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Centered triangular numbers (A005448) which are also centered heptagonal numbers (A069099).
3

%I #11 Sep 08 2022 08:46:10

%S 1,316,7246,3818431,87657571,46195373386,1060481282176,

%T 558871623400861,12829702464103141,6761228853708238456,

%U 155213739350238513106,81797346113290645435291,1877775805829483067448711,989584286517361374767907526,22717331543711346799755988036

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

%H Colin Barker, <a href="/A253689/b253689.txt">Table of n, a(n) for n = 1..490</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,12098,-12098,-1,1).

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

%F 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)).

%e 316 is in the sequence because it is the 15th centered triangular number and the 10th centered heptagonal number.

%t 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 *)

%o (PARI) 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))

%o (Magma) 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

%Y Cf. A005448, A069099, A253476, A253477.

%K nonn,easy

%O 1,2

%A _Colin Barker_, Jan 09 2015