A214247 -id:A214247 - cp's OEIS frontend

A370929 Number of compositions of n with parts (p_1, ..., p_i) such that the set of adjacent differences is a subset of {-k,k} for some k > 0 and the number of parts equals ceiling(p_1/k).

1, 1, 1, 2, 2, 3, 4, 3, 5, 5, 7, 5, 9, 7, 7, 10, 11, 9, 13, 9, 13, 17, 11, 14, 19, 15, 13, 20, 19, 18, 23, 19, 20, 26, 21, 20, 32, 22, 25, 27, 33, 25, 37, 21, 34, 36, 35, 24, 50, 26, 40, 37, 44, 32, 51, 31, 48, 46, 49, 34, 65, 40, 45, 54, 56, 48, 63, 42, 58

Offset: 0

John Tyler Rascoe, Mar 06 2024

			The compositions for n = 6 and n = 8 are:
6: [6], [5,1], [4,2], [3,2,1].
8: [8], [7,1], [6,2], [3,2,3], [3,5].

John Tyler Rascoe, Table of n, a(n) for n = 0..1000

Compositions such that no adjacent parts are equal is A003242.
Compositions such that the set of adjacent differences is a subset of {-1,1} is A173258 and {-2,2} is A214254.
The array A214247 counts compositions such that the set of adjacent differences is a subset of {-k,k}.

{ my(N=75, x='x+O('x^N));
my(gf= 1 + sum(p=1, N, sum(k=1, p, x^(p*ceil(p/k)) * prod(j=1, ceil(p/k)-1, (x^(-j*k) + x^(j*k))))));
Vec(gf) }

G.f.: 1 + Sum_{p>0} Sum_{k=1..p} x^(p*i) * Product_{j=1..i-1} (x^(-j*k) + x^(j*k)), where i = ceiling(p/k).

A383620 Number of weak compositions of n such that the set of adjacent differences is a subset of {-1,1}.

Original entry on oeis.org

1, 4, 5, 9, 13, 20, 30, 45, 66, 102, 152, 229, 344, 518, 780, 1180, 1775, 2676, 4037, 6088, 9182, 13852, 20891, 31512, 47536, 71706, 108166, 163172, 246140, 371303, 560118, 844943, 1274606, 1922767, 2900522, 4375493, 6600511, 9956990, 15020307, 22658428

Offset: 0

John Tyler Rascoe, May 02 2025

nonn

			a(0) = 1: (0).
a(1) = 4: (0,1), (0,1,0), (1,0), (1).
...
a(4) = 13: (0,1,0,1,0,1,0,1), (0,1,0,1,0,1,0,1,0), (1,0,1,0,1,0,1,0), (1,0,1,0,1,0,1), (0,1,0,1,2), (1,0,1,2), (2,1,0,1,0), (2,1,0,1), (0,1,2,1,0), (0,1,2,1), (1,2,1,0), (1,2,1), (4).

Alois P. Heinz, Table of n, a(n) for n = 0..5598

Cf. A007318, A173258, A214247, A214249, A227310.

M(k) = matrix(k+1,k+1, i,j, if(i==j,1,if(i==j-1, -x^(i-1), if(i==j+1, -x^(i-1), 0))))
A_x(N) = {my(k=N+1,x='x+O('x^k)); Vec(vecsum(M(k)^(-1) * vector(k+1,i,x^(i-1))~))}
A_x(10)

A383549 Number of rises in all compositions of n with parts in {1,2,3} and adjacent differences in {-1,1}.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 5, 3, 9, 11, 10, 24, 21, 30, 50, 43, 75, 93, 96, 161, 170, 215, 312, 323, 456, 574, 639, 906, 1046, 1276, 1710, 1935, 2501, 3135, 3642, 4760, 5699, 6893, 8823, 10401, 12952, 16079, 19104, 24002, 29097, 35165, 43865, 52628, 64503, 79363, 95329

Offset: 0

John Tyler Rascoe, Apr 29 2025

A rise is any pair of parts (p_{i-1},p_i) with p_{i-1} < p_i.

By reversal a(n) is also the number of descents in all compositions of n of this kind.

			For n = 6 the following compositions have 5 rises: (1,2,1,2), (1,2,3), (2,1,2,1), (3,2,1).

Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,2,-1,0,-2,0,-1).

Cf. A011782, A076118, A124760, A151842, A173258, A214247, A220062, A238343.

A_x(N) = {my(x='x+O('x^N)); concat([0,0,0], Vec(x^3*(1 + x^2)^2*(1 + x + x^3)/(1 - x^3 - x^5)^2))}
A_x(40)

G.f.: x^3*(1 + x^2)^2*(1 + x + x^3)/(1 - x^3 - x^5)^2.

cp's OEIS Frontend

A370929 Number of compositions of n with parts (p_1, ..., p_i) such that the set of adjacent differences is a subset of {-k,k} for some k > 0 and the number of parts equals ceiling(p_1/k).

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A383620 Number of weak compositions of n such that the set of adjacent differences is a subset of {-1,1}.

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PARI

A383549 Number of rises in all compositions of n with parts in {1,2,3} and adjacent differences in {-1,1}.

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