A309025 Expansion of x * Product_{k>=0} (1 + x^(2^k) + x^(2^(k+3))).

0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 2, 2, 1, 4, 2, 2, 1, 4, 2, 2, 1, 5, 3, 3, 2, 4, 2, 2, 1, 6, 3, 4, 2, 4, 2, 2, 1, 7, 4, 4, 2, 4, 2, 2, 1, 8, 4, 4, 2, 4, 2, 2, 1, 9, 5, 5, 3, 5, 3, 3, 2, 9, 4, 5, 2, 5, 2, 3, 1, 10, 6, 5, 3, 6, 4, 3, 2, 10, 4, 5, 2, 6, 2, 3, 1, 11, 7, 6, 4

Offset: 0

Author

Ilya Gutkovskiy, Jul 07 2019

Keywords

• nonn

Comments

a(n+1) is the number of representations of n=sum_i c_i*2^i with c_i in {0,1,8} [Anders]. See A120562 if c_i in {0,1,3} or A000012 if c_i in {0,1}. - R. J. Mathar, Mar 01 2023

Links

• K. Anders, Counting Non-Standard Binary Representations, JIS vol 19 (2016) #16.3.3 example 5.

Crossrefs

Cf. A002487, A112970, A309026.

Programs

Mathematica

nmax = 100; CoefficientList[Series[x Product[(1 + x^(2^k) + x^(2^(k + 3))), {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x]

Formula

a(n) = 0 for n <= 0, a(1) = 1; a(2*n) = a(n), a(2*n+1) = a(n-3) + a(n+1).