A214825 a(n) = a(n-1) + a(n-2) + a(n-3), with a(0) = 1, a(1) = a(2) = 3.

1, 3, 3, 7, 13, 23, 43, 79, 145, 267, 491, 903, 1661, 3055, 5619, 10335, 19009, 34963, 64307, 118279, 217549, 400135, 735963, 1353647, 2489745, 4579355, 8422747, 15491847, 28493949, 52408543, 96394339, 177296831, 326099713, 599790883, 1103187427

Offset: 0

Author

Abel Amene, Jul 28 2012

Keywords

• nonn
• easy

Comments

Part of a group of sequences defined by a(0), a(1)=a(2), a(n) = a(n-1) + a(n-2) + a(n-3) which is a subgroup of sequences with linear recurrences and constant coefficients listed in the index. See Comments in A214727.

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..3770
• Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
• Index entries for linear recurrences with constant coefficients, signature (1,1,1).

Crossrefs

Cf. A000073, A000213, A000288, A000322, A000383, A001644, A060455, A136175, A141036, A141523, A214825-A214831.

Programs

GAP

a:=[1,3,3];; for n in [4..40] do a[n]:=a[n-1]+a[n-2]+a[n-3]; od; a; # G. C. Greubel, Apr 23 2019

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+2*x-x^2)/(1-x-x^2-x^3) )); // G. C. Greubel, Apr 23 2019

Mathematica

LinearRecurrence[{1,1,1},{1,3,3},40] (* Harvey P. Dale, Oct 05 2013 *)

PARI

a(n)=([0,1,0; 0,0,1; 1,1,1]^n*[1;3;3])[1,1] \\ Charles R Greathouse IV, Mar 22 2016

PARI

my(x='x+O('x^40)); Vec((1+2*x-x^2)/(1-x-x^2-x^3)) \\ G. C. Greubel, Apr 23 2019

SageMath

((1+2*x-x^2)/(1-x-x^2-x^3)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 23 2019

Formula

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

a(n) = K(n) - 2*T(n+1) + 4*T(n), where K(n) = A001644(n), and T(n) = A000073(n+1). - G. C. Greubel, Apr 23 2019