A014769 Squares of odd pentagonal numbers.
1, 25, 1225, 2601, 13689, 21025, 61009, 82369, 180625, 227529, 423801, 511225, 855625, 1002001, 1555009, 1782225, 2614689, 2948089, 4141225, 4609609, 6255001, 6890625, 9090225, 9928801, 12794929, 13875625, 17530969, 18896409, 23474025, 25170289, 30813601
Offset: 0
Links
- Colin Barker, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (1,4,-4,-6,6,4,-4,-1,1).
Programs
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Mathematica
Select[Table[PolygonalNumber[5,n]^2, {n,0, 61}],OddQ] (* James C. McMahon, Dec 24 2023 *) Select[PolygonalNumber[5,Range[70]],OddQ]^2 (* or *) LinearRecurrence[{1,4,-4,-6,6,4,-4,-1,1},{1,25,1225,2601,13689,21025,61009,82369,180625},40] (* Harvey P. Dale, Apr 07 2025 *) -
PARI
Vec((1 + 24*x + 1196*x^2 + 1280*x^3 + 6294*x^4 + 1976*x^5 + 2828*x^6 + 176*x^7 + 49*x^8) / ((1 - x)^5*(1 + x)^4) + O(x^40)) \\ Colin Barker, Nov 20 2018
Formula
a(n) = A014632(n)^2. - Sean A. Irvine, Nov 20 2018
From Colin Barker, Nov 20 2018: (Start)
G.f.: (1 + 24*x + 1196*x^2 + 1280*x^3 + 6294*x^4 + 1976*x^5 + 2828*x^6 + 176*x^7 + 49*x^8) / ((1 - x)^5*(1 + x)^4).
a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) - 6*a(n-4) + 6*a(n-5) + 4*a(n-6) - 4*a(n-7) - a(n-8) + a(n-9) for n>8.
a(n) = 36*n^4 + 60*n^3 + 37*n^2 + 10*n + 1 for n even.
a(n) = 36*n^4 - 12*n^3 + n^2 for n odd.
(End)
Extensions
More terms from Scott Lindhurst (ScottL(AT)alumni.princeton.edu)