A075676 Sequences A001644 and A000073 interleaved.

3, 1, 3, 2, 11, 7, 39, 24, 131, 81, 443, 274, 1499, 927, 5071, 3136, 17155, 10609, 58035, 35890, 196331, 121415, 664183, 410744, 2246915, 1389537, 7601259, 4700770, 25714875, 15902591, 86992799, 53798080, 294294531, 181997601

Offset: 0

Author

Mario Catalani (mario.catalani(AT)unito.it), Sep 24 2002

Keywords

• easy
• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0, 3, 0, 1, 0, 1).

Crossrefs

Cf. A000073, A001644, A005013, A005247, A075536.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (3+x- 6*x^2-x^3-x^4)/(1-3*x^2-x^4-x^6) )); // G. C. Greubel, Apr 21 2019

Mathematica

CoefficientList[Series[(3+x-6x^2-x^3-x^4)/(1-3x^2-x^4-x^6), {x, 0, 40}], x]
LinearRecurrence[{0,3,0,1,0,1},{3,1,3,2,11,7},40] (* Harvey P. Dale, May 01 2014 *)

PARI

my(x='x+O('x^40)); Vec((3+x-6*x^2-x^3-x^4)/(1-3*x^2-x^4-x^6)) \\ G. C. Greubel, Apr 21 2019

Sage

((3+x-6*x^2-x^3-x^4)/(1-3*x^2-x^4-x^6)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 21 2019

Formula

a(n) = A000073(n) if n odd, a(n) = A001644(n) if n even.

a(n) = ((1-(-1)^n)*T(n) + (1+(-1)^n)*S(n))/2, where T(n) = A000073(n), S(n) = A001644(n).

a(n) = 3*a(n-2) + a(n-4) + a(n-6), a(0)=3, a(1)=1, a(2)=3, a(3)=2, a(4)=11, a(5)=7.

O.g.f.: (3 + x - 6*x^2 - x^3 - x^4)/(1 - 3*x^2 - x^4 - x^6).

a(n) = T(n) + (1+(-1)^n)*(T(n-1) + (3/2)*T(n-2)).