A079958 -id:A079958 - cp's OEIS frontend

A224341 Apparently solves the identity: Find sequence A that represents the numbers of ordered compositions of n into the elements of the set {B}; and vice versa.

1, 2, 4, 7, 13, 25, 46, 86, 161, 301, 562, 1051, 1964, 3670, 6859, 12819, 23956, 44772, 83673

Offset: 0

Gary W. Adamson, Apr 03 2013

nonn

Represents the numbers of ordered compositions of n using the terms of A224342: (1, 2, 3, 6, 10, 18, 32, ...); such that the latter represents the numbers of ordered compositions of n using the terms of A224341.

It appears that given any sequence of real terms pulled out of a hat S(n); repeated iterates of S(n) -> characteristic function of S(n) -> INVERT transform of the latter -> next sequence, (repeat); will converge upon two alternating sequences A224341 and A224342 as a fixed limit, as to absolute values.

			Given the sequence (1, 0, 0, 0, ...), a few iterates using the rules rapidly converge upon A224341 and A224342.

Cf. A224342, A079958.

The sequences are obtained by taking iterates as described in the comments. There is no known generating function at the date of this submission.

A224342 Apparently solves the identity: find sequence B that represents the numbers of ordered compositions of n using the terms of A, and vice versa.

Original entry on oeis.org

1, 2, 3, 6, 10, 18, 32, 57, 101, 179, 318, 564, 1002, 1778, 3157, 5604, 9949, 17661, 31352, 55657

Offset: 1

Gary W. Adamson, Apr 03 2013

It appears that given any sequence of real numbers taken out of a hat, S(n); repeated iterates of the operation: S(n) -> characteristic function of S(n) -> INVERT transform of the latter -> new sequence, then (repeat), will converge upon two sequences A = A224341 and B = A224342 as a 2-cycle fixed limit.

Alternatively as a conjecture, A and B solve the unique identity as described in the heading as to ordered compositions with A = A224341 and B = A224342. The INVERT transform of the characteristic function of A = B, and the INVERT transform of the characteristic function of B = A.

			Given the sequence (1, 0, 0, 0, ...) and following the iterative rules, the sequences converge upon A224341 and A224342 as an alternating fixed limit.

Cf. A224341, A079958.

Repeated trials of any sequence of real numbers pulled out of a hat will apparently converge upon A224341 and A224342 as a 2-cycle fixed limit (absolute values of terms). There is no known generating function at the date of this submission.

A306489 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of 1/(1 - Sum_{d|k} x^d).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 2, 2, 5, 1, 1, 1, 1, 3, 3, 8, 1, 1, 1, 2, 1, 6, 4, 13, 1, 1, 1, 1, 4, 1, 10, 6, 21, 1, 1, 1, 2, 1, 7, 2, 18, 9, 34, 1, 1, 1, 1, 3, 1, 13, 3, 31, 13, 55, 1, 1, 1, 2, 2, 6, 1, 25, 4, 55, 19, 89, 1, 1, 1, 1, 3, 3, 10, 1, 46, 5, 96, 28, 144, 1

Offset: 0

Ilya Gutkovskiy, Feb 19 2019

A(n,k) is the number of compositions (ordered partitions) of n into divisors of k.

			Square array begins:
  1,  1,  1,   1,  1,   1,  ...
  1,  1,  1,   1,  1,   1,  ...
  1,  2,  1,   2,  1,   2,  ...
  1,  3,  2,   3,  1,   4,  ...
  1,  5,  3,   6,  1,   7,  ...
  1,  8,  4,  10,  2,  13,  ...

Columns k=1..7 give A000012, A000045 (for n > 0), A000930, A060945, A003520, A079958, A005709.

Cf. A100346, A214575.

Table[Function[k, SeriesCoefficient[1/(1 - Sum[x^d, {d, Divisors[k]}]), {x, 0, n}]][i - n + 1], {i, 0, 12}, {n, 0, i}] // Flatten

G.f. of column k: 1/(1 - Sum_{d|k} x^d).

cp's OEIS Frontend

A224341 Apparently solves the identity: Find sequence A that represents the numbers of ordered compositions of n into the elements of the set {B}; and vice versa.

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A224342 Apparently solves the identity: find sequence B that represents the numbers of ordered compositions of n using the terms of A, and vice versa.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A306489 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of 1/(1 - Sum_{d|k} x^d).

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