A214249 -id:A214249 - 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

A214258 Number T(n,k) of compositions of n where the difference between largest and smallest parts equals k; triangle T(n,k), n>=1, 0<=k

Original entry on oeis.org

1, 2, 0, 2, 2, 0, 3, 3, 2, 0, 2, 9, 3, 2, 0, 4, 11, 12, 3, 2, 0, 2, 25, 20, 12, 3, 2, 0, 4, 35, 49, 23, 12, 3, 2, 0, 3, 60, 95, 58, 23, 12, 3, 2, 0, 4, 96, 188, 123, 61, 23, 12, 3, 2, 0, 2, 157, 366, 266, 132, 61, 23, 12, 3, 2, 0, 6, 241, 714, 557, 294, 135, 61, 23, 12, 3, 2, 0

Offset: 1

Alois P. Heinz, Jul 08 2012

For fixed k > 0, limit_{n->infinity} T(n,k)^(1/n) = d, where d > 1 is the real root of the equation d^(k+2) - 2*d^(k+1) + 1 = 0. - Vaclav Kotesovec, Jan 07 2019

			T(4,0) = 3: [4], [2,2], [1,1,1,1].
T(5,1) = 9: [3,2], [2,3], [2,2,1], [2,1,2], [2,1,1,1], [1,2,2], [1,2,1,1], [1,1,2,1], [1,1,1,2].
T(5,2) = 3: [3,1,1], [1,3,1], [1,1,3].
T(5,3) = 2: [4,1], [1,4].
T(6,2) = 12: [4,2], [3,2,1], [3,1,2], [3,1,1,1], [2,4], [2,3,1], [2,1,3], [1,3,2], [1,3,1,1], [1,2,3], [1,1,3,1], [1,1,1,3].
Triangle T(n,k) begins:
  1;
  2,  0;
  2,  2,  0;
  3,  3,  2,  0;
  2,  9,  3,  2,  0;
  4, 11, 12,  3,  2,  0;
  2, 25, 20, 12,  3,  2,  0;
  4, 35, 49, 23, 12,  3,  2,  0;

Alois P. Heinz, Rows n = 1..150, flattened

Columns k=0-10 give: A000005, A214259, A323119, A323120, A323121, A323122, A323123, A323124, A323125, A323126, A323127.
Row sums give: A011782.
T(2n,n) gives A323111.
Cf. A214246, A214247, A214248, A214249, A214257, A214268, A214269.

b:= proc(n, k, s, t) option remember;
      `if`(n<0, 0, `if`(n=0, 1, add(b(n-j, k,
       min(s,j), max(t,j)), j=max(1, t-k+1)..s+k-1)))
    end:
A:= proc(n, k) option remember;
      `if`(n=0, 1, add(b(n-j, k+1, j, j), j=1..n))
    end:
T:= (n, k)-> A(n, k) -`if`(k=0, 0, A(n, k-1)):
seq(seq(T(n,k), k=0..n-1), n=1..15);
# second Maple program:
b:= proc(n, s, t) option remember; `if`(n=0, x^(t-s),
      add(b(n-j, min(s, j), max(t, j)), j=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n-1))(b(n$2, 0)):
seq(T(n), n=1..15);  # Alois P. Heinz, Jan 05 2019

b[n_, k_, s_, t_] := b[n, k, s, t] = If[n < 0, 0, If[n == 0, 1, Sum [b[n-j, k, Min[s, j], Max[t, j]], {j, Max[1, t-k+1], s+k-1}]]]; A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[b[n-j, k+1, j, j], {j, 1, n}]]; T[n_, k_] := A[n, k] - If[k == 0, 0, A[n, k-1]]; Table[Table[T[n, k], {k, 0, n-1}], {n, 1, 15}] // Flatten (* Jean-François Alcover, Jan 15 2014, translated from Maple *)

T(n,0) = A214257(n,0), T(n,k) = A214257(n,k)-A214257(n,k-1) for k>0.

