A141523 Expansion of (3-2*x-3*x^2)/(1-x-x^2-x^3).

3, 1, 1, 5, 7, 13, 25, 45, 83, 153, 281, 517, 951, 1749, 3217, 5917, 10883, 20017, 36817, 67717, 124551, 229085, 421353, 774989, 1425427, 2621769, 4822185, 8869381, 16313335, 30004901, 55187617, 101505853, 186698371, 343391841

Offset: 0

Author

Roger L. Bagula and Gary W. Adamson, Aug 11 2008

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Martin Burtscher, Igor Szczyrba, 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, A001644.

Programs

Magma

I:=[3, 1, 1]; [n le 3 select I[n] else Self(n-1)+Self(n-2) +Self(n-3): n in [1..40]]; // Vincenzo Librandi, Oct 17 2012

Mathematica

a[0]=3; a[1]=1; a[2]=1; a[n_]:= a[n]=a[n-1]+a[n-2]+a[n-3]; Table[a[n], {n, 0, 40}]
LinearRecurrence[{1, 1, 1}, {3, 1, 1}, 40] (* Vincenzo Librandi, Oct 17 2012 *)

PARI

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

PARI

my(x='x+O('x^40)); Vec((3-2*x-3*x^2)/(1-x-x^2-x^3)) \\ G. C. Greubel, Apr 22 2019

Sage

((3-2*x-3*x^2)/(1-x-x^2-x^3)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 22 2019

Formula

a(0)=3; a(1)=1; a(2)=1; thereafter a(n) = a(n-1) + a(n-2) + a(n-3).

From R. J. Mathar, Aug 22 2008: (Start)

O.g.f.: (3-2*x-3*x^2)/(1-x-x^2-x^3).

a(n) = A001644(n) - 2*A000073(n). (End)

Extensions

Edited by N. J. A. Sloane, Oct 17 2012