A074678 a(n) = Sum_{j=0..floor(n/2)} (-1)^(j+floor(n/2))*S(2j+q), where S(n) are generalized tribonacci numbers (A001644) and q = (1-(-1)^n)/2.

3, 1, 0, 6, 11, 15, 28, 56, 103, 185, 340, 630, 1159, 2127, 3912, 7200, 13243, 24353, 44792, 82390, 151539, 278719, 512644, 942904, 1734271, 3189817, 5866988, 10791078, 19847887, 36505951, 67144912, 123498752, 227149619, 417793281

Offset: 0

Author

Mario Catalani (mario.catalani(AT)unito.it), Aug 30 2002

Keywords

• easy
• nonn

Comments

a(n) is the convolution of S(n) with the sequence (1,0,-1,0,1,0,-1,0,....) A056594.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,0,2,1,1).

Crossrefs

Cf. A001644, A056594.

Programs

GAP

a:=[3,1,0,6,11];; for n in [6..40] do a[n]:=a[n-1]+2*a[n-3]+a[n-4] +a[n-5]; od; a; # G. C. Greubel, Apr 02 2019

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (3-2*x-x^2)/(1-x-2*x^3-x^4-x^5) )); // G. C. Greubel, Apr 02 2019

Mathematica

CoefficientList[Series[(3-2*x-x^2)/(1-x-2*x^3-x^4-x^5), {x, 0, 40}], x]
LinearRecurrence[{1,0,2,1,1}, {3,1,0,6,11}, 40] (* G. C. Greubel, Apr 02 2019 *)

PARI

my(x='x+O('x^40)); Vec((3-2*x-x^2)/(1-x-2*x^3-x^4-x^5)) \\ G. C. Greubel, Apr 02 2019

Sage

((3-2*x-x^2)/(1-x-2*x^3-x^4-x^5)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 02 2019

Formula

a(n) = Sum_{j=0..floor(n/2)} (-1)^(j+floor(n/2))*S(2j+q), where S(n) are generalized tribonacci numbers (A001644) and q = (1-(-1)^n)/2.

a(n) = a(n-1) + 2*a(n-3) + a(n-4) + a(n-5), a(0)=3, a(1)=1, a(2)=0, a(3)=6, a(4)=11.

G.f.: (3 - 2*x - x^2)/(1 - x - 2*x^3 - x^4 - x^5).