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A041004 The sequence b, given that c is a left shift by one place of b. 3
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 (list; graph; refs; listen; history; text; internal format)
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,changed

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

STATUS

approved

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Last modified April 17 06:46 EDT 2021. Contains 343059 sequences. (Running on oeis4.)