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Numbers n such that hexagonal number H(n) is the sum of two consecutive hexagonal numbers.
3

%I #15 Mar 02 2016 11:24:10

%S 1,19,637,21631,734809,24961867,847968661,28805972599,978555099697,

%T 33242067417091,1129251737081389,38361316993350127,

%U 1303155526036822921,44268926568258629179,1503840347794756569157,51086302898453464722151,1735430458199623043983969

%N Numbers n such that hexagonal number H(n) is the sum of two consecutive hexagonal numbers.

%C Also nonnegative integers y in the solutions to 8*x^2-4*y^2+4*x+2*y+2 = 0, the corresponding values of x being A251601.

%H Colin Barker, <a href="/A251602/b251602.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*(7*x^2-16*x+1) / ((x-1)*(x^2-34*x+1)).

%F a(n) = (2+(27-19*sqrt(2))*(17+12*sqrt(2))^n+(17+12*sqrt(2))^(-n)*(27+19*sqrt(2)))/8. - _Colin Barker_, Mar 02 2016

%e 19 is in the sequence because H(19) = 703 = 325 + 378 = H(13) + H(14).

%o (PARI) Vec(-x*(7*x^2-16*x+1)/((x-1)*(x^2-34*x+1)) + O(x^20))

%Y Cf. A000384, A251601.

%K nonn,easy

%O 1,2

%A _Colin Barker_, Dec 05 2014