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 1-10 of 13 results. Next

A331844 Number of compositions (ordered partitions) of n into distinct squares.

Original entry on oeis.org

1, 1, 0, 0, 1, 2, 0, 0, 0, 1, 2, 0, 0, 2, 6, 0, 1, 2, 0, 0, 2, 6, 0, 0, 0, 3, 8, 0, 0, 8, 30, 0, 0, 0, 2, 6, 1, 2, 6, 24, 2, 8, 6, 0, 0, 8, 30, 0, 0, 7, 32, 24, 2, 8, 30, 120, 6, 24, 2, 6, 0, 8, 36, 24, 1, 34, 150, 0, 2, 12, 30, 24, 0, 2, 38, 150, 0, 12, 78, 144, 2
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 29 2020

Keywords

Examples

			a(14) = 6 because we have [9,4,1], [9,1,4], [4,9,1], [4,1,9], [1,9,4] and [1,4,9].

Links

Crossrefs

Cf. A000290, A006456, A032020, A032021, A032022, A033461, A218396, A219107, A331843, A331845, A331846, A331847.

Programs

  • Maple
    b:= proc(n, i, p) option remember;
      `if`(i*(i+1)*(2*i+1)/6n, 0, b(n-i^2, i-1, p+1))+b(n, i-1, p)))
    end:
a:= n-> b(n, isqrt(n), 0):
seq(a(n), n=0..82);  # Alois P. Heinz, Jan 30 2020
  • Mathematica
    b[n_, i_, p_] := b[n, i, p] = If[i(i+1)(2i+1)/6 < n, 0, If[n == 0, p!, If[i^2 > n, 0, b[n - i^2, i - 1, p + 1]] + b[n, i - 1, p]]];
a[n_] := b[n, Sqrt[n] // Floor, 0];
a /@ Range[0, 82] (* Jean-François Alcover, Oct 29 2020, after Alois P. Heinz *)

A331845 Number of compositions (ordered partitions) of n into distinct cubes.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 2, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 2, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 6, 0, 0, 0, 0, 0, 0, 6, 24
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 29 2020

Keywords

Examples

			a(36) = 6 because we have [27,8,1], [27,1,8], [8,27,1], [8,1,27], [1,27,8] and [1,8,27].

Links

Crossrefs

Cf. A000578, A023358, A032020, A032021, A032022, A218396, A219107, A279329, A331843, A331844, A331846, A331847.

Programs

  • Maple
    b:= proc(n, i, p) option remember;
      `if`((i*(i+1)/2)^2n, 0, b(n-i^3, i-1, p+1))+b(n, i-1, p)))
    end:
a:= n-> b(n, iroot(n, 3), 0):
seq(a(n), n=0..100);  # Alois P. Heinz, Jan 30 2020
  • Mathematica
    b[n_, i_, p_] := b[n, i, p] = If[(i(i+1)/2)^2 < n, 0, If[n == 0, p!, If[i^3 > n, 0, b[n-i^3, i-1, p+1]] + b[n, i-1, p]]];
a[n_] := b[n, Floor[n^(1/3)], 0];
a /@ Range[0, 100] (* Jean-François Alcover, Oct 31 2020, after Alois P. Heinz *)

A331846 Number of compositions (ordered partitions) of n into distinct squarefree parts.

Original entry on oeis.org

1, 1, 1, 3, 2, 3, 9, 5, 12, 16, 21, 41, 42, 49, 59, 79, 130, 231, 230, 295, 226, 495, 609, 699, 1472, 1042, 1377, 2308, 2982, 3425, 3879, 4877, 7156, 7189, 13531, 14797, 13570, 19551, 27667, 30327, 36382, 47519, 60783, 70561, 78330, 136988, 121659, 174851
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 29 2020

Keywords

Examples

			a(7) = 5 because we have [7], [6, 1], [5, 2], [2, 5] and [1, 6].

