A041004 The sequence b, given that c is a left shift by one place of b.

1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1

Offset

0,1

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)

a(n) = A041003(n+1) mod 2.

G.f.: A(x) = 1 + Sum_{k>=0} x^A041002(k). (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); v} \\ Andrew Howroyd, Apr 14 2021

Crossrefs

Cf. A041002, A041003.

Sequence in context: A217206 A189097 A189094 * A141736 A190188 A278513

Adjacent sequences: A041001 A041002 A041003 * A041005 A041006 A041007

Keyword

nonn,easy,eigen

Author

N. J. A. Sloane

Extensions

Terms a(53) and beyond from Andrew Howroyd, Apr 14 2021

Status

approved