A048919 - OEIS

%I #31 Feb 16 2025 08:32:40

%S 1,88,12445,1767052,250908889,35627295136,5058825000373,

%T 718317522757780,101996029406604337,14482717858215058024,

%U 2056443939837131635021,292000556739014477114908

%N Indices of 9-gonal numbers which are also heptagonal.

%C As n increases, this sequence is approximately geometric with common ratio r = lim_{n->oo} a(n)/a(n-1) = (6 + sqrt(35))^2 = 71 + 12*sqrt(35). - _Ant King_, Jan 01 2012

%H Vincenzo Librandi, <a href="/A048919/b048919.txt">Table of n, a(n) for n = 1..200</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/NonagonalHeptagonalNumber.html">Nonagonal Heptagonal number.</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (143,-143,1).

%F G.f.: -x*(1 - 55*x + 4*x^2) / ( (x-1)*(x^2 - 142*x + 1) ). - _R. J. Mathar_, Dec 21 2011

%F a(n) = (50 + (25-3r)*(6+r)^(2n-1) + (25+3r)*(6-r)^(2n-1))/140, where r=sqrt(35). - _Bruno Berselli_, Dec 21 2011

%F From _Ant King_, Jan 01 2012: (Start)

%F a(n) = 142*a(n-1) - a(n-2) - 50.

%F a(n) = ceiling(1/140*(45 + 7*sqrt(35))*(6 + sqrt(35))^(2*n - 2)). (End)

%t LinearRecurrence[{143,-143,1},{1,88,12445},30] (* _Vincenzo Librandi_, Dec 21 2011 *)

%o (Magma) I:=[1, 88, 12445]; [n le 3 select I[n] else 143*Self(n-1)-143*Self(n-2)+Self(n-3): n in [1..30]]; // _Vincenzo Librandi_, Dec 21 2011

%o (Maxima) makelist(expand((50+(25-3*sqrt(35))*(6+sqrt(35))^(2*n-1)+(25+3*sqrt(35))*(6-sqrt(35))^(2*n-1))/140), n, 1, 12); /* _Bruno Berselli_, Dec 21 2011 */

%Y Cf. A048920, A048921.

%K nonn,easy,changed

%O 1,2

%A _Eric W. Weisstein_