A060945 -id:A060945 - cp's OEIS frontend

A245367 Compositions of n into parts 3, 5 and 7.

1, 0, 0, 1, 0, 1, 1, 1, 2, 1, 3, 3, 3, 6, 5, 8, 10, 11, 17, 18, 25, 32, 37, 52, 61, 79, 102, 123, 163, 200, 254, 326, 402, 519, 649, 819, 1045, 1305, 1664, 2096, 2643, 3358, 4220, 5352, 6759, 8527, 10806, 13622, 17237, 21785, 27501, 34802, 43934, 55544, 70209, 88672, 112131, 141644, 179018, 226274, 285860, 361358

Offset: 0

David Neil McGrath, Aug 20 2014

			a(16) = 10: the compositions are the permutations of [5533] (there are 4!/2!2!=6 of them) and the permutations of [7333] (there are 4!/3!=4).

Index entries for linear recurrences with constant coefficients, signature (0,0,1,0,1,0,1).

Cf. A000073, A013979, A017818, A060945, A079956, A079957, A079971, A079973, A120400, A121833.

LinearRecurrence[{0,0,1,0,1,0,1},{1,0,0,1,0,1,1},70] (* Harvey P. Dale, Jan 27 2017 *)

Vec(1/(1-x^3-x^5-x^7) +O(x^66)) \\ Joerg Arndt, Aug 20 2014

G.f: 1/(1-x^3-x^5-x^7).

a(n) = a(n-3) + a(n-5) + a(n-7).

A347493 a(0) = 1, a(1) = 0, a(2) = a(3) = 1; thereafter, a(n) = a(n-1) + a(n-2) + a(n-4).

Original entry on oeis.org

1, 0, 1, 1, 3, 4, 8, 13, 24, 41, 73, 127, 224, 392, 689, 1208, 2121, 3721, 6531, 11460, 20112, 35293, 61936, 108689, 190737, 334719, 587392, 1030800, 1808929, 3174448, 5570769, 9776017, 17155715, 30106180, 52832664, 92714861, 162703240, 285524281, 501060185, 879299327, 1543062752

Offset: 0

Greg Dresden and Yichen P. Wang, Sep 03 2021

a(n) is also the number of ways to tile a strip of length n with squares, dominoes, and tetrominoes such that the first tile is NOT a square. As such, it completes the set of such tilings with A005251 (first tile is NOT a domino), A005314 (first tile is NOT a tetromino), and A060945 (no restrictions on first tile).

Michael De Vlieger, Table of n, a(n) for n = 0..4096
Jean-Luc Baril, Sergey Kirgizov, and Armen Petrossian, Dyck Paths with catastrophes modulo the positions of a given pattern, Australasian J. Comb. (2022) Vol. 84, No. 2, 398-418.
Index entries for linear recurrences with constant coefficients, signature (1,1,0,1).

Cf. A005251, A005314, A060945.

CoefficientList[Series[(1 - x)/(1 - x - x^2 - x^4), {x, 0, 40}], x] (* Michael De Vlieger, Mar 04 2022 *)
LinearRecurrence[{1,1,0,1},{1,0,1,1},60] (* Harvey P. Dale, Aug 17 2023 *)

a(n) = 2*A060945(n) - A005251(n) - A005314(n).
G.f.: (1 - x)/(1 - x - x^2 - x^4).
Sum_{k=0..n} a(k)*F(n-k) = a(n+3) - F(n+2) for F(n)=A000045(n) the Fibonacci numbers.
5*a(n) = 2*(-1)^n + 3*A005314(n+1) -4*A005314(n) +2*A005314(n-1). - R. J. Mathar, Sep 30 2021

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

A245367 Compositions of n into parts 3, 5 and 7.

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A347493 a(0) = 1, a(1) = 0, a(2) = a(3) = 1; thereafter, a(n) = a(n-1) + a(n-2) + a(n-4).

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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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