|
%I #16 Mar 02 2016 11:34:43
%S 0,13,450,15295,519588,17650705,599604390,20368898563,691942946760,
%T 23505691291285,798501560956938,27125547381244615,921470109401359980,
%U 31302858172264994713,1063375707747608460270,36123471205246422654475
%N Numbers n such that hexagonal numbers H(n) and H(n+1) sum to another hexagonal number.
%C Also nonnegative integers x in the solutions to 8*x^2-4*y^2+4*x+2*y+2 = 0, the corresponding values of y being A251602.
%H Colin Barker, <a href="/A251601/b251601.txt">Table of n, a(n) for n = 1..654</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (35,-35,1).
%F a(n) = 35*a(n-1) - 35*a(n-2) + a(n-3).
%F G.f.: x^2*(5*x-13) / ((x-1)*(x^2-34*x+1)).
%F a(n) = (-4+(17+12*sqrt(2))^n*(-38+27*sqrt(2))-(17+12*sqrt(2))^(-n)*(38+27*sqrt(2)))/16. - _Colin Barker_, Mar 02 2016
%e 13 is in the sequence because H(13) + H(14) = 325 + 378 = 703 = H(19).
%o (PARI) concat(0, Vec(x^2*(5*x-13)/((x-1)*(x^2-34*x+1)) + O(x^20)))
%Y Cf. A000384, A251602.
%K nonn,easy
%O 1,2
%A _Colin Barker_, Dec 05 2014
| 