Links

Crossrefs

Cf. A005117, A032020, A032021, A032022, A087188, A218396, A219107, A280194, A331843, A331844, A331845, A331847.

A331847 Number of compositions (ordered partitions) of n into distinct prime powers (1 excluded).

Original entry on oeis.org

1, 0, 1, 1, 1, 3, 2, 5, 3, 11, 10, 13, 18, 19, 52, 30, 61, 77, 114, 109, 146, 260, 318, 341, 356, 631, 666, 927, 848, 1849, 1978, 2305, 2213, 3560, 4302, 4748, 5588, 6779, 13952, 9044, 15534, 16897, 25084, 20731, 29524, 34882, 49360, 50765, 55112, 106903, 83652, 128552, 106638
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 29 2020

Keywords

Examples

			a(10) = 10 because we have [8, 2], [7, 3], [5, 3, 2], [5, 2, 3], [3, 7], [3, 5, 2], [3, 2, 5], [2, 8], [2, 5, 3] and [2, 3, 5].

Links

Crossrefs

Cf. A032020, A032021, A032022, A035942, A054685, A218396, A219107, A246655, A280195, A331843, A331844, A331845, A331846.

A331919 Number of compositions (ordered partitions) of n into distinct tetrahedral numbers.

Original entry on oeis.org

1, 1, 0, 0, 1, 2, 0, 0, 0, 0, 1, 2, 0, 0, 2, 6, 0, 0, 0, 0, 1, 2, 0, 0, 2, 6, 0, 0, 0, 0, 2, 6, 0, 0, 6, 25, 2, 0, 0, 2, 6, 0, 0, 0, 0, 2, 6, 0, 0, 6, 24, 0, 0, 0, 0, 2, 7, 2, 0, 6, 26, 6, 0, 0, 0, 6, 26, 6, 0, 24, 126, 24, 0, 0, 0, 0, 2, 6, 0, 0, 6, 24, 0, 0, 1, 2, 6, 24, 2, 6, 24
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 01 2020

Keywords

Examples

			a(15) = 6 because we have [10, 4, 1], [10, 1, 4], [4, 10, 1], [4, 1, 10], [1, 10, 4] and [1, 4, 10].

Links

Crossrefs

Cf. A000292, A023361, A068980, A279278, A282582, A298857, A331843, A331984.

Programs

  • Maple
    N:= 200: # for a(0)..a(N)
G:= mul(1+t*x^(i*(i+1)*(i+2)/6), i=1..floor((6*N)^(1/3))):
F:= proc(n) local R, k, v;
  R:= coeff(G, x, n);
  add(k!*coeff(R, t, k), k=1..degree(R, t))
end proc:
F(0):= 1:
map(F, [$0..N]); # Robert Israel, Feb 03 2020
  • Mathematica
    M = 100;
G = Product[1 + t x^(i(i+1)(i+2)/6), {i, 1, Floor[(6M)^(1/3)]}];
F[n_] := Module[{R, k, v}, R = Coefficient[G, x, n]; Sum[k! Coefficient[R, t, k], {k, 1, Exponent[R, t]}]];
F[0] = 1;
F /@ Range[0, M] (* Jean-François Alcover, Jun 20 2020, after Robert Israel *)

A332007 Number of compositions (ordered partitions) of n into distinct pentagonal numbers.

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 2, 6, 0, 0, 0, 1, 2, 0, 0, 0, 2, 6, 0, 0, 0, 0, 0, 2, 7, 2, 0, 0, 6, 26, 6, 0, 0, 0, 0, 0, 2, 6, 0, 0, 1, 8, 24, 0, 0, 2, 8, 6, 0, 0, 0, 6, 26, 6, 0, 0, 0, 6, 30, 25, 2, 0, 2, 30, 122, 6, 0, 6, 24
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 04 2020

Keywords

Examples

			a(18) = 6 because we have [12, 5, 1], [12, 1, 5], [5, 12, 1], [5, 1, 12], [1, 12, 5] and [1, 5, 12].

