A048896 -id:A048896 - cp's OEIS frontend

A339146 a(n) = a(floor(n / 5)) * (n mod 5 + 1); initial terms are 1.

1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 2, 4, 6, 8, 10, 3, 6, 9, 12, 15, 4, 8, 12, 16, 20, 5, 10, 15, 20, 25, 1, 2, 3, 4, 5, 2, 4, 6, 8, 10, 3, 6, 9, 12, 15, 4, 8, 12, 16, 20, 5, 10, 15, 20, 25, 1, 2, 3, 4, 5, 2, 4, 6, 8, 10, 3, 6, 9, 12, 15, 4, 8, 12, 16, 20, 5, 10, 15, 20, 25

Offset: 0

Robert Dougherty-Bliss, Nov 25 2020

nonn

If a(n) is arranged in a table with row lengths 5, then the first column is the transpose of the first row, followed the transpose of the second row, followed by the transpose of the third row, and so on. The remainder of each row (except the first) is an arithmetic progression whose start and step size equals the first entry of the row.
a(n) = O(n).
limsup_n a(n) = +oo.

			a(10) = a(2) * 1 = 1.
a(13) = a(2) * 4 = 4.

Cf. A194459.

Cf. A048896 (with 2 instead of 5, but shifted).

a(n) = if (n < 5, 1, a(n\5)*(n % 5 + 1)); \\ Michel Marcus, Nov 26 2020

def a(n):
    if n < 5:
        return 1
    q, r = divmod(n, 5)
    return a(q) * (r + 1)

A358333 By concatenating the standard compositions for each part of the n-th standard composition, we get a sequence of length a(n). Row-lengths of A357135.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 2, 3, 1, 3, 2, 3, 3, 3, 3, 4, 2, 2, 3, 4, 3, 3, 3, 4, 2, 4, 3, 4, 4, 4, 4, 5, 2, 3, 2, 3, 4, 4, 4, 5, 2, 4, 3, 4, 4, 4, 4, 5, 3, 3, 4, 5, 4, 4, 4, 5, 3, 5, 4, 5, 5, 5, 5, 6, 3, 3, 3, 4, 3, 3, 3, 4, 3, 5, 4, 5, 5, 5, 5, 6, 3, 3, 4, 5, 4, 4, 4

Offset: 0

Gus Wiseman, Nov 10 2022

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			Composition 92 in standard order is (2,1,1,3), with compositions ((2),(1),(1),(1,1)) so a(92) = 5.

Gus Wiseman, Statistics, classes, and transformations of standard compositions

See link for sequences related to standard compositions (A066099).
Dominates A000120.
Row-lengths of A357135, which is ranked by A357134.
A related sequence is A358330.
Cf. A001511, A029931, A048896, A058891, A070939, A096111, A333766, A357137, A357139, A357186, A357187.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Length/@Table[Join@@stc/@stc[n],{n,0,100}]

Sum of A000120 over row n of A066099.

A358525 Number of distinct permutations of the n-th composition in standard order.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 3, 2, 3, 3, 1, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 1, 1, 2, 2, 3, 1, 6, 6, 4, 2, 6, 1, 6, 6, 6, 6, 5, 2, 3, 6, 4, 6, 6, 6, 5, 3, 4, 6, 5, 4, 5, 5, 1, 1, 2, 2, 3, 2, 6, 6, 4, 2, 3, 3, 12, 3, 12, 12, 5, 2, 6, 3, 12, 3, 4

Offset: 0

Gus Wiseman, Nov 21 2022

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The a(45) = 6 permutations are: (2121), (2112), (2211), (1221), (1212), (1122).

Gus Wiseman, Statistics, classes, and transformations of standard compositions

See link for sequences related to standard compositions.
Positions of 1's are A272919.
Cf. A000120, A048896, A070939, A329395, A333766, A358371, A358372, A358379, A358505, A358507, A358508.

stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
Table[Length[Permutations[stc[n]]],{n,0,100}]

cp's OEIS Frontend

A339146 a(n) = a(floor(n / 5)) * (n mod 5 + 1); initial terms are 1.

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A358333 By concatenating the standard compositions for each part of the n-th standard composition, we get a sequence of length a(n). Row-lengths of A357135.

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A358525 Number of distinct permutations of the n-th composition in standard order.

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Mathematica