A214246 -id:A214246 - cp's OEIS frontend

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

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.

A034297 Number of ordered positive integer solutions (m_1, m_2, ..., m_k) (for some k) to Sum_{i=1..k} m_i=n with |m_i-m_{i-1}| <= 1 for i = 2 ... k.

Original entry on oeis.org

1, 1, 2, 4, 6, 11, 17, 29, 47, 78, 130, 215, 357, 595, 990, 1651, 2748, 4584, 7643, 12744, 21256, 35451, 59133, 98636, 164531, 274463, 457837, 763746, 1274060, 2125356, 3545491, 5914545, 9866602, 16459421, 27457549, 45804648, 76411272, 127469285, 212644336

Offset: 0

Erich Friedman

nonn

Compositions of n where successive parts differ by at most 1, see example. [Joerg Arndt, Dec 10 2012]

			From _Joerg Arndt_, Dec 10 2012: (Start)
The a(6) = 17 such compositions of 6 are
[ #]     composition
[ 1]    [ 1 1 1 1 1 1 ]
[ 2]    [ 1 1 1 1 2 ]
[ 3]    [ 1 1 1 2 1 ]
[ 4]    [ 1 1 2 1 1 ]
[ 5]    [ 1 1 2 2 ]
[ 6]    [ 1 2 1 1 1 ]
[ 7]    [ 1 2 1 2 ]
[ 8]    [ 1 2 2 1 ]
[ 9]    [ 1 2 3 ]
[10]    [ 2 1 1 1 1 ]
[11]    [ 2 1 1 2 ]
[12]    [ 2 1 2 1 ]
[13]    [ 2 2 1 1 ]
[14]    [ 2 2 2 ]
[15]    [ 3 2 1 ]
[16]    [ 3 3 ]
[17]    [ 6 ]
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..4500
Jia Huang, Compositions with restricted parts, arXiv:1812.11010 [math.CO], 2018.

Cf. A003116, A034296.
Column k=1 of A214246, A214248.
Row sums of A309939.

b:= proc(n, i) option remember;
      `if`(n=i, 1, `if`(n<0 or i<1, 0, add(b(n-i, i+j), j=-1..1)))
    end:
a:= n-> add(b(n, k), k=0..n):
seq(a(n), n=0..50);  # Alois P. Heinz, Jul 06 2012

b[n_, i_] := b[n, i] = If[n == i, 1, If[n<0 || i<1, 0, Sum[b[n-i, i+j], {j, -1, 1}] ]]; a[n_] := Sum[b[n, k], {k, 1, n}]; Array[a, 50] (* Jean-François Alcover, Mar 13 2015, after Alois P. Heinz *)

N=70;  nil=-1;
T = matrix(N, N, i, j, nil);
doIt(last, left) = my(c); c = T[last, left]; if (c == nil, c = 0; for (i = max(1, last - 1), last + 1, c += b(i, left - i)); T[last, left] = c); c;
b(last, left) = if (left == 0, return(1)); if (left < 0, return(0)); doIt(last, left);
a(n) = sum (i = 1, n, b(i, n - i));
vector(N,n,a(n))  \\ David Wasserman, Feb 02 2006

from sympy.core.cache import cacheit
@cacheit
def b(n, i): return 1 if n==i else 0 if n<0 or i<1 else sum(b(n - i, i + j) for j in range(-1, 2))
def a(n): return sum(b(n, k) for k in range(n + 1))
print([a(n) for n in range(51)]) # Indranil Ghosh, Aug 14 2017, after Maple code

a(n) ~ c * d^n, where d = 1.668202067018461116361070469945501401879811945303435230637248..., c = 0.762436680050402638439806786781869262562176911054246754543346... . - Vaclav Kotesovec, Sep 02 2014

More terms from David Wasserman, Feb 02 2006

a(0)=1 prepended by Alois P. Heinz, Aug 14 2017

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 *)

A214249 Number A(n,k) of compositions of n where differences between neighboring parts are in {-k,...,k} \ {0}; 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, 5, 1, 1, 1, 1, 3, 4, 7, 11, 5, 1, 1, 1, 1, 3, 4, 7, 12, 14, 7, 1, 1, 1, 1, 3, 4, 7, 14, 20, 18, 10, 1, 1, 1, 1, 3, 4, 7, 14, 21, 30, 36, 9, 1, 1, 1, 1, 3, 4, 7, 14, 23, 36, 50, 49, 14, 1

Offset: 0

Alois P. Heinz, Jul 08 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].
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,  5, 11, 12, 14, 14, 14, 14, ...
  1,  5, 14, 20, 21, 23, 23, 23, ...

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

Columns k=0-2 give: A000012, A173258, A214256.
Main diagonal gives: A003242.
Cf. A214246, A214247, A214248, 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} minus{0})))
    end:
A:= (n, k)-> `if`(n=0, 1, add(b(n, j, min(n, 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, Range[-k, -1] ~Join~ Range[k]}]]]; A[n_, k_] := If[n == 0, 1, Sum[b[n, j, Min[n, k]], {j, 1, n}]]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 15}] // Flatten (* Jean-François Alcover, Jan 15 2014, 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 *)

A214253 Number of compositions of n where differences between neighboring parts are in {-2,0,2}.

Original entry on oeis.org

1, 1, 2, 2, 5, 5, 10, 10, 21, 22, 42, 47, 87, 103, 179, 224, 380, 491, 802, 1074, 1721, 2354, 3696, 5157, 7995, 11305, 17328, 24778, 37680, 54320, 82071, 119076, 179061, 261046, 391087, 572275, 854975, 1254578, 1870298, 2750361, 4093539, 6029538, 8962963

Offset: 0

Alois P. Heinz, Jul 08 2012

nonn

			a(5) = 5: [5], [3,1,1], [1,3,1], [1,1,3], [1,1,1,1,1].
a(6) = 10: [6], [4,2], [3,3], [3,1,1,1], [2,4], [2,2,2], [1,3,1,1], [1,1,3,1], [1,1,1,3], [1,1,1,1,1,1].

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

Column k=2 of A214246.

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, 0, 2])))
    end:
a:= n-> `if`(n=0, 1, add(b(n, j), j=1..n)):
seq(a(n), n=0..60);

b[n_, i_] := b[n, i] = If[n < 1 || i < 1, 0, If[n == i, 1, Sum[b[n-i, i+j], {j, {-2, 0, 2}}]]]; a[n_] := If[n == 0, 1, Sum[b[n, j], {j, 1, n}]]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Nov 06 2014, after Alois P. Heinz *)

a(n) ~ c * d^n, where d = 1.480632733359847628849916564959539381483927975663120268887..., c = 0.6193575859000249187293498067457554927448225891538342... . - Vaclav Kotesovec, Sep 02 2014

cp's OEIS Frontend

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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A034297 Number of ordered positive integer solutions (m_1, m_2, ..., m_k) (for some k) to Sum_{i=1..k} m_i=n with |m_i-m_{i-1}| <= 1 for i = 2 ... 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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A214249 Number A(n,k) of compositions of n where differences between neighboring parts are in {-k,...,k} \ {0}; 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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