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%I #53 Jul 26 2023 17:44:12
%S 1,6,9,27,54,135,297,702,1593,3699,8478,19575,45009,103734,238761,
%T 549963,1266246,2916135,6714873,15463278,35607897,81997731,188821422,
%U 434814615,1001278881,2305722726,5309559369,12226727547,28155405654,64835588295,149301805257,343808570142
%N a(n) = a(n-1) + 3*a(n-2) with n>1, a(0)=1, a(1)=6.
%C a(n)/a(n+1) converges to 1/A209927 as n approaches infinity.
%H Bruno Berselli, <a href="/A274977/b274977.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 G.f.: (1 + 5*x)/(1 - x - 3*x^2).
%F a(n) = ((13 + 11*sqrt(13))*(1 + sqrt(13))^n + (13 - 11*sqrt(13))*(1 - sqrt(13))^n)/(26*2^n).
%F 3*a(n) + a(n+1) = 9*A105476(n+1).
%F 3*a(n) - a(n+1) = 27*A006130(n-3) with n>1, A006130(-1) = 0.
%F a(n+1) - a(n) = 27*A105476(n-3) with n>2.
%F a(n) = 3^((n-1)/2)*( sqrt(3)*Fibonacci(n+1, 1/sqrt(3)) + 5*Fibonacci(n, 1/sqrt(3)) ). - _G. C. Greubel_, Jan 15 2020
%F E.g.f.: (1/13)*exp(x/2)*(13*cosh((sqrt(13)*x)/2) + 11*sqrt(13)*sinh((sqrt(13)*x)/2)). - _Stefano Spezia_, Jan 15 2020
%e Table of similar sequences (not extendable on the left side) where this recurrence can be applied to the first two terms:
%e ----------------------------------------------------------------------
%e (*) - - 1, -1, 2, -1, 5, 2, 17, 23, 74, 143, 365, ...
%e A052533: - - 1, 0, 3, 3, 12, 21, 57, 120, 291, 651, 1524, ...
%e (^) - 0, 1, 1, 4, 7, 19, 40, 97, 217, 508, 1159, 2683, ...
%e A006138: - - 1, 2, 5, 11, 26, 59, 137, 314, 725, 1667, 3842, ...
%e A105476: - - 1, 3, 6, 15, 33, 78, 177, 411, 942, 2175, 5001, ...
%e (^) 0, 1, 1, 4, 7, 19, 40, 97, 217, 508, 1159, 2683, 6160, ...
%e A105963: - - 1, 5, 8, 23, 47, 116, 257, 605, 1376, 3191, 7319, ...
%e A274977: - - 1, 6, 9, 27, 54, 135, 297, 702, 1593, 3699, 8478, ...
%e A075118: - 2, 1, 7, 10, 31, 61, 154, 337, 799, 1810, 4207, 9637, ...
%e ----------------------------------------------------------------------
%e (*) see version A140165.
%e (^) see A006130 and the signed versions A140167, A182228.
%p seq(coeff(series((1+5*x)/(1-x-3*x^2), x, n+1), x, n), n = 0..40); # _G. C. Greubel_, Jan 15 2020
%t RecurrenceTable[{a[n]==a[n-1] +3a[n-2], a[0]==1, a[1]==6}, a, {n,0,40}]
%t Table[Round[Sqrt[3]^(n-1)*(Sqrt[3]*Fibonacci[n+1, 1/Sqrt[3]] + 5*Fibonacci[n, 1/Sqrt[3]])], {n,0,40}] (* _G. C. Greubel_, Jan 15 2020 *)
%t LinearRecurrence[{1,3},{1,6},40] (* _Harvey P. Dale_, Jul 11 2023 *)
%o (PARI) v=vector(40); v[1]=1; v[2]=6; for(n=3, #v, v[n]=v[n-1]+3*v[n-2]); v
%o (Sage)
%o from sage.combinat.sloane_functions import recur_gen2
%o a = recur_gen2(1, 6, 1, 3)
%o [next(a) for n in range(40)]
%o (Magma) [n le 2 select 5*n-4 else Self(n-1)+3*Self(n-2): n in [1..40]];
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 32); Coefficients(R!((1 + 5*x)/(1- x-3*x^2))); // _Marius A. Burtea_, Jan 15 2020
%o (GAP) a:=[1,6];; 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, A006138, A052533, A075118, A105476, A105963.
%K nonn,easy
%O 0,2
%A _Bruno Berselli_, Sep 13 2016
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