A253477 -id:A253477 - cp's OEIS frontend

A253476 Indices of centered triangular numbers (A005448) which are also centered heptagonal numbers (A069099).

1, 15, 70, 1596, 7645, 175491, 840826, 19302360, 92483161, 2123084055, 10172306830, 233519943636, 1118861268085, 25685070715851, 123064567182466, 2825124258799920, 13535983528803121, 310737983397275295, 1488835123601160790, 34178353049441482476

Offset: 1

Colin Barker, Jan 02 2015

Also positive integers x in the solutions to 3*x^2 - 7*y^2 - 3*x + 7*y = 0, the corresponding values of y being A253477.

			15 is in the sequence because the 15th centered triangular number is 316, which is also the 10th 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. A005448, A069099, A253477, A253689.

LinearRecurrence[{1,110,-110,-1,1},{1,15,70,1596,7645},30] (* Harvey P. Dale, Jun 14 2016 *)

Vec(x*(14*x^3+55*x^2-14*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*(14*x^3+55*x^2-14*x-1) / ((x-1)*(x^4-110*x^2+1)).

A253689 Centered triangular numbers (A005448) which are also centered heptagonal numbers (A069099).

Original entry on oeis.org

1, 316, 7246, 3818431, 87657571, 46195373386, 1060481282176, 558871623400861, 12829702464103141, 6761228853708238456, 155213739350238513106, 81797346113290645435291, 1877775805829483067448711, 989584286517361374767907526, 22717331543711346799755988036

Offset: 1

Colin Barker, Jan 09 2015

			316 is in the sequence because it is the 15th centered triangular number and the 10th 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. A005448, A069099, A253476, A253477.

I:=[1,316,7246,3818431,87657571]; [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 10 2015

LinearRecurrence[{1, 12098, -12098, -1, 1}, {1, 316, 7246, 3818431, 87657571}, 20] (* or *) CoefficientList[Series[(x^4 + 315 x^3 - 5168 x^2 + 315 x + 1) / ((1 - x) (x^2 - 110 x + 1)(x^2 + 110 x + 1)), {x, 0, 20}], x] (* Vincenzo Librandi, Jan 10 2015 *)

Vec(-x*(x^4+315*x^3-5168*x^2+315*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+315*x^3-5168*x^2+315*x+1) / ((x-1)*(x^2-110*x+1)*(x^2+110*x+1)).

cp's OEIS Frontend

A253476 Indices of centered triangular numbers (A005448) which are also centered heptagonal numbers (A069099).

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