A274845 a(0)=1, a(1)=0, a(4n+2) = a(4n+3) = a(4n+5) = (4^(n+1) +(-1)^n)/5, a(4n+4) = (2*4^(n+1) -3*(-1)^n)/5.

1, 0, 1, 1, 1, 1, 3, 3, 7, 3, 13, 13, 25, 13, 51, 51, 103, 51, 205, 205, 409, 205, 819, 819, 1639, 819, 3277, 3277, 6553, 3277, 13107, 13107, 26215, 13107, 52429, 52429, 104857, 52429, 209715, 209715, 419431, 209715, 838861, 838861, 1677721, 838861, 3355443

Offset: 0

Author

Paul Curtz, Jul 08 2016

Keywords

• nonn
• easy

Comments

Antidiagonals of the array in A274613 written as a triangle:

1,

0, 1/2,

0, 1/2, 1/4,

0, 0, 1/2, 1/8,

0, 0, 1/4, 3/8, 1/16,

... .

a(n) is the numerators of the antidiagonal sums i.e. 1, 0, 1/2, 1/2, 1/4, 1/2, 3/8, 3/8, 7/16, 3/8, 13/32, 13/32, 25/64, 13/32, ... = a(n)/b(n).

The denominators b(n) are A173300(n).

a(0)+a(1) = 1, a(4n+2) +a(4n+3) +a(4n+4) +a(4n+5) = 4, 16, 64, 256, ... .

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 3, 0, 0, 0, 4).

Crossrefs

Cf. A000302, A102900, A015521, A173300, A274613.

Programs

Mathematica

LinearRecurrence[{0,0,0,3,0,0,0,4}, {1,0,1,1,1,1,3,3}, 50] (* G. C. Greubel, Jul 08 2016 *)

PARI

Vec((1+x^2+x^3-2*x^4+x^5)/((1-2*x^2)*(1+2*x^2)*(1+x^4)) + O(x^60)) \\ Colin Barker, Jul 22 2016

Formula

a(4n) = A102900(n), a(4n+1) = A015521(n), a(4n+2) = a(4n+3) = A015521(n+1).

a(n) = 3*a(n-4) + 4*a(n-8). - G. C. Greubel, Jul 08 2016

G.f.: (1+x^2+x^3-2*x^4+x^5) / ((1-2*x^2)*(1+2*x^2)*(1+x^4)). - Colin Barker, Jul 22 2016

Extensions

More terms from Colin Barker, Jul 22 2016