cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Views

Author

Alois P. Heinz, Jul 09 2012

Keywords

Examples

			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;

Links

Crossrefs

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.

Programs

  • Maple
    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);
  • Mathematica
    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 *)

Formula

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

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

Views

Author

Alois P. Heinz, Jul 09 2012

Keywords

Examples

			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, ...

Links

Crossrefs

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

Programs

  • Maple
    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);
  • Mathematica
    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 *)

A368557 Number of compositions of n such that the set of absolute differences is a subset of the set of parts.

Original entry on oeis.org

1, 1, 1, 3, 2, 2, 11, 10, 13, 27, 58, 87, 157, 253, 438, 850, 1462, 2474, 4472, 7716, 13544, 24115, 42360, 74013, 131038, 229009, 401946, 707293, 1242059, 2177682, 3828831, 6716062, 11777179, 20678592, 36267148, 63586772, 111556751, 195610763, 342949281
Offset: 0

Views

Author

John Tyler Rascoe, Dec 29 2023

Keywords

Examples

			For n=12, composition [2,1,2,4,3] of 12 has the set of absolute differences {1,2}, which is a subset of the set of parts {1,2,3,4}, so it counts towards a(12) = 157.
a(3) = 3 compositions: [3], [2,1], [1,2].
a(6) = 11 compositions: [6], [4,2], [2,4], [3,2,1], [3,1,2], [2,3,1], [2,1,3], [1,3,2], [1,2,3], [2,1,2,1], [1,2,1,2].

Links

Crossrefs

Cf. A003242, A032020, A173258, A214248, A214270.

Programs

  • Mathematica
    g[0] = {{}}; g[n_Integer] := g[n] = Flatten[ParallelTable[Append[#, i] & /@ g[n - i], {i, 1, n}], 1];
isC[p_List] := Module[{d}, d = Abs[Differences[p]]; Union[d] === Union[Select[d, MemberQ[p, #] &]]];
a[n_Integer] := a[n] = Count[g[n], p_ /; isC[p]];
Monitor[Table[a[n], {n, 0, 19}], {n, Table[a[m], {m, 0, n - 1}]}] (* Robert P. P. McKone, Jan 02 2024 *)

Extensions

a(24)-a(38) from Alois P. Heinz, Dec 30 2023
Showing 1-3 of 3 results.

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