A362500 -id:A362500 - cp's OEIS frontend

A173258 Number of compositions of n where differences between neighboring parts are in {-1,1}.

1, 1, 1, 3, 2, 4, 5, 5, 7, 10, 9, 14, 16, 19, 24, 31, 35, 45, 55, 66, 84, 104, 124, 156, 192, 236, 292, 363, 444, 551, 681, 839, 1040, 1287, 1586, 1967, 2430, 3001, 3717, 4597, 5683, 7034, 8697, 10758, 13312, 16469, 20369, 25204, 31180, 38574, 47726, 59047

Offset: 0

Alois P. Heinz, Jul 08 2012

nonn

			a(3) = 3: [3], [2,1], [1,2].
a(4) = 2: [4], [1,2,1].
a(5) = 4: [5], [3,2], [2,3], [2,1,2].
a(6) = 5: [6], [3,2,1], [2,1,2,1], [1,2,3], [1,2,1,2].

Alois P. Heinz, Table of n, a(n) for n = 0..5000
John Tyler Rascoe, Illustration of n = 1..16

Column k=1 of A214247, A214249.
Row sums of A309938, A364039.
Cf. A227310, A291904, A291905, A343795, A362500, A363718, A364529.

b:= proc(n, i) option remember;
      `if`(n<1 or i<1, 0, `if`(n=i, 1, add(b(n-i, i+j), j=[-1, 1])))
    end:
a:= n-> `if`(n=0, 1, add(b(n, j), j=1..n)):
seq(a(n), n=0..70);

b[n_, i_] := b[n, i] = If[n < 1 || i < 1, 0, If[n == i, 1, Sum[b[n - i, i + j], {j, {-1, 1}}]]]; a[n_] := If[n == 0, 1, Sum[b[n, j], {j, 1, n}]]; Table[a[n], {n, 0, 70}] // Flatten (* Jean-François Alcover, Dec 13 2013, translated from Maple *)

step(R,n)={matrix(n, n, i, j, if(i>j, if(j>1, R[i-j, j-1]) + if(j+1<=n, R[i-j, j+1])) )}
a(n)={my(R=matid(n), t=(n==0), m=0); while(R, m++; t+=vecsum(R[n,]); R=step(R,n)); t} \\ Andrew Howroyd, Aug 23 2019

a(n) ~ c * d^n, where d=1.23729141259673487395949649334678514763130846902468..., c=1.134796087242490181499736234755111281606636700030106.... - Vaclav Kotesovec, May 01 2014

G.f.: 1 + Sum_{k>0} G(x,k) where G(x,k) = x^k*(1 + G(x,k+1) + G(x,k-1)) for k > 0 and G(x,0) = 0. - John Tyler Rascoe, Sep 16 2023

A364039 Triangle read by rows: T(n,k) is the number of integer compositions of n with first part k and differences between neighboring parts in {-1,1}.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 2, 1, 0, 1, 0, 2, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 3, 2, 0, 0, 0, 0, 1, 0, 3, 2, 1, 2, 1, 0, 0, 0, 1, 0, 2, 3, 2, 1, 0, 0, 0, 0, 0, 1, 0, 3, 4, 3, 1, 1, 1, 0, 0, 0, 0, 1, 0, 4, 4, 4, 2, 1, 0, 0, 0, 0, 0, 0, 1

Offset: 0

John Tyler Rascoe, Aug 06 2023

			Triangle begins:
  1;
  0, 1;
  0, 0, 1;
  0, 1, 1, 1;
  0, 1, 0, 0, 1;
  0, 0, 2, 1, 0, 1;
  0, 2, 1, 1, 0, 0, 1;
  0, 1, 1, 1, 1, 0, 0, 1;
  0, 1, 3, 2, 0, 0, 0, 0, 1;
  0, 3, 2, 1, 2, 1, 0, 0, 0, 1;
  0, 2, 3, 2, 1, 0, 0, 0, 0, 0, 1;
  ...
For n = 6 there are a total of 5 compositions:
  T(6,1) = 2: (123), (1212)
  T(6,2) = 1: (2121)
  T(6,3) = 1: (321)
  T(6,6) = 1: (6)

Alois P. Heinz, Rows n = 0..200, flattened

Cf. A291905 (column k=1), A173258 (row sums).

Cf. A227310, A291904, A309938, A362500, A364529.

T:= proc(n, i) option remember; `if`(n<1 or i<1, 0,
     `if`(n=i, 1, add(T(n-i, i+j), j=[-1, 1])))
    end: T(0$2):=1:
seq(seq(T(n, k), k=0..n), n=0..14);  # Alois P. Heinz, Aug 08 2023

def A364039_rowlist(row_max):
    A = []
    for n in range(0,row_max+1):
        A.append([])
        for k in range(0,n+1):
            z = 0
            if n==k: z += 1
            elif k > 1 and k-1 <= n-k: z += A[n-k][k-1]
            if k+1 <= n-k and k != 0: z += A[n-k][k+1]
            A[n].append(z)
        print(A[n])
A364039_rowlist(12)

T(n,n) = 1.
T(n,k) = T(n-k,k+1) + T(n-k,k-1) for 0 < k < n.
T(n,k) = 0 for n < k.
T(n,0) = 0 for 0 < n.

A372647 Number of compositions such that their set adjacent differences are a subset of {-1,1} and contain 1 as a part of the composition itself.

Original entry on oeis.org

0, 1, 0, 2, 1, 1, 4, 1, 5, 5, 5, 10, 8, 14, 15, 22, 26, 31, 44, 47, 69, 80, 101, 131, 156, 203, 246, 315, 388, 484, 609, 746, 945, 1163, 1453, 1812, 2242, 2799, 3464, 4319, 5351, 6652, 8261, 10235, 12732, 15763, 19577, 24276, 30092, 37338, 46254, 57376, 71081

Offset: 0

John Tyler Rascoe, May 08 2024

nonn

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

Cf. (row sums of A372646).

Cf. A003242, A173258, A362500.

cp's OEIS Frontend

A173258 Number of compositions of n where differences between neighboring parts are in {-1,1}.

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A364039 Triangle read by rows: T(n,k) is the number of integer compositions of n with first part k and differences between neighboring parts in {-1,1}.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Python

Formula

A372647 Number of compositions such that their set adjacent differences are a subset of {-1,1} and contain 1 as a part of the composition itself.

Views

Author

Keywords

Examples

Crossrefs