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 1-5 of 5 results.

A123641 Triangular array related to sequence A123640 with row sum A001045.

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 0, 1, 0, 4, 1, 0, 2, 0, 8, 0, 1, 0, 4, 0, 16, 1, 0, 2, 0, 8, 0, 32, 0, 1, 0, 4, 0, 16, 0, 64, 1, 0, 2, 0, 8, 0, 32, 0, 128, 0, 1, 0, 4, 0, 16, 0, 64, 0, 256, 1, 0, 2, 0, 8, 0, 32, 0, 128, 0, 512, 0, 1, 0, 4, 0, 16, 0, 64, 0, 256, 0, 1024, 1
Offset: 1

Views

Author

Alford Arnold, Oct 04 2006

Keywords

Examples

			For n>0 A123640 begins
1
0 1
1 0 1 1
0 1 0 0 1 1 1 1
suppressing multiple zeros and counting contiguous ones yields
1
0 1
1 0 2
0 1 0 4
1 0 2 0 8
0 1 0 4 0 16
...

Crossrefs

Cf. A000079, A001045, A059841, A123640.

Programs

  • PARI
    a001045(n)=(2^n-(-1)^n)/3;
lista(nn) = {last = v[2]; nb = 1; s = 0; j = 1; for (i=3, nn, if ((v[i] == last) && (s+v[i] < a001045(j)), nb++; s += v[i];, if (last, w = nb, w = 0); print1(w, ", "); nb = 1; last = v[i]; if (s + v[i] >= a001045(j), s = 0; j++;, s += v[i];);););} \\ Michel Marcus, Feb 09 2014

Extensions

More terms from Michel Marcus, Feb 09 2014

A123639 Consider the 2^n compositions of n and count only those ending in an even part.

Original entry on oeis.org

0, 1, 2, 6, 18, 61, 224, 890, 3784, 17113, 81950, 414230, 2204110, 12314109, 72049548, 440379770, 2805266692, 18584809833, 127812870474, 910990458022, 6719535098378, 51223251471453, 403044829472760, 3269538955148698, 27314067026782976, 234749040898160153
Offset: 1

Views

Author

Alford Arnold, Oct 04 2006

Keywords

Comments

Compositions ending in an even part yield sequence 0 1 2 6 18 ... (this sequence). and A123638(n)+a(n) = A047970(n). Ending parity of compositions can be detected using mod(A065120,2)

Examples

			4
31 32 33
211 221 222
1111
Consider the above multisets- permute and note the parity of the ending part of each of the 14 compositions.
4
31 13 32 23 33
211 121 112 221 212 122 222
1111
4 is even
31 13 23 and 33 are odd
32 is even
etc
there are 1+1+4+0 even compositions therefore a(4)=6.

Links

Crossrefs

Cf. A001045, A047970, A065120, A123638, A123640, A123641.

Programs

  • Maple
    g:= proc(b,t,l,m) option remember; if t=0 then b*(1-l) else add (g(b, t-1, irem(k, 2), m), k=1..m-1) +g(1, t-1, irem(m, 2), m) fi end: a:= n-> add (g(0, k, 0, n+1-k), k=1..n): seq (a(n), n=1..30); # Alois P. Heinz, Nov 06 2009
  • Mathematica
    g[b_, t_, l_, m_] := g[b, t, l, m] = If[ t == 0 , b*(1-l), Sum[g[b, t-1, Mod[k, 2], m], {k, 1, m-1}] + g[1, t-1, Mod[m, 2], m]]; a[n_] := Sum[g[0, k, 0, n+1-k], {k, 1, n}]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Nov 04 2013, translated from Alois P. Heinz's Maple program *)

Extensions

More terms from Alois P. Heinz, Nov 06 2009

A123638 Consider the 2^n compositions of n and count only those ending in an odd part with row sum A001045.

