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A083425 a(n) = (5*5^n + (-1)^n)/6.

%I A083425 #37 Sep 08 2022 08:45:10
%S A083425 1,4,21,104,521,2604,13021,65104,325521,1627604,8138021,40690104,
%T A083425 203450521,1017252604,5086263021,25431315104,127156575521,
%U A083425 635782877604,3178914388021,15894571940104,79472859700521,397364298502604,1986821492513021,9934107462565104
%N A083425 a(n) = (5*5^n + (-1)^n)/6.
%C A083425 Binomial transform of A083424. Inverse binomial transform of A052934.
%C A083425 Primes occur at indices n = 4, 66, 100, 102, 228, 346, ..., see A138647. - _R. J. Mathar_, Jan 19 2011
%C A083425 Sum_{i=0..m} (-1)^(m+i)*5^i, for m >= 0, gives all terms of the sequence. - _Bruno Berselli_, Aug 28 2013
%H A083425 Vincenzo Librandi, <a href="/A083425/b083425.txt">Table of n, a(n) for n = 0..1000</a>
%H A083425 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,5).
%F A083425 a(n) = (5*5^n + (-1)^n)/6.
%F A083425 G.f.: 1/((1+x)*(1-5x)).
%F A083425 E.g.f.: (5*exp(5x) + exp(-x))/6.
%F A083425 a(n) = Sum_{k=0..n} C(n-k,k)*4^(n-2k)*5^k. - _Paul Barry_, Jul 29 2004
%F A083425 a(n) = A015531(n+1). - _R. J. Mathar_, Sep 17 2008
%F A083425 a(n) = 4*a(n-1) + 5*a(n-2). - _Vincenzo Librandi_, Jun 23 2012
%p A083425 seq(coeff(series(factorial(n)*(5*exp(5*x)+exp(-x))/6,x,n+1), x, n), n = 0 .. 25); # _Muniru A Asiru_, Sep 21 2018
%t A083425 LinearRecurrence[{4,5},{1,4},40] (* _Vincenzo Librandi_, Jun 23 2012 *)
%o A083425 (Magma) [n le 2 select n^2 else 4*Self(n-1)+5*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Jun 23 2012
%o A083425 (PARI) a(n)=(5*5^n+(-1)^n)/6 \\ _Charles R Greathouse IV_, Oct 07 2015
%o A083425 (GAP) List([0..25],n->(5*5^n+(-1)^n)/6); # _Muniru A Asiru_, Sep 21 2018
%K A083425 nonn,easy
%O A083425 0,2
%A A083425 _Paul Barry_, Apr 30 2003