A046198 - OEIS

%I #23 Jun 27 2019 01:29:09

%S 1,42,2585,160210,9930417,615525626,38152658377,2364849293730,

%T 146582503552865,9085750370983882,563169940497447801,

%U 34907450560470779762,2163698764808690897425,134114415967578364860570,8312930091225049930457897,515267551239985517323529026

%N Indices of heptagonal numbers (A000566) which are also pentagonal.

%C As n increases, this sequence is approximately geometric with common ratio r = lim_{n->infinity} a(n)/a(n-1) = (4+sqrt(15))^2 = 31 + 8*sqrt(15). - _Ant King_, Dec 15 2011

%H Colin Barker, <a href="/A046198/b046198.txt">Table of n, a(n) for n = 1..558</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HeptagonalPentagonalNumber.html">Heptagonal Pentagonal Number.</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (63,-63,1)

%F From _Ant King_, Dec 15 2011: (Start)

%F a(n) = 63*a(n-1) - 63*a(n-2) + a(n-3).

%F a(n) = 62*a(n-1) - a(n-2) - 18.

%F a(n) = (1/60)*((9-sqrt(15))*(4+sqrt(15))^(2*n-1) + (9+sqrt(15))*(4-sqrt(15))^(2*n-1)+18).

%F a(n) = ceiling((1/60)*(9-sqrt(15))*(4+sqrt(15))^(2*n-1)).

%F G.f.: x*(1-21*x+2*x^2)/((1-x)*(1-62*x+x^2)).

%F (End)

%t LinearRecurrence[{63, -63, 1}, {1, 42, 2585}, 14] (* _Ant King_, Dec 15 2011 *)

%o (PARI) Vec(-x*(2*x^2-21*x+1)/((x-1)*(x^2-62*x+1)) + O(x^30)) \\ _Colin Barker_, Jun 23 2015

%Y Cf. A046199, A048900.

%K nonn,easy

%O 1,2

%A _Eric W. Weisstein_