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.

A002703 Sets with a congruence property.

Original entry on oeis.org

0, 0, 0, 2, 6, 14, 24, 46, 88, 162, 300, 562, 1056, 1982, 3742, 7082, 13438, 25574, 48768, 93198, 178480, 342392, 657918, 1266202, 2440318, 4709374, 9099504, 17602322, 34087010, 66076414, 128207976, 248983550, 483939976, 941362694, 1832519262, 3569842946, 6958934352
Offset: 3

Views

Author

N. J. A. Sloane

Keywords

Comments

a(n) is the sequence k(n) in Table 3 of the first 1965 paper. - N. J. A. Sloane, Oct 20 2015
See English summary at the end of the first 1965 paper, which is repeated in the Zentralblatt review. - Jonathan Sondow, Nov 02 2013

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A002704, A002705.
See A262567, A262568, A262569 for other versions.
Tables 1 and 2 of the first Rosa-Znám 1965 paper are A053632 and A178666 respectively.

Programs

  • Maple
    A002703 := proc(n)
    A262568(n)-2 ;
end proc: # R. J. Mathar, Oct 21 2015
  • Mathematica
    A178666[r_, s_] := SeriesCoefficient[Product[ (1 + x^(2i+1)), {i, 0, Floor[(s-1)/2]}], {x, 0, r}];
kstart[n_, m_] := Ceiling[Binomial[n+1, 2]/m];
kend[n_, m_] := Floor[Binomial[3n+1, 2]/3/m];
A262568[n_] := Module[{s = 2n-1, m = 2n+1, Q=0, vi, k}, For[k = kstart[n, m], k <= kend[n, m], k++, vi = m k - Binomial[n+1, 2]; Q += A178666[vi, s] ]; Q];
a[n_] := A262568[n] - 2;
a /@ Range[3, 39] (* Jean-François Alcover, Mar 24 2020, after R. J. Mathar in A262568 *)

Extensions

More terms from R. J. Mathar, Oct 21 2015

A262568 a(n) = A002703(n) + 2.

Original entry on oeis.org

2, 2, 2, 4, 8, 16, 26, 48, 90, 164, 302, 564, 1058, 1984, 3744, 7084, 13440, 25576, 48770, 93200, 178482, 342394, 657920, 1266204, 2440320, 4709376, 9099506, 17602324, 34087012, 66076416, 128207978, 248983552, 483939978, 941362696, 1832519264, 3569842948
Offset: 3

Views

Author

N. J. A. Sloane, Oct 20 2015

Keywords

Links

Crossrefs

Cf. A002703, A262567, A262569.
Tables 1 and 2 of the first Rosa-Znám 1965 paper are A053632 and A178666 respectively.

Programs

  • Maple
    A178666 := proc(r,s)
    product( (1+x^(2*i+1)),i=0..floor((s-1)/2)) ;
    expand(%) ;
    coeftayl(%,x=0,r) ;
end proc:
kstart := proc(n,m)
    ceil(binomial(n+1,2)/m) ;
end proc:
kend := proc(n,m)
    floor(binomial(3*n+1,2)/3/m) ;
end proc:
A262568 := proc(n)
    local s,m,Q ,vi,k;
    s := 2*n-1 ;
    m := 2*n+1 ;
    Q := 0 ;
    for k from kstart(n,m) to kend(n,m) do
        vi := m*k-binomial(n+1,2) ;
        Q := Q+A178666(vi,s) ;
    end do:
    Q ;
end proc: # R. J. Mathar, Oct 21 2015
  • Mathematica
    A178666[r_, s_] := SeriesCoefficient[Product[(1 + x^(2i+1)), {i, 0, Floor[ (s - 1)/2]}], {x, 0, r}];
kstart [n_, m_] := Ceiling[Binomial[n+1, 2]/m];
kend[n_, m_] := Floor[Binomial[3n+1, 2]/3/m];
a[n_] := Module[{s = 2n-1, m = 2n+1, Q=0, vi, k}, For[k = kstart[n, m], k <= kend[n, m], k