A110036 -id:A110036 - cp's OEIS frontend

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

N. J. A. Sloane, Dec 20 2003

nonn

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

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.

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

Cf. A110036, A110037.

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 *)

{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 */

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

A110037 Signed version of A090678 and congruent to A088567 mod 2.

Original entry on oeis.org

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

Paul D. Hanna, Jul 09 2005

a(n) = (-1)^[n/2]*A090678(n) = A088567(n) mod 2, where A088567(n) equals the number of "non-squashing" partitions of n. a(n) = -A110036(n)/2 for n>=2, where the A110036 gives the partial quotients of the continued fraction expansion of 1 + Sum_{n>=0} 1/x^(2^n).

Cf. A110036, A090678, A088567, A073089.

{a(n)=polcoeff(A=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)}

G.f.: A(x) = 1+x - x^2*(1+x)/(1+x^2) + Sum_{k>=1} x^(3*2^(k-1))/Product_{j=0..k} (1+x^(2^j)).

Conjecture: a(n) = A073089(n) - A073089(n+1) for n >= 2. - Alan Michael Gómez Calderón, Aug 19 2025

cp's OEIS Frontend

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

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