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

Author

Joshua Giambalvo, Aug 06 2016

Keywords

• nonn
• easy

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

Crossrefs

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

Programs

Magma

[n^2*(2*n + 3): n in [0..30]]; // G. C. Greubel, Oct 19 2018

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 *)

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

Python

for n in range(0,50): print(n**2*(2*n+3), end=' ') # Stefano Spezia, Oct 19 2018

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)