A090678 a(n) = A088567(n) mod 2.

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

Offset: 0

Author

N. J. A. Sloane, Dec 20 2003

Keywords

• nonn

Comments

a(n) = -(-1)^[n/2]*A110036(n)/2 for n>=2, where A110036 gives the partial quotients of the continued fraction expansion of 1 + Sum_{n>=0} 1/x^(2^n). - Paul D. Hanna, Jul 09 2005

Links

• O. J. Rodseth and J. A. Sellers, On a Restricted m-Non-Squashing Partition Function, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.4.
• N. J. A. Sloane and J. A. Sellers, On non-squashing partitions, arXiv:math/0312418 [math.CO], 2003.
• N. J. A. Sloane and J. A. Sellers, On non-squashing partitions, Discrete Math., 294 (2005), 259-274.

Crossrefs

b(8m) is (apart from the first term) A038189(m).

Cf. A110036, A110037.

Programs

Mathematica

nmax = 104; f = 1 + x/(1 - x) + Sum[x^(3*2^(k - 1))/Product[1 - x^(2^j), {j, 0, k}], {k, 1, Log[2, nmax]}];
a[n_] := Mod[SeriesCoefficient[f, {x, 0, n}], 2];
Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Jul 26 2018 *)

PARI

{a(n)=(-1)^(n\2)*polcoeff(1+x-x^2*(1+x)/(1+x^2)+ sum(k=1,#binary(n),x^(3*2^(k-1))/prod(j=0,k,1+x^(2^j)+x*O(x^n))),n)} /* Paul D. Hanna */

Formula

b(0) == 1; if n is odd, b(n) == b(n-1) + 1; b(8m+2) == 1; b(8m+6) == 0; b(16m+4) == 0; b(16m+12) == 1; for m>0, b(16m) == b(8m), b(32m+8) == 0, b(32m+24) == 1. In other words, for m>0, b(8m) is the value of the bit immediately to the left of the rightmost 1 when m is written in binary.

a(n) = (-1)^floor(n/2)*A110037(n). - Paul D. Hanna, Jul 09 2005