Original entry on oeis.org

1, 1, 3, 8, 25, 83, 299, 1158, 4813, 21373, 100955, 504916, 2662761, 14754311, 85643459, 519493938, 3285790317, 21628225041, 147887079907, 1048634836288, 7698589399833, 58432476430139, 457901993065915, 3700291495531166
Offset: 1

Views

Author

Alford Arnold, Oct 04 2006

Keywords

Comments

Compositions ending in an even part yield sequence 0 1 2 6 18 ... A123639. and a(n)+A123639(n) = A047970(n). Ending parity of compositions can be detected using mod(A065120,2)

Examples

			4
31 32 33
211 221 222
1111
Consider the above multisets: permute and note the parity of the ending part of each of the 14 compositions.
4
31 13 32 23 33
211 121 112 221 212 122 222
1111
4 is even
31 13 23 and 33 are odd
32 is even
etc
there are 0 + 4 + 3 + 1 = 8 odd compositions therefore a(4)=8.

Crossrefs

Cf. A001045 A047970 A065120 A123639 A123640 A123641.

Programs

  • Maple
    g:= proc(b,t,l,m) option remember; if t=0 then b*l else add (g(b, t-1, irem(k, 2), m), k=1..m-1) +g(1, t-1, irem(m, 2), m) fi end: a:= n-> add (g(0, k, 0, n+1-k), k=1..n): seq (a(n), n=1..30);
  • Mathematica
    g[b_, t_, l_, m_] := g[b, t, l, m] = If[t == 0 , b*l , Sum[g[b, t-1, Mod[k, 2], m], {k, 1, m-1}] + g[1, t-1, Mod[m, 2], m]]; a[n_] := Sum[g[0, k, 0, n+1-k], {k, 1, n}]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Nov 04 2013, translated from Alois P. Heinz's Maple program *)

Extensions

Offset corrected, Maple program and more terms added by Alois P. Heinz, Nov 06 2009

A123682 First in an infinite series of triangular arrays which, when taken together, sum to 1,1,3,5,11,21,43,85,... cf. A001045.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 0, 2, 2, 1, 1, 1, 3, 3, 1, 0, 2, 2, 4, 4, 1, 1, 1, 3, 3, 5, 5, 1, 0, 2, 2, 4, 4, 6, 6, 1, 1, 1, 3, 3, 5, 5, 7, 7, 1
Offset: 1

Views

Author

Alford Arnold, Oct 06 2006

Keywords

Comments

The starting row for each triangle is determined by the applicable source integer partition; thus for partitions 1,22,33,222,44,332,2222,333,... the starting rows are 1,4,6,6,8,8,8,9,...

Examples

			Summing the row sums of the triangular arrays we have
1 1
1 1
3 3
5 5 0
11 9 2
21 13 7 1 0
43 19 18 4 2
85 25 36 12 10 2
...

Crossrefs

Cf. A059841 A001045 A123640.

A123683 Second in an infinite series of triangular arrays which, when taken together, sum to 1,1,3,5,11,21,43,85,..., cf. A001045.

Original entry on oeis.org

0, 1, 1, 0, 4, 3, 1, 2, 9, 6, 0, 4, 6, 16, 10, 1, 2, 9, 12, 25, 15, 0, 4, 6, 16, 20, 36, 21, 1, 2, 9, 12, 25, 30, 49, 28
Offset: 1

Views

Author

Alford Arnold, Oct 06 2006, Oct 19 2006

Keywords

Comments

The starting row for each triangle is determined by the applicable source integer partition; thus for partitions 1,22,33,222,44,332,2222,333,... the starting rows are 1,4,6,6,8,8,8,9,...

Examples

			Summing the row sums of the triangular arrays we have
1 1
1 1
3 3
5 5 0
11 9 2
21 13 7 1 0
43 19 18 4 2
85 25 36 12 10 2
...

Crossrefs

Cf. A000041 A059841 A000217 A000290 A001045 A123640 A123682.
Cf. A000035 A002378.
Showing 1-5 of 5 results.

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