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.

3, 10, 26, 62, 140, 306, 654, 1376, 2862, 5902, 12092, 24650, 50054, 101328, 204630, 412454, 830076, 1668514, 3350558, 6723008, 13481438, 27020190, 54133116, 108416282, 217075350, 434543536, 869722694, 1740473846, 3482611772, 6967916082, 13940188782, 27887426720

Offset

0,1

Comments

Row sums of triangle A316938.

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (3,-1,-2).

Formula

G.f.: (3 + x - x^2) / ((1 - 2*x)*(1 - x - x^2)).

a(n) = 13*2^n - 2*Fibonacci(n+5) for n>0.

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

Maple

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

Mathematica

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}]
f[n_] := 13*2^n - 2 Fibonacci[n + 5]; Array[f, 32, 0] (* or *)
LinearRecurrence[{3, -1, -2}, {3, 10, 26}, 32] (* Robert G. Wilson v, Jul 21 2018 *)

Prog

(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]];
(GAP) List([0..35], n->13*2^n-2*Fibonacci(n+5)); # Muniru A Asiru, Jul 22 2018
(PARI) Vec((3 + x - x^2) / ((1 - 2*x)*(1 - x - x^2)) + O(x^40)) \\ Colin Barker, Jul 22 2018
(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

Crossrefs

Cf. A000045, A167821, A316528, A316938.

Sequence in context: A355902 A301410 A333897 * A269965 A110158 A301308

Adjacent sequences: A316934 A316935 A316936 * A316938 A316939 A316940

Keyword

nonn,easy

Author

Vincenzo Librandi, Jul 17 2018

Status

approved