A218694 -id:A218694 - cp's OEIS frontend

A301428 Number of compositions (ordered partitions) of n into prime parts such that no two adjacent parts are equal (Carlitz compositions).

1, 0, 1, 1, 0, 3, 0, 4, 3, 3, 10, 3, 16, 12, 18, 35, 24, 64, 57, 90, 137, 136, 259, 270, 416, 573, 679, 1088, 1264, 1869, 2491, 3199, 4691, 5834, 8341, 11053, 14685, 20595, 26636, 37199, 49449, 66572, 91377, 120733, 166151, 221912, 300038, 407775, 544843, 743318, 996752

Offset: 0

Ilya Gutkovskiy, Mar 21 2018

nonn

			a(7) = 4 because we have [7], [5, 2], [2, 5] and [2, 3, 2].

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for sequences related to compositions

Cf. A003242, A023360, A218694, A219107.

nmax = 50; CoefficientList[Series[1/(1 - Sum[x^Prime[k]/(1 + x^Prime[k]), {k, 1, nmax}]), {x, 0, nmax}], x]

G.f.: 1/(1 - Sum_{k>=1} x^prime(k)/(1 + x^prime(k))).

A298732 Number of compositions (ordered partitions) of n into parts > 1 such that no two adjacent parts are equal (Carlitz compositions).

Original entry on oeis.org

1, 0, 1, 1, 1, 3, 3, 6, 7, 14, 18, 30, 45, 66, 107, 157, 245, 369, 569, 862, 1325, 2020, 3078, 4717, 7183, 10991, 16769, 25626, 39117, 59763, 91264, 139362, 212893, 325060, 496525, 758258, 1158079, 1768634, 2701162, 4125320, 6300303, 9622247, 14695253, 22443451, 34276405, 52348435

Offset: 0

Ilya Gutkovskiy, Jan 25 2018

nonn

			a(7) = 6 because we have [7], [5, 2], [4, 3], [3, 4], [2, 5] and [2, 3, 2].

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for sequences related to compositions

Cf. A000045, A002865, A003242, A011782, A025147, A032020, A114900, A155822, A212804, A218694.

b:= proc(n, i) option remember; `if`(n=0, 1,
      add(`if`(j=i, 0, b(n-j, `if`(j<=n-j, j, 0))), j=2..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 25 2018

nmax = 45; CoefficientList[Series[1/(1 - Sum[x^k/(1 + x^k), {k, 2, nmax}]), {x, 0, nmax}], x]

G.f.: 1/(1 - Sum_{k>=2} x^k/(1 + x^k)).

cp's OEIS Frontend

A301428 Number of compositions (ordered partitions) of n into prime parts such that no two adjacent parts are equal (Carlitz compositions).

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