A073711 G.f. satisfies: A(x) = A(x^2) + x*A(x^2)^2.

1, 1, 1, 2, 1, 3, 2, 6, 1, 7, 3, 12, 2, 16, 6, 26, 1, 31, 7, 42, 3, 59, 12, 72, 2, 104, 16, 116, 6, 184, 26, 186, 1, 303, 31, 282, 7, 497, 42, 406, 3, 783, 59, 612, 12, 1224, 72, 840, 2, 1856, 104, 1232, 16, 2784, 116, 1656, 6, 4136, 184, 2376, 26, 6008, 186, 3138, 1

Offset: 0

Author

Paul D. Hanna, Aug 05 2002

Keywords

• easy
• nice
• nonn

Comments

This sequence interlaced with its self-convolution yields the original sequence.

Examples

			a(0)=1, a(2^k)=1, a(3*2^k)=2, a(5*2^k)=3, a(7*2^k)=6, a(9*2^k)=7, for k>=0.
Self-convolution of [1,1,1,2,1,3,2,6,1,7,3,12,2,16,...] = [1,2,3,6,7,12,16,...], which forms the terms found at odd-indexed positions.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A073712 (self convolution), A194279, A211604.

Programs

Haskell

import Data.List (transpose)
a073711 n = a073711_list !! n
a073711_list = 1 :
(tail $ concat $ transpose [a073711_list, a073712_list])
-- Reinhard Zumkeller, Dec 20 2012

Mathematica

For[A = 1; n = 1, n <= 65, n++, A = (Normal[A] /. x -> x^2) + x*(Normal[A] /. x -> x^2)^2 + O[x]^n]; CoefficientList[A, x] (* Jean-François Alcover, Mar 06 2013, updated Apr 23 2016 *)

PARI

a(n)=local(A=1); for(i=0,#binary(n), A=subst(A,x,x^2+x*O(x^n))+x*subst(A,x,x^2+x*O(x^n))^2); polcoeff(A,n)
for(n=0,65,print1(a(n),", ")) \\ Paul D. Hanna, Dec 21 2012

Formula

a(2^k) = 1 and a(2^k*n) = a(n), with a(0) = 1, for k>=0 and n>=0.

a(2^n-1) = A211604(n) for n>=0.

Extensions

Name changed and entry revised by Paul D. Hanna, Dec 21 2012