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%I #37 Mar 08 2024 12:15:46
%S 1,5,8,23,47,116,257,605,1376,3191,7319,16892,38849,89525,206072,
%T 474647,1092863,2516804,5795393,13345805,30731984,70769399,162965351,
%U 375273548,864169601,1989990245,4582499048,10552469783,24299966927
%N Expansion of (1+4*x)/(1-x-3*x^2).
%C Inversion of the periodic sequence with initial period (1,4,-1,-4). Sequence appears to have the property: for m > n, if s divides both a(n) and a(m) then s also divides a(2*m-n). E.g., 23 divides both a(3) = 23 and a(25) = 1989990245; 23 also divides a(2*25-3) = a(47) = 185518234185384428 = (2)^2*(23)*(131)*(15393149202239).
%C Floretion Algebra Multiplication Program, FAMP Code: 1jesforseq[.5'k + .5k' + 2'kk' + 2e]
%H Vincenzo Librandi, <a href="/A105963/b105963.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,3).
%F a(n) = A006130(n) + 4*A006130(n-1) = A006130(n+1) + A006130(n-1). - _R. J. Mathar_, Dec 12 2009
%F From _Colin Barker_, May 01 2019: (Start)
%F a(n) = (2^(-1-n)*((1-sqrt(13))^n*(-9+sqrt(13)) + (1+sqrt(13))^n*(9+sqrt(13)))) / sqrt(13).
%F a(n) = a(n-1) + 3*a(n-2) for n > 1. (End)
%F a(n) = 3^((n-1)/2)*( sqrt(3)*Fibonacci(n+1, 1/sqrt(3)) + 4*Fibonacci(n, 1/sqrt(3)) ). - _G. C. Greubel_, Jan 15 2020
%p seq(coeff(series((1+4*x)/(1-x-3*x^2), x, n+1), x, n), n = 0 .. 40); # _G. C. Greubel_, Jan 15 2020
%t CoefficientList[Series[(1+4x)/(1-x-3x^2), {x,0,40}], x] (* _Vincenzo Librandi_, Jul 20 2013 *)
%t Table[Round[3^((n-1)/2)*(Sqrt[3]*Fibonacci[n+1, 1/Sqrt[3]] + 4*Fibonacci[n, 1/Sqrt[3]] )], {n,0,40}] (* _G. C. Greubel_, Jan 15 2020 *)
%o (PARI) Vec((1+4*x)/(1-x-3*x^2)+O(x^40)) \\ _Charles R Greathouse IV_, Sep 27 2012
%o (Magma) I:=[ 1,5]; [n le 2 select I[n] else Self(n-1)+3*Self(n-2): n in [1..40]]; // _Vincenzo Librandi_, Jul 20 2013
%o (SageMath)
%o def A077952_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( (1+4*x)/(1-x-3*x^2) ).list()
%o A077952_list(30) # _G. C. Greubel_, Jan 15 2020
%o (GAP) a:=[1,5];; for n in [3..40] do a[n]:=a[n-1]+3*a[n-2]; od; a; # _G. C. Greubel_, Jan 15 2020
%Y Cf. A006130, A105476, A274977.
%K nonn,easy
%O 0,2
%A _Creighton Dement_, Apr 28 2005
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