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.

A123640 a(n) = A065120(n) modulo 2.

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Alford Arnold, Oct 04 2006

Keywords

Comments

Previous name was: Consider the 2^n compositions of n per row and mark only those ending in an odd part.

Examples

			A065120 begins 0, 1, 2, 1, 3, 2, 1, 1, 4, 3, 2, 2, 1, 1, 1, 1, ...
Therefore this sequence begins 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, ...

Crossrefs

Cf. A001045, A065120, A123638, A123639, A123641.

Programs

  • PARI
    lista(nn) = {my(v = vector(nn)); v[1] = 1; for (i=2, nn, v[i] = mg(i-1)*v[(i+1)\2];); for (i=1, nn, print1(valuation(v[i], 2) % 2, ", "););} \\ Michel Marcus, Feb 09 2014

Extensions

More terms from Nathaniel Johnston, Apr 30 2011
New name using given formula, Joerg Arndt, Jan 24 2024

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

A124220 Transpose table A123685 then subtract, term by term, from table A047969.

Original entry on oeis.org

0, 1, 0, 0, 2, 0, 1, 1, 4, 0, 0, 5, 5, 8, 0, 1, 2, 23, 19, 16, 0, 0, 8, 18, 101, 65, 32, 0, 1, 3, 58, 110, 431, 211, 64, 0
Offset: 1

Views

Author

Alford Arnold, Oct 29 2006

Keywords

Examples

			Row six of A047969 is 1 9 37 65 31 1
Row six of A123685^T is 0 7 14 46 15 1
so row six of a(n) is 1 2 23 19 16 0

Crossrefs

The sequence of row sums is 0 1 2 6 18 61 224 ... = A123639. The sequence of row sums for table A123685 is 1 1 3 8 25 83 299 ... = A123638. The sequence of row sums for table A047969 is 1 2 5 14 43 144 523 ... = A047970
Cf. A047970 A123638 A123639.

Formula

A124220(n) = A047969(n) - Trn(A123685).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.