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A048920 Indices of heptagonal numbers (A000566) which are also 9-gonal.

%I A048920 #30 Feb 16 2025 08:32:40
%S A048920 1,104,14725,2090804,296879401,42154784096,5985682462189,
%T A048920 849924754846700,120683329505769169,17136182865064375256,
%U A048920 2433217283509635517141,345499718075503179058724
%N A048920 Indices of heptagonal numbers (A000566) which are also 9-gonal.
%C A048920 As n increases, this sequence is approximately geometric with common ratio r = lim_{n->infinity} a(n)/a(n-1) = (6 + sqrt(35))^2 = 71 + 12*sqrt(35). - _Ant King_, Dec 31 2011
%H A048920 Vincenzo Librandi, <a href="/A048920/b048920.txt">Table of n, a(n) for n = 1..200</a>
%H A048920 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/NonagonalHeptagonalNumber.html">Nonagonal Heptagonal Number</a>.
%H A048920 <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (143,-143,1).
%F A048920 From _Bruno Berselli_, Dec 20 2011: (Start)
%F A048920 G.f.: x*(1 - 39*x - 4*x^2)/((1-x)*(1 - 142*x + x^2)).
%F A048920 a(n) = (42 + (-21+5r)*(6+r)^(2n-1) - (21+5r)*(6-r)^(2n-1))/140, where r=sqrt(35). (End)
%F A048920 From _Ant King_, Dec 31 2011: (Start)
%F A048920 a(n) = 142*a(n-1) - a(n-2) - 42.
%F A048920 a(n) = ceiling(1/140*(49+9*sqrt(35))*(6+sqrt(35))^(2*n-2)).
%F A048920 (End)
%t A048920 LinearRecurrence[{143,-143,1},{1,104,14725},30] (* _Vincenzo Librandi_, Dec 21 2011 *)
%o A048920 (Maxima) makelist(expand((42+(-21+5*sqrt(35))*(6+sqrt(35))^(2*n-1)-(21+5*sqrt(35))*(6-sqrt(35))^(2*n-1))/140), n, 1, 12); /* _Bruno Berselli_, Dec 20 2011 */
%o A048920 (Magma) I:=[1, 104, 14725]; [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
%Y A048920 Cf. A000566, A048919, A048921.
%K A048920 nonn,easy
%O A048920 1,2
%A A048920 _Eric W. Weisstein_