A277792 Squares that are also pentagonal pyramidal numbers.

0, 1, 196, 2601, 15376, 60025, 181476, 461041, 1032256, 2099601, 3960100, 7027801, 11861136, 19193161, 29964676, 45360225, 66846976, 96216481, 135629316, 187662601, 255360400, 342287001, 452583076, 591024721, 763085376, 975000625, 1233835876, 1547556921, 1925103376, 2376465001, 2912760900

Offset: 0

Author

Ilya Gutkovskiy, Oct 31 2016

Keywords

• nonn
• easy

Comments

Intersection of A000290 and A002411.

Examples

			a(2) = 196 because 196 = 14^2 is a perfect square and 196 = 7^2*(7 + 1)/2 is the 7th pentagonal pyramidal number.
		

Links

• Daniel Mondot, Table of n, a(n) for n = 0..1000
• Eric Weisstein's World of Mathematics, Pentagonal Pyramidal Number
• Eric Weisstein's World of Mathematics, Square Number
• Index to sequences related to polygonal numbers
• Index to sequences related to pyramidal numbers
• Index entries for two-way infinite sequences
• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Crossrefs

Cf. A000290, A001653, A002411, A007588, A036353, A056220, A254333.

Programs

Magma

[n^2*(2*n^2-1)^2: n in [0..30]]; // Vincenzo Librandi, Nov 01 2016

Mathematica

Table[n^2 (2 n^2 - 1)^2, {n, 0, 30}]
LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,1,196,2601,15376,60025,181476},40] (* Harvey P. Dale, Nov 01 2024 *)

Formula

O.g.f.: x*(1 + 189*x + 1250*x^2 + 1250*x^3 + 189*x^4 + x^5)/(1 - x)^7.

E.g.f.: x*(1 + 97*x + 336*x^2 + 256*x^3 + 60*x^4 + 4*x^5)*exp(x).

a(n) = a(-n).

a(n) = n^2*(2*n^2 - 1)^2.

a(n) = A000290(A007588(n)).

a(n) = A000290(n)*A000290(A056220(n)).

Sum_{n>=1} 1/a(n) = (2*Pi^2+9*sqrt(2)*Pi*cot(Pi/sqrt(2))+3*Pi^2*csc(Pi/sqrt(2))^2-24)/12 = 1.0055779712856...