A102396 A mod 2 related Jacobsthal sequence.

0, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 3, 3, 5, 1, 1, 1, 3, 1, 3, 3, 5, 1, 3, 3, 5, 3, 5, 5, 11, 1, 1, 1, 3, 1, 3, 3, 5, 1, 3, 3, 5, 3, 5, 5, 11, 1, 3, 3, 5, 3, 5, 5, 11, 3, 5, 5, 11, 5, 11, 11, 21, 1, 1, 1, 3, 1, 3, 3, 5, 1, 3, 3, 5, 3, 5, 5, 11, 1, 3, 3, 5, 3, 5, 5, 11, 3, 5, 5, 11, 5, 11, 11, 21, 1, 3

Offset

0,8

Comments

Conjecture: all the terms are Jacobsthal numbers.

The conjecture is true since a(n) = A001045(A000120(n)). - Paul Barry, Jan 07 2005

Links

Amiram Eldar, Table of n, a(n) for n = 0..10000

Formula

a(2^n-1) = A001045(n).

A001316(n) = A102395(n) + 2*a(n).

a(n) = Sum_{k=0..n} if(n+k == 1 (mod 3), C(n, k) mod 2, 0).

a(n) = Sum_{k=0..n} if(n+k == 2 (mod 3), C(n, k) mod 2, 0).

a(n) = (1/2) * Sum_{k=0..n} ({F(n+k)*binomial(n, k)} mod 2) where F = A000045.

Recurrence : a(0) = 0 then a(2n) = a(n) and a(2n+1) = 2*a(n) + 1 - 2*t(n) where t(n) = A010060(n) is the Thue-Morse sequence. - Benoit Cloitre, May 15 2005

Mathematica

j[n_] := (2^n - (-1)^n) / 3; a[n_] := j[DigitCount[n, 2, 1]]; Array[a, 100, 0] (* Amiram Eldar, Jul 22 2023 *)

Crossrefs

Cf. A000045, A000120, A001045, A001316, A010060, A102395.

Sequence in context: A337322 A090176 A351544 * A344852 A095960 A316314

Adjacent sequences: A102393 A102394 A102395 * A102397 A102398 A102399

Keyword

easy,nonn

Author

Paul Barry, Jan 06 2005

Status

approved