A096748 Expansion of (1+x)^2/(1-x^2-x^4).

1, 2, 2, 2, 3, 4, 5, 6, 8, 10, 13, 16, 21, 26, 34, 42, 55, 68, 89, 110, 144, 178, 233, 288, 377, 466, 610, 754, 987, 1220, 1597, 1974, 2584, 3194, 4181, 5168, 6765, 8362, 10946, 13530, 17711, 21892, 28657, 35422, 46368, 57314, 75025, 92736, 121393, 150050

Offset: 0

Author

Paul Barry, Jul 07 2004

Keywords

• easy
• nonn

Comments

The ratio a(n+1) / a(n) increasingly approximates two constants connected to the golden ratio phi = (1 + sqrt(5))/2: (phi+1)/2 = 1.30901699... = A239798 and (phi-1)*2 = 1.23606797... = A134972, according to whether n is odd or even. - Davide Rotondo, Jul 31 2020

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Davide Rotondo, Perfino I Capelli Sono Tutti Contati (in Italian), see p. 11.
• Index entries for linear recurrences with constant coefficients, signature (0,1,0,1).

Crossrefs

Cf. A000045, A079977.

Cf. A134972 and A239798 (limiting ratios for a(n+1)/a(n)).

Programs

Mathematica

CoefficientList[Series[(1+x)^2/(1-x^2-x^4),{x,0,50}],x] (* or *) LinearRecurrence[{0,1,0,1},{1,2,2,2},50] (* Harvey P. Dale, Jan 29 2012 *)

Formula

a(n) = a(n-2) + a(n-4).

a(n) = 2*F((n+1)/2)*(1-(-1)^n)/2 + F((n+4)/2)*(1+(-1)^n)/2.

a(2*n) = A000045(n+2); a(2*n+1) = 2*A000045(n+1).

a(n) = Sum_{k=0..n} binomial(floor((n-k)/2), floor(k/2)). - Paul Barry, Jul 24 2004

a(n) = A079977(n) + A079977(n-2) + 2*A079977(n-1). - R. J. Mathar, Jul 15 2013