A214269 Number T(n,k) of compositions of n where the difference between largest and smallest parts equals k and adjacent parts are unequal; triangle T(n,k), n>=1, 0<=k

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 1, 2, 0, 1, 3, 1, 2, 0, 1, 2, 8, 1, 2, 0, 1, 4, 7, 8, 1, 2, 0, 1, 2, 13, 12, 8, 1, 2, 0, 1, 4, 25, 18, 12, 8, 1, 2, 0, 1, 4, 27, 46, 23, 12, 8, 1, 2, 0, 1, 4, 43, 69, 51, 23, 12, 8, 1, 2, 0, 1, 3, 71, 111, 90, 56, 23, 12, 8, 1, 2, 0

Offset: 1

Alois P. Heinz, Jul 09 2012

			T(7,0) = 1: [7].
T(7,1) = 4: [4,3], [3,4], [2,3,2], [1,2,1,2,1].
T(7,2) = 7: [3,1,3], [3,1,2,1], [2,1,3,1], [1,3,2,1], [1,3,1,2], [1,2,3,1], [1,2,1,3].
T(7,3) = 8: [5,2], [4,2,1], [4,1,2], [2,5], [2,4,1], [2,1,4], [1,4,2], [1,2,4].
T(7,4) = 1: [1,5,1].
T(7,5) = 2: [6,1], [1,6].
Triangle T(n,k) begins:
  1;
  1,  0;
  1,  2,  0;
  1,  1,  2,  0;
  1,  3,  1,  2,  0;
  1,  2,  8,  1,  2,  0;
  1,  4,  7,  8,  1,  2,  0;
  1,  2, 13, 12,  8,  1,  2,  0;

Alois P. Heinz, Rows n = 1..150, flattened

Columns k=0-10 give: A000012, A214270, A214271, A214272, A214273, A214274, A214275, A214276, A214277, A214278, A214279.
Row sums give: A003242.
Cf. A214246, A214247, A214248, A214249, A214257, A214258, A214268.

b:= proc(n, k, s, t, l) option remember;
      `if`(n<0, 0, `if`(n=0, 1, add(`if`(j=l, 0, b(n-j, k,
       min(s, j), max(t, j), j)), j=max(1, t-k+1)..s+k-1)))
    end:
A:= proc(n, k) option remember;
      `if`(n=0, 1, add(b(n-j, k+1, j, j, j), j=1..n))
    end:
T:= (n, k)-> A(n, k) -`if`(k=0, 0, A(n, k-1)):
seq(seq(T(n, k), k=0..n-1), n=1..14);

b[n_, k_, s_, t_, l_] := b[n, k, s, t, l] = If[n < 0, 0, If[n == 0, 1, Sum [If[j == l, 0, b[n-j, k, Min[s, j], Max[t, j], j]], {j, Max[1, t-k+1], s+k-1}] ] ]; a[n_, k_] := a[n, k] = If[n == 0, 1, Sum[b[n - j, k+1, j, j, j], {j, 1, n}]]; t[n_, k_] := a[n, k] - If[k == 0, 0, a[n, k-1]]; Table[Table[t[n, k], {k, 0, n-1}], {n, 1, 12}] // Flatten (* Jean-François Alcover, Dec 11 2013, translated from Maple *)

T(n,0) = 1, T(n,k) = A214268(n,k) - A214268(n,k-1) for k>0.

A214247 Number A(n,k) of compositions of n where differences between neighboring parts are in {-k,k}; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 3, 4, 4, 1, 1, 1, 1, 1, 2, 5, 2, 1, 1, 1, 1, 1, 3, 3, 5, 4, 1, 1, 1, 1, 1, 1, 2, 2, 7, 3, 1, 1, 1, 1, 1, 1, 3, 3, 6, 10, 4, 1, 1, 1, 1, 1, 1, 1, 2, 1, 4, 9, 2

Offset: 0

Alois P. Heinz, Jul 08 2012

			A(5,0) = 2: [5], [1,1,1,1,1].
A(5,1) = 4: [5], [3,2], [2,3], [2,1,2].
A(5,2) = 2: [5], [1,3,1].
A(5,3) = 3: [5], [4,1], [1,4].
Square array A(n,k) begins:
  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
  2,  1,  1,  1,  1,  1,  1,  1,  1, ...
  2,  3,  1,  1,  1,  1,  1,  1,  1, ...
  3,  2,  3,  1,  1,  1,  1,  1,  1, ...
  2,  4,  2,  3,  1,  1,  1,  1,  1, ...
  4,  5,  3,  2,  3,  1,  1,  1,  1, ...
  2,  5,  2,  3,  2,  3,  1,  1,  1, ...
  4,  7,  6,  1,  3,  2,  3,  1,  1, ...

