171.
MATHEMATICA
Table[(17n^2-17n+2)/2, {n, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {1, 18, 52}, 50] (* Harvey P. Dale, Jun 05 2011 *)
PROG
(Magma) [ (17*n^2 - 17*n + 2)/2 : n in [1..50] ]; // Wesley Ivan Hurt, Jun 09 2014
CROSSREFS
Cf. centered polygonal numbers listed in A069190.
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