A339422 -id:A339422 - cp's OEIS frontend

A339426 Number of compositions (ordered partitions) of n into an even number of powers of 2.

1, 0, 1, 2, 2, 6, 9, 14, 30, 48, 86, 156, 268, 478, 849, 1486, 2638, 4660, 8214, 14532, 25664, 45304, 80078, 141412, 249768, 441276, 779376, 1376696, 2431924, 4295534, 7587753, 13403102, 23674870, 41819588, 73870046, 130483396, 230486384, 407130332, 719153726

Offset: 0

Ilya Gutkovskiy, Dec 03 2020

nonn

			a(5) = 6 because we have [4, 1], [1, 4], [2, 1, 1, 1], [1, 2, 1, 1], [1, 1, 2, 1] and [1, 1, 1, 2].

Index entries for sequences related to compositions

Cf. A000079, A023359, A034008, A040039, A339422, A339427.

b:= proc(n, t) option remember; `if`(n=0, t,
      add(b(n-2^i, 1-t), i=0..ilog2(n)))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..42);  # Alois P. Heinz, Dec 03 2020

nmax = 38; CoefficientList[Series[(1/2) (1/(1 - Sum[x^(2^k), {k, 0, Floor[Log[2, nmax]] + 1}]) + 1/(1 + Sum[x^(2^k), {k, 0, Floor[Log[2, nmax]] + 1}])), {x, 0, nmax}], x]

G.f.: (1/2) * (1 / (1 - Sum_{k>=0} x^(2^k)) + 1 / (1 + Sum_{k>=0} x^(2^k))).
a(n) = (A023359(n) + A339422(n)) / 2.
a(n) = Sum_{k=0..n} A023359(k) * A339422(n-k).

A339427 Number of compositions (ordered partitions) of n into an odd number of powers of 2.

Original entry on oeis.org

0, 1, 1, 1, 4, 4, 9, 17, 26, 50, 88, 150, 274, 478, 841, 1497, 2634, 4650, 8234, 14518, 25654, 45340, 80040, 141414, 249822, 441192, 779422, 1376752, 2431772, 4295678, 7587761, 13402881, 23675186, 41819442, 73869802, 130483966, 230485902, 407130212, 719154602

Offset: 0

Ilya Gutkovskiy, Dec 03 2020

nonn

			a(5) = 4 because we have [2, 2, 1], [2, 1, 2], [1, 2, 2] and [1, 1, 1, 1, 1].

Index entries for sequences related to compositions

Cf. A000079, A023359, A040039, A166444, A339422, A339426.

b:= proc(n, t) option remember; `if`(n=0, t,
      add(b(n-2^i, 1-t), i=0..ilog2(n)))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..42);  # Alois P. Heinz, Dec 03 2020

nmax = 38; CoefficientList[Series[(1/2) (1/(1 - Sum[x^(2^k), {k, 0, Floor[Log[2, nmax]] + 1}]) - 1/(1 + Sum[x^(2^k), {k, 0, Floor[Log[2, nmax]] + 1}])), {x, 0, nmax}], x]

G.f.: (1/2) * (1 / (1 - Sum_{k>=0} x^(2^k)) - 1 / (1 + Sum_{k>=0} x^(2^k))).
a(n) = (A023359(n) - A339422(n)) / 2.
a(n) = -Sum_{k=0..n-1} A023359(k) * A339422(n-k).

cp's OEIS Frontend

A339426 Number of compositions (ordered partitions) of n into an even number of powers of 2.

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