Links

Crossrefs

Cf. A000326, A181324, A218379, A218380, A331843, A331844.

A331900 Number of compositions (ordered partitions) of the n-th triangular number into distinct triangular numbers.

Original entry on oeis.org

1, 1, 1, 1, 7, 1, 3, 13, 3, 55, 201, 159, 865, 1803, 7093, 43431, 14253, 22903, 130851, 120763, 1099693, 4527293, 4976767, 7516897, 14349685, 72866239, 81946383, 167841291, 897853735, 455799253, 946267825, 5054280915, 3941268001, 17066300985, 49111862599
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 31 2020

Keywords

Examples

			a(6) = 3 because we have [21], [15, 6] and [6, 15].

Links

Crossrefs

Cf. A000217, A072964, A126683, A196010, A224677, A288126, A298858, A307614, A331843, A331884.

Programs

  • Maple
    b:= proc(n, i, p) option remember; (t->
      `if`(t*(i+2)/3n, 0, b(n-t, i-1, p+1)))))((i*(i+1)/2))
    end:
a:= n-> b(n*(n+1)/2, n, 0):
seq(a(n), n=0..37);  # Alois P. Heinz, Jan 31 2020
  • Mathematica
    b[n_, i_, p_] := b[n, i, p] = With[{t = i(i+1)/2}, If[t(i+2)/3 < n, 0, If[n == 0, p!, b[n, i-1, p] + If[t > n, 0, b[n-t, i-1, p+1]]]]];
a[n_] := b[n(n+1)/2, n, 0];
a /@ Range[0, 37] (* Jean-François Alcover, Nov 17 2020, after Alois P. Heinz *)

Formula

a(n) = A331843(A000217(n)).

Extensions

More terms from Alois P. Heinz, Jan 31 2020

A332014 Number of compositions (ordered partitions) of n into distinct hexagonal numbers.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 2, 6, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 2, 6, 0, 0, 0, 0, 0, 0, 0, 2, 6, 1, 2, 0, 0, 6, 24, 2, 6, 0, 0, 0, 0, 0, 0, 0, 2, 6, 0, 0, 0, 0, 7, 26, 0, 0, 0, 0, 2, 8, 6, 0, 0, 0, 0, 6, 24
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 04 2020

Keywords

Examples

			a(22) = 6 because we have [15, 6, 1], [15, 1, 6], [6, 15, 1], [6, 1, 15], [1, 15, 6] and [1, 6, 15].

Links

Crossrefs

Cf. A000384, A278949, A279279, A322798, A331843, A331844, A332007, A332015, A332016.

A332015 Number of compositions (ordered partitions) of n into distinct heptagonal numbers.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 2, 6, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 2, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 6, 0, 1, 2, 0, 0, 6, 24, 0, 2, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 6, 0, 0, 0, 0, 0, 6, 25, 2
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 04 2020

Keywords

Examples

			a(26) = 6 because we have [18, 7, 1], [18, 1, 7], [7, 18, 1], [7, 1, 18], [1, 18, 7] and [1, 7, 18].

Links

Crossrefs

Cf. A000566, A279012, A279280, A322799, A331843, A331844, A332007, A332014, A332016.

A332016 Number of compositions (ordered partitions) of n into distinct octagonal numbers.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 2, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 2, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 6, 0, 0, 1, 2, 0, 0, 6, 24, 0, 0, 2, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 6
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 04 2020

Keywords

Examples

			a(30) = 6 because we have [21, 8, 1], [21, 1, 8], [8, 21, 1], [8, 1, 21], [1, 21, 8] and [1, 8, 21].

Links

Crossrefs

Cf. A000567, A279041, A279281, A322800, A331843, A331844, A332007, A332014, A332015.
Showing 1-10 of 13 results. Next

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