A254653 -id:A254653 - cp's OEIS frontend

A254652 Indices of pentagonal numbers (A000326) which are also centered heptagonal numbers (A069099).

1, 4, 88, 421, 9661, 46288, 1062604, 5091241, 116876761, 559990204, 12855381088, 61593831181, 1413975042901, 6774761439688, 155524399338004, 745162164534481, 17106269952137521, 81961063337353204, 1881534170335789288, 9014971804944317941

Offset: 1

Colin Barker, Feb 04 2015

Also positive integers x in the solutions to 3*x^2 - 7*y^2 - x + 7*y - 2 = 0, the corresponding values of y being A254653.

			4 is in the sequence because the 4th pentagonal number is 22, which is also the 3rd centered heptagonal number.

Colin Barker, Table of n, a(n) for n = 1..980
Index entries for linear recurrences with constant coefficients, signature (1,110,-110,-1,1).

Cf. A000326, A069099, A254653, A254654.

LinearRecurrence[{1,110,-110,-1,1},{1,4,88,421,9661},30] (* Harvey P. Dale, Dec 09 2018 *)

Vec(-x*(x^2-4*x+1)*(x^2+7*x+1)/((x-1)*(x^4-110*x^2+1)) + O(x^100))

a(n) = a(n-1)+110*a(n-2)-110*a(n-3)-a(n-4)+a(n-5).

G.f.: -x*(x^2-4*x+1)*(x^2+7*x+1) / ((x-1)*(x^4-110*x^2+1)).

A254654 Pentagonal numbers (A000326) which are also centered heptagonal numbers (A069099).

Original entry on oeis.org

1, 22, 11572, 265651, 139997551, 3213845272, 1693690359922, 38881099834501, 20490265834338301, 470383542583947322, 247891234370134405072, 5690700059299494866551, 2998988132919620198222251, 68846088847021746311586172, 36281758184170330787958387022

Offset: 1

Colin Barker, Feb 04 2015

			22 is in the sequence because it is the 4th pentagonal number and the 3rd centered heptagonal number.

Colin Barker, Table of n, a(n) for n = 1..490
Index entries for linear recurrences with constant coefficients, signature (1,12098,-12098,-1,1).

Cf. A000326, A069099, A254652, A254653.

I:=[1,22,11572,265651,139997551]; [n le 5 select I[n] else Self(n-1)+12098*Self(n-2)-12098*Self(n-3)-Self(n-4)+Self(n-5): n in [1..20]]; // Vincenzo Librandi, Jan 20 2017

CoefficientList[Series[(x^4 + 21*x^3 - 548*x^2 + 21*x + 1)/((1 - x)*(x^2 - 110*x + 1)*(x^2 + 110*x + 1)), {x, 0, 20}], x] (* Wesley Ivan Hurt, Jan 19 2017 *)
LinearRecurrence[{1,12098,-12098,-1,1},{1,22,11572,265651,139997551},20] (* Harvey P. Dale, Jan 10 2025 *)

Vec(-x*(x^4+21*x^3-548*x^2+21*x+1)/((x-1)*(x^2-110*x+1)*(x^2+110*x+1)) + O(x^100))

a(n) = a(n-1)+12098*a(n-2)-12098*a(n-3)-a(n-4)+a(n-5).

G.f.: -x*(x^4+21*x^3-548*x^2+21*x+1) / ((x-1)*(x^2-110*x+1)*(x^2+110*x+1)).

cp's OEIS Frontend

A254652 Indices of pentagonal numbers (A000326) which are also centered heptagonal numbers (A069099).

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