Alois P. Heinz, Antidiagonals n = 0..140

Columns k=0-2 give: A000005, A173258, A214254.
Rows n=0, 1 and main diagonal give: A000012.
Cf. A000108, A214246, A214248, A214249, A214257, A214258, A214268, A214269, A227543.

b:= proc(n, i, k) option remember;
      `if`(n<1 or i<1, 0, `if`(n=i, 1, add(b(n-i, i+j, k), j={-k, k})))
    end:
A:= (n, k)-> `if`(n=0, 1, add(b(n, j, k), j=1..n)):
seq(seq(A(n, d-n), n=0..d), d=0..15);

b[n_, i_, k_] := b[n, i, k] = If[n < 1 || i < 1, 0, If[n == i, 1, Sum[b[n - i, i + j, k], { j, Union[{-k, k}]}]]]; a[n_, k_] := If[n == 0, 1, Sum[b[n, j, k], {j, 1, n}]]; Table[Table[a[n, d - n], {n, 0, d}], {d, 0, 15}] // Flatten (* Jean-François Alcover, Dec 13 2013, translated from Maple *)

tri(k) = {(k*(k+1)/2)}
ra(n) = {(sqrt(1+8*n)-1)/2}
C(q,n) = {c = if(n<=1, return(1)); return(sum(i=0, n-1, C(q, i) * C(q, n-1-i) * q^((i+1)*(n-1 -i))));}
Cw_q(i,k) = {return(q^(((k+2)*i)+1) * polrecip(C(q^(2*k),i)))}
A_qt(k,N) = {1 + sum(i=0,N/(k+1), t^((2*i)+1) * Cw_q(i,k) * (sum(j=0,ra(N+2)+1, prod(u=1,j, sum(v=0,(N-(tri(u)*k))/(k+2), t^((2*v)+1) * q^(((2*v)+1)*u*k) * Cw_q(v,k)))))^2)}
A_q(k,N) = {my(q='q+O('q^N), Aqt = A_qt(k,N), Aq = 1); for(i=1,poldegree(Aqt,t), Aq += polcoef(Aqt,i,t)/(1-q^i)); Aq}
A214247_array(maxrow,maxcolumn) = {my(m=concat([1],vector(maxrow,n,numdiv(n)))~); for(k=1,maxcolumn, m = matconcat([m,Vec(A_q(k,maxrow))~])); m}
A214247_array(10,10) \\ John Tyler Rascoe, Oct 15 2024

G.f. for column k > 0: A(k,q) is A(k,q,t) = Sum_{n,m>=0} (q^n)*(t^m) under the transform (q^n)*(t^m) -> (q^n)/(1-q^m) for all m > 0 where A(k,q,t) = 1 + Sum_{i>=0} ( t^((2*i)+1) * Cw(i,k,q) * (Sum_{j>=0} (Product_{u=1..j} (Sum_{v>=0} t^((2*v)+1) * q^(((2*v)+1)*k*u) * Cw(v,k,q))))^2 ), Cw(i,k,q) = q^(((k+2)*i)+1) * Ca(i,q^(2*k)), and Ca(i,q) is the i-th Carlitz-Riordan q-Catalan number (i-th row polynomial of A227543). - John Tyler Rascoe, Sep 13 2024

A214248 Number A(n,k) of compositions of n where differences between neighboring parts are in {-k,...,k}; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 4, 3, 1, 1, 2, 4, 6, 2, 1, 1, 2, 4, 8, 11, 4, 1, 1, 2, 4, 8, 14, 17, 2, 1, 1, 2, 4, 8, 16, 27, 29, 4, 1, 1, 2, 4, 8, 16, 30, 49, 47, 3, 1, 1, 2, 4, 8, 16, 32, 59, 92, 78, 4, 1, 1, 2, 4, 8, 16, 32, 62, 113, 170, 130, 2

Offset: 0

Alois P. Heinz, Jul 08 2012

			A(3,0) = 2: [3], [1,1,1].
A(4,1) = 6: [4], [2,2], [2,1,1], [1,2,1], [1,1,2], [1,1,1,1].
A(5,2) = 14: [5], [3,2], [3,1,1], [2,3], [2,2,1], [2,1,2], [2,1,1,1], [1,3,1], [1,2,2], [1,2,1,1], [1,1,3], [1,1,2,1], [1,1,1,2], [1,1,1,1,1].
Square array A(n,k) begins:
  1,  1,  1,  1,  1,  1,  1,  1, ...
  1,  1,  1,  1,  1,  1,  1,  1, ...
  2,  2,  2,  2,  2,  2,  2,  2, ...
  2,  4,  4,  4,  4,  4,  4,  4, ...
  3,  6,  8,  8,  8,  8,  8,  8, ...
  2, 11, 14, 16, 16, 16, 16, 16, ...
  4, 17, 27, 30, 32, 32, 32, 32, ...
  2, 29, 49, 59, 62, 64, 64, 64, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-2 give: A000005, A034297, A214255.
Main diagonal gives: A011782.
Cf. A214246, A214247, A214249, A214257, A214258, A214268, A214269.

b:= proc(n, i, k) option remember;
      `if`(n<1 or i<1, 0, `if`(n=i, 1, add(b(n-i, i+j, k), j={$-k..k})))
    end:
A:= (n, k)-> `if`(n=0, 1, add(b(n, j, k), j=1..n)):
seq(seq(A(n, d-n), n=0..d), d=0..15);

b[n_, i_, k_] := b[n, i, k] = If[n < 1 || i < 1, 0, If[n == i, 1, Sum[b[n - i, i + j, k], {j, -k, k}]]]; A[n_, k_] := If[n == 0, 1, Sum[b[n, j, k], {j, 1, n}]]; Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 15}] // Flatten (* Jean-François Alcover, Dec 27 2013, translated from Maple *)

A214246 Number A(n,k) of compositions of n where differences between neighboring parts are in {-k,0,k}; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 4, 3, 1, 1, 2, 2, 6, 2, 1, 1, 2, 2, 5, 11, 4, 1, 1, 2, 2, 3, 5, 17, 2, 1, 1, 2, 2, 3, 4, 10, 29, 4, 1, 1, 2, 2, 3, 2, 7, 10, 47, 3, 1, 1, 2, 2, 3, 2, 6, 8, 21, 78, 4, 1, 1, 2, 2, 3, 2, 4, 5, 9, 22, 130, 2

Offset: 0

Alois P. Heinz, Jul 08 2012

			A(3,0) = 2: [3], [1,1,1].
A(4,1) = 6: [4], [2,2], [2,1,1], [1,2,1], [1,1,2], [1,1,1,1].
A(5,2) = 5: [5], [3,1,1], [1,3,1], [1,1,3], [1,1,1,1,1].
A(6,3) = 7: [6], [4,1,1], [3,3], [2,2,2], [1,4,1], [1,1,4], [1,1,1,1,1,1].
Square array A(n,k) begins:
  1,  1,  1,  1,  1,  1,  1,  1, ...
  1,  1,  1,  1,  1,  1,  1,  1, ...
  2,  2,  2,  2,  2,  2,  2,  2, ...
  2,  4,  2,  2,  2,  2,  2,  2, ...
  3,  6,  5,  3,  3,  3,  3,  3, ...
  2, 11,  5,  4,  2,  2,  2,  2, ...
  4, 17, 10,  7,  6,  4,  4,  4, ...
  2, 29, 10,  8,  5,  4,  2,  2, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Column k=0 and main diagonal give: A000005.
Columns k=1, 2 give: A034297, A214253.
Cf. A214247, A214248, A214249, A214257, A214258, A214268, A214269.

b:= proc(n, i, k) option remember;
      `if`(n<1 or i<1, 0, `if`(n=i, 1, add(b(n-i, i+j, k), j={-k, 0, k})))
    end:
A:= (n, k)-> `if`(n=0, 1, add(b(n, j, k), j=1..n)):
seq(seq(A(n, d-n), n=0..d), d=0..15);

b[n_, i_, k_] := b[n, i, k] = If[n < 1 || i < 1, 0, If[n == i, 1, Sum[b[n - i, i + j, k], {j, Union[{-k, 0, k}]}]]]; A[n_, k_] := If[n == 0, 1, Sum[b[n, j, k], {j, 1, n}]]; Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 15}] // Flatten (* Jean-François Alcover, Dec 27 2013, translated from Maple *)

A214257 Number A(n,k) of compositions of n where the difference between largest and smallest parts is <= k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 4, 3, 1, 1, 2, 4, 6, 2, 1, 1, 2, 4, 8, 11, 4, 1, 1, 2, 4, 8, 14, 15, 2, 1, 1, 2, 4, 8, 16, 27, 27, 4, 1, 1, 2, 4, 8, 16, 30, 47, 39, 3, 1, 1, 2, 4, 8, 16, 32, 59, 88, 63, 4, 1, 1, 2, 4, 8, 16, 32, 62, 111, 158, 100, 2

Offset: 0

Alois P. Heinz, Jul 08 2012

			A(3,0) =  2: [3], [1,1,1].
A(4,1) =  6: [4], [2,2], [2,1,1], [1,2,1], [1,1,2], [1,1,1,1].
A(5,1) =  8: [5], [3,2], [2,3], [2,2,1], [2,1,2], [2,1,1,1], [1,2,2], [1,2,1,1], [1,1,2,1], [1,1,1,2], [1,1,1,1,1],
A(5,2) = 14: [5], [3,2], [3,1,1], [2,3], [2,2,1], [2,1,2], [2,1,1,1], [1,3,1], [1,2,2], [1,2,1,1], [1,1,3], [1,1,2,1], [1,1,1,2], [1,1,1,1,1].
Square array A(n,k) begins:
  1,  1,  1,  1,  1,  1,  1,  1, ...
  1,  1,  1,  1,  1,  1,  1,  1, ...
  2,  2,  2,  2,  2,  2,  2,  2, ...
  2,  4,  4,  4,  4,  4,  4,  4, ...
  3,  6,  8,  8,  8,  8,  8,  8, ...
  2, 11, 14, 16, 16, 16, 16, 16, ...
  4, 15, 27, 30, 32, 32, 32, 32, ...
  2, 27, 47, 59, 62, 64, 64, 64, ...

Alois P. Heinz, Antidiagonals n = 0..150, flattened

Columns k=0-1 give: A000005, A072951.
Main diagonal gives: A011782.
Cf. A214246, A214247, A214248, A214249, A214258, A214268, A214269.

b:= proc(n, k, s, t) option remember;
      `if`(n<0, 0, `if`(n=0, 1, add(b(n-j, k,
       min(s,j), max(t,j)), j=max(1, t-k+1)..s+k-1)))
    end:
A:= (n, k)-> `if`(n=0, 1, add(b(n-j, k+1, j, j), j=1..n)):
seq(seq(A(n, d-n), n=0..d), d=0..11);
# second Maple program:
b:= proc(n, s, t) option remember; `if`(n=0, x^(t-s),
      add(b(n-j, min(s, j), max(t, j)), j=1..n))
    end:
T:= (n, k)-> coeff(b(n$2, 0), x, k):
A:= proc(n, k) option remember; `if`(k<0, 0,
      `if`(k>n, A(n$2), A(n, k-1)+T(n, k)))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..11);  # Alois P. Heinz, Jan 05 2019

b[n_, k_, s_, t_] := b[n, k, s, t] = If[n < 0, 0, If[n == 0, 1, Sum [b[n - j, k, Min[s, j], Max[t, j]], {j, Max[1, t - k + 1], s + k - 1}]]]; A[n_, k_] := If[n == 0, 1, Sum[b[n - j, k + 1, j, j], {j, 1, n}]]; Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 11}] // Flatten (* Jean-François Alcover, Dec 27 2013, translated from Maple *)

T(n,k) = Sum_{i=0..k} A214258(n,i).

A214268 Number A(n,k) of compositions of n where the difference between largest and smallest parts is <= k and adjacent parts are unequal; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 3, 4, 4, 1, 1, 1, 1, 3, 4, 5, 3, 1, 1, 1, 1, 3, 4, 7, 11, 5, 1, 1, 1, 1, 3, 4, 7, 12, 12, 3, 1, 1, 1, 1, 3, 4, 7, 14, 20, 16, 5, 1, 1, 1, 1, 3, 4, 7, 14, 21, 28, 30, 5, 1

Offset: 0

Alois P. Heinz, Jul 09 2012

			A(3,0) = 1: [3].
A(4,1) = 2: [4], [1,2,1].
A(5,2) = 5: [5], [3,2], [2,3], [2,1,2], [1,3,1].
A(6,3) = 12: [6], [4,2], [3,2,1], [3,1,2], [2,4], [2,3,1], [2,1,3], [2,1,2,1], [1,4,1], [1,3,2], [1,2,3], [1,2,1,2].
A(7,4) = 21: [7], [5,2], [4,3], [4,2,1], [4,1,2], [3,4], [3,1,3], [3,1,2,1], [2,5], [2,4,1], [2,3,2], [2,1,4], [2,1,3,1], [1,5,1], [1,4,2], [1,3,2,1], [1,3,1,2], [1,2,4], [1,2,3,1], [1,2,1,3], [1,2,1,2,1].
Square array A(n,k) begins:
  1,  1,  1,  1,  1,  1,  1,  1, ...
  1,  1,  1,  1,  1,  1,  1,  1, ...
  1,  1,  1,  1,  1,  1,  1,  1, ...
  1,  3,  3,  3,  3,  3,  3,  3, ...
  1,  2,  4,  4,  4,  4,  4,  4, ...
  1,  4,  5,  7,  7,  7,  7,  7, ...
  1,  3, 11, 12, 14, 14, 14, 14, ...
  1,  5, 12, 20, 21, 23, 23, 23, ...

Alois P. Heinz, Antidiagonals n = 0..140

Columns k=0, 1 give: A000012, 1+A214270(n).
Main diagonal gives: A003242.
Cf. A214246, A214247, A214248, A214249, A214257, A214258, A214269.

b:= proc(n, k, s, t, l) option remember;
      `if`(n<0, 0, `if`(n=0, 1, add(`if`(j=l, 0, b(n-j, k,
       min(s, j), max(t, j), j)), j=max(1, t-k+1)..s+k-1)))
    end:
A:= (n, k)-> `if`(n=0, 1, add(b(n-j, k+1, j, j, j), j=1..n)):
seq(seq(A(n,d-n), n=0..d), d=0..14);

b[n_, k_, s_, t_, l_] := b[n, k, s, t, l] = If[n < 0, 0, If[n == 0, 1, Sum[If[j == l, 0, b[n - j, k, Min[s, j], Max[t, j], j]], {j, Max[1, t - k + 1], s + k - 1}]]]; A[n_, k_] := If[n == 0, 1, Sum[b[n - j, k + 1, j, j, j], {j, 1, n}]]; Table[Table[A [n, d - n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Dec 27 2013, translated from Maple *)

A214256 Number of compositions of n where differences between neighboring parts are in {-2,-1,1,2}.

Original entry on oeis.org

1, 1, 1, 3, 4, 5, 11, 14, 18, 36, 49, 66, 118, 169, 238, 401, 586, 846, 1371, 2042, 2998, 4731, 7114, 10566, 16419, 24809, 37118, 57139, 86558, 130151, 199193, 302109, 455737, 695084, 1054761, 1594484, 2426813, 3683310, 5575665, 8475607, 12864385, 19490762

Offset: 0

Alois P. Heinz, Jul 08 2012

nonn

			a(4) = 4: [4], [3,1], [1,3], [1,2,1].
a(5) = 5: [5], [3,2], [2,3], [2,1,2], [1,3,1].
a(6) = 11: [6], [4,2], [3,2,1], [3,1,2], [2,4], [2,3,1], [2,1,3], [2,1,2,1], [1,3,2], [1,2,3], [1,2,1,2].

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column k=2 of A214249.

b:= proc(n, i) option remember; `if`(n<1 or i<1, 0,
     `if`(n=i, 1, add(b(n-i, i+j), j=[-2, -1, 1, 2])))
    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, {-2, -1, 1, 2}}]]]; a[n_] := If[n == 0, 1, Sum[b[n, j], {j, 1, n}]]; Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Nov 06 2014, after Alois P. Heinz *)

a(n) ~ c * d^n, where d = 1.517398641547133636808941933115508174163721854365273284016518..., c = 0.733002948674062952076493110095119568735789170592785923762247... . - Vaclav Kotesovec, Sep 02 2014

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)

cp's OEIS Frontend

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

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A214258 Number T(n,k) of compositions of n where the difference between largest and smallest parts equals k; triangle T(n,k), n>=1, 0<=k

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A214269 Number T(n,k) of compositions of n where the difference between largest and smallest parts equals k and adjacent parts are unequal; triangle T(n,k), n>=1, 0<=k

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A214247 Number A(n,k) of compositions of n where differences between neighboring parts are in {-k,k}; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A214248 Number A(n,k) of compositions of n where differences between neighboring parts are in {-k,...,k}; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A214246 Number A(n,k) of compositions of n where differences between neighboring parts are in {-k,0,k}; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A214257 Number A(n,k) of compositions of n where the difference between largest and smallest parts is <= k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A214268 Number A(n,k) of compositions of n where the difference between largest and smallest parts is <= k and adjacent parts are unequal; square array A(n,k), n>=0, k>=0, read by antidiagonals.