login
Indices of hexagonal numbers (A000384) which are also centered heptagonal numbers (A069099).
3

%I #7 Jun 13 2015 00:55:21

%S 1,2857,45529,184300369,2937241777,11889953986681,189493215939721,

%T 767068491312421537,12224965330197902689,49486656636639609035209,

%U 788681413122894278122297,3192582165489099245985035761,50880992673985436128583949841

%N Indices of hexagonal numbers (A000384) which are also centered heptagonal numbers (A069099).

%C Also positive integers x in the solutions to 4*x^2-7*y^2-2*x+7*y-2 = 0, the corresponding values of y being A253715.

%H Colin Barker, <a href="/A253714/b253714.txt">Table of n, a(n) for n = 1..416</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,64514,-64514,-1,1).

%F a(n) = a(n-1)+64514*a(n-2)-64514*a(n-3)-a(n-4)+a(n-5).

%F G.f.: -x*(x^4+2856*x^3-21842*x^2+2856*x+1) / ((x-1)*(x^2-254*x+1)*(x^2+254*x+1)).

%e 2857 is in the sequence because the 2857th hexagonal number is 16322041, which is also the 2160th centered heptagonal number.

%o (PARI) Vec(-x*(x^4+2856*x^3-21842*x^2+2856*x+1)/((x-1)*(x^2-254*x+1)*(x^2+254*x+1)) + O(x^100))

%Y Cf. A000384, A069099, A253715, A253716.

%K nonn,easy

%O 1,2

%A _Colin Barker_, Jan 10 2015