A041003 The sequence e, given that c is a left shift by one place of b.

1, 1, 1, 2, 3, 3, 4, 6, 7, 8, 10, 12, 14, 16, 20, 23, 26, 30, 36, 41, 47, 55, 64, 73, 83, 96, 111, 125, 144, 165, 187, 211, 241, 272, 306, 346, 391, 439, 493, 553, 622, 695, 779, 871, 974, 1086, 1211, 1348, 1502, 1671, 1857, 2061, 2288, 2533, 2808, 3107, 3439, 3800, 4199, 4634, 5113

Offset

0,4

Comments

Map a binary sequence b=[ b_1,... ] to a binary sequence c=[ c_1,... ] so that C=1/Product (1-x^i)^c_i == 1+Sum b_i*x^i mod 2.

This produces 2 new sequences: d={i:c_i=1} and e=[ 1,e_1,... ] where C=1+Sum e_i*x^i.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Formula

From Andrew Howroyd, Apr 15 2021: (Start)

Euler transform of A041004.

G.f.: A(x) = 1/Product_{k>=1} (1 - x^k)^(a(k+1) mod 2). (End)

Prog

(PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
seq(n)={my(v=vector(n+1)); v[1]=1; for(n=2, #v, v[n]=(v[n-1]+EulerT(v[2..n])[n-1])%2); concat([1], EulerT(v[2..n+1]))} \\ Andrew Howroyd, Apr 14 2021

Crossrefs

Cf. A041002, A041004.

Sequence in context: A265109 A301277 A029033 * A067592 A101195 A036018

Adjacent sequences: A041000 A041001 A041002 * A041004 A041005 A041006

Keyword

nonn,easy,eigen

Author

N. J. A. Sloane

Extensions

Terms a(50) and beyond from Andrew Howroyd, Apr 14 2021

Status

approved