A275709 a(n) = 2*n^3 + 3*n^2.

0, 5, 28, 81, 176, 325, 540, 833, 1216, 1701, 2300, 3025, 3888, 4901, 6076, 7425, 8960, 10693, 12636, 14801, 17200, 19845, 22748, 25921, 29376, 33125, 37180, 41553, 46256, 51301, 56700, 62465, 68608, 75141, 82076, 89425, 97200, 105413, 114076, 123201, 132800, 142885

Offset

0,2

Comments

Apart from the initial zero this sequence gives the 2nd pentagonal number, the 4th hexagonal number, the 6th heptagonal number, the 8th octagonal number, the 10th nonagonal number, etc. as well as the 5th nonnegative number, the 7th triangular number, the 9th square, the 11th pentagonal number, the 13th hexagonal number, etc. This is a reliable pattern that does not seem to appear on any other pairs of polygonal numbers (see link).

a(n) is the maximal determinant of a 3 X 3 matrix with integer elements from {1, ..., n+1}, so (for example) the maximum determinant of a 3 X 3 matrix with integer elements from {1, ..., 5} = det(1, 5, 5; 5, 1, 5; 5, 5, 1) = a(4) = 176. - Matthew Scroggs, Dec 31 2022

Links

Carauleanu Marc and Colin Barker, Table of n, a(n) for n = 0..3030 (first 1000 terms from Colin Barker)

Joshua Giambalvo, Illustration of initial terms in a square array, Imgur, (2016).

Petro Kolosov, Another power identity involving binomial theorem and Faulhaber's formula, arXiv:1603.02468 [math.NT], 2018. (In particular p. 3)

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Formula

From Colin Barker, Aug 06 2016: (Start)

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 4.

G.f.: x*(5+8*x-x^2) / (1-x)^4.

(End)

a(n) = A033431(n) + A033428(n). - Omar E. Pol, Aug 09 2016

a(n) = A000290(n) * A005408(n+1). - Robert Israel, Aug 09 2016

a(n) = A320047(1, n, 0). - Kolosov Petro, Oct 04 2018

E.g.f.: x*(5 + 9*x + 2*x^2)*exp(x). - G. C. Greubel, Oct 19 2018

From Amiram Eldar, Jan 21 2023: (Start)

Sum_{n>=1} 1/a(n) = Pi^2/18 + 4*log(2)/9 - 16/27.

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/36 + Pi/9 -2*log(2)/9 - 8/27. (End)

Maple

seq(2*n^3+3*n^2, n=0..30); # Robert Israel, Aug 09 2016

Mathematica

Table[2 n^3 + 3 n^2, {n, 0, 41}] (* or *)
CoefficientList[Series[x (5 + 8 x - x^2)/(1 - x)^4, {x, 0, 41}], x] (* Michael De Vlieger, Aug 11 2016 *)

Prog

(PARI) concat(0, Vec(x*(5+8*x-x^2)/(1-x)^4 + O(x^50))) \\ Colin Barker, Aug 28 2016
(PARI) a(n)=n^2*(2*n+3) \\ Charles R Greathouse IV, Aug 28 2016
(Magma) [n^2*(2*n + 3): n in [0..30]]; // G. C. Greubel, Oct 19 2018
(Python) for n in range(0, 50): print(n**2*(2*n+3), end=' ') # Stefano Spezia, Oct 19 2018

Crossrefs

Cf. A000290, A005408, A005563, A033428, A033431, A057145, A139600.

Sequence in context: A161165 A063140 A316560 * A257093 A225261 A088727

Adjacent sequences: A275706 A275707 A275708 * A275710 A275711 A275712

Keyword

nonn,easy

Author

Joshua Giambalvo, Aug 06 2016

Status

approved