A252630 - OEIS

%I #10 Mar 03 2016 05:31:05

%S 50,16503,5314316,1711193649,550999041062,177419980028715,

%T 57128682570205568,18395258367626164581,5923216065693054789914,

%U 1907257177894796016188127,614130888066058624157787380,197748238700092982182791348633,63674318730541874204234656472846

%N Numbers n such that the sum of the hexagonal numbers X(n), X(n+1), X(n+2) and X(n+3) is equal to the heptagonal number H(m) for some m.

%C Also positive integers x in the solutions to 16*x^2-5*y^2+40*x+3*y+44 = 0, the corresponding values of y being A252631.

%H Colin Barker, <a href="/A252630/b252630.txt">Table of n, a(n) for n = 1..398</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (323,-323,1).

%F a(n) = 323*a(n-1)-323*a(n-2)+a(n-3).

%F G.f.: x*(3*x^2-353*x-50) / ((x-1)*(x^2-322*x+1)).

%F a(n) =-5/4+1/80*(161+72*sqrt(5))^(-n)*(-70-37*sqrt(5)+(-70+37*sqrt(5))*(161+72*sqrt(5))^(2*n)). - _Colin Barker_, Mar 03 2016

%e 50 is in the sequence because X(50)+X(51)+X(52)+X(53) = 4950+5151+5356+5565 = 21022 = H(92).

%o (PARI) Vec(x*(3*x^2-353*x-50)/((x-1)*(x^2-322*x+1)) + O(x^100))

%Y Cf. A000384, A000566, A252631.

%K nonn,easy

%O 1,1

%A _Colin Barker_, Dec 19 2014