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%I #22 Sep 08 2022 08:46:01
%S 1,12376,118837251,1141075274626,10956604668124501,
%T 105205316882256186876,1010181441746819238261751,
%U 9699762098447641443533149126,93137114659112811393986059649001,894302565257039116557412701216561376,8587093138460974938071465363095362686251
%N Numbers which are both decagonal and pentagonal.
%C As n increases, this sequence is approximately geometric with common ratio r = lim(n->Infinity, a(n)/a(n-1)) = (sqrt(3)+sqrt(2))^8 = 4801+1960*sqrt(6).
%C Intersection of A000326 and A001107. - _Michel Marcus_, Jun 20 2015
%H Vincenzo Librandi, <a href="/A202563/b202563.txt">Table of n, a(n) for n = 1..100</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (9603,-9603,1).
%F G.f.: x*(1+2773*x+126*x^2) / ((1-x)*(1-9602*x+x^2)).
%F a(n) = 9602*a(n-1)-a(n-2)+2900.
%F a(n) = 9603*a(n-1)-9603*a(n-2)+a(n-3).
%F a(n) = 1/192*(25*((sqrt(3)+sqrt(2))^(8*n-6)+(sqrt(3)-sqrt(2))^(8*n-6))-58).
%F a(n) = floor(25/192*(sqrt(3)+sqrt(2))^(8*n-6)).
%e The second natural number which is both pentagonal and decagonal is 12376. Hence a(2) = 12376.
%t LinearRecurrence[{9603, -9603, 1}, {1, 12376, 118837251}, 11]
%o (Maxima) makelist(expand((25*((sqrt(3)+sqrt(2))^(8*n-6)+(sqrt(3)-sqrt(2))^(8*n-6))-58)/192), n, 1, 11); \\ _Bruno Berselli_, Dec 22 2011
%o (Magma) I:=[1, 12376, 118837251]; [n le 3 select I[n] else 9603*Self(n-1)-9603*Self(n-2)+1*Self(n-3): n in [1..15]]; // _Vincenzo Librandi_, Jan 24 2012
%Y Cf. A202564, A202565, A000326, A001107.
%K nonn,easy
%O 1,2
%A _Ant King_, Dec 21 2011
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