A194279 -id:A194279 - cp's OEIS frontend

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

Paul D. Hanna, Aug 05 2002

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

			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.

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

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

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

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 *)

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

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.

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

A073712 Self-convolution of A073711.

Original entry on oeis.org

1, 2, 3, 6, 7, 12, 16, 26, 31, 42, 59, 72, 104, 116, 184, 186, 303, 282, 497, 406, 783, 612, 1224, 840, 1856, 1232, 2784, 1656, 4136, 2376, 6008, 3138, 8735, 4362, 12345, 5754, 17693, 7756, 24432, 10170, 34471, 13302, 46771, 17688, 65144, 22296, 87008

Offset: 0

Paul D. Hanna, Aug 05 2002

The g.f. G(x) of A073711 satisfies: G(x) = G(x^2) + x*G(x^2)^2.

The terms of this sequence found at odd-indexed positions are equal to twice that of A194279, which equals the self-convolution cube of A073711.

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

Cf. A073711, A194279.

a073712 n = a073712_list !! n
a073712_list = map (g a073711_list) [1..] where
g xs k = sum $ zipWith (*) xs $ reverse $ take k xs
-- Reinhard Zumkeller, Dec 20 2012

nmax = 46; max = 2*nmax+1; f[x_] := Sum[ a[k]*x^k, {k, 0, max}]; a[0] = a[1] = a[2] = 1; coes = CoefficientList[ Series[ f[x] - f[x^2] - x*f[x^2]^2, {x, 0, max}], x]; sol = Solve[ Thread[ coes == 0]] // First; Table[ a[2*n+1], {n, 0, nmax}] /. sol (* Jean-François Alcover, Mar 06 2013 *)

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^2,n)
for(n=0,65,print1(a(n),", ")) \\ Paul D. Hanna, Dec 21 2012

a(n) = A073711(2*n+1) for n>=0.

a(2*n+1) = 2*A194279(n) for n>=0, where A194279 equals the self-convolution cube of A073711.

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

cp's OEIS Frontend

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

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