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A316937 a(n) = 3*a(n-1) - a(n-2) - 2*a(n-3) for n > 2, a(0)=3, a(1)=10, a(2)=26.

%I A316937 #36 Sep 08 2022 08:46:22
%S A316937 3,10,26,62,140,306,654,1376,2862,5902,12092,24650,50054,101328,
%T A316937 204630,412454,830076,1668514,3350558,6723008,13481438,27020190,
%U A316937 54133116,108416282,217075350,434543536,869722694,1740473846,3482611772,6967916082,13940188782,27887426720
%N A316937 a(n) = 3*a(n-1) - a(n-2) - 2*a(n-3) for n > 2, a(0)=3, a(1)=10, a(2)=26.
%C A316937 Row sums of triangle A316938.
%H A316937 Colin Barker, <a href="/A316937/b316937.txt">Table of n, a(n) for n = 0..1000</a>
%H A316937 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-2).
%F A316937 G.f.: (3 + x - x^2) / ((1 - 2*x)*(1 - x - x^2)).
%F A316937 a(n) = 13*2^n - 2*Fibonacci(n+5) for n>0.
%F A316937 a(n) = (2^(-n)*(65*4^n + (1-sqrt(5))^n*(-25+11*sqrt(5)) - (1+sqrt(5))^n*(25+11*sqrt(5)))) / 5. - _Colin Barker_, Jul 22 2018
%p A316937 seq(coeff(series((3+x-x^2)/((1-2*x)*(1-x-x^2)), x,n+1),x,n),n=0..35); # _Muniru A Asiru_, Jul 22 2018
%t A316937 CoefficientList[Series[(3 + x - x^2) / ((1 - 2 x) (1 - x - x^2)), {x, 0, 33}], x] (* or *) RecurrenceTable[{a[n]==3 a[n-1] - a[n-2] - 2 a[n-3], a[0]==3, a[1]==10, a[2]==26}, a, {n, 0, 40}]
%t A316937 f[n_] := 13*2^n - 2 Fibonacci[n + 5]; Array[f, 32, 0] (* or *)
%t A316937 LinearRecurrence[{3, -1, -2}, {3, 10, 26}, 32] (* _Robert G. Wilson v_, Jul 21 2018 *)
%o A316937 (Magma) I:=[3, 10, 26]; [n le 3 select I[n] else 3*Self(n-1)-Self(n-2)-2*Self(n-3): n in [1..40]];
%o A316937 (GAP) List([0..35],n->13*2^n-2*Fibonacci(n+5)); # _Muniru A Asiru_, Jul 22 2018
%o A316937 (PARI) Vec((3 + x - x^2) / ((1 - 2*x)*(1 - x - x^2)) + O(x^40)) \\ _Colin Barker_, Jul 22 2018
%o A316937 (Magma) [((2^(-n)*(65*4^n + (1-Sqrt(5))^n*(-25 + 11*Sqrt(5)) - (1 + Sqrt(5))^n*(25 + 11*Sqrt(5)))) / 5): n in [0..20]]; // _Vincenzo Librandi_, Aug 24 2018
%Y A316937 Cf. A000045, A167821, A316528, A316938.
%K A316937 nonn,easy
%O A316937 0,1
%A A316937 _Vincenzo Librandi_, Jul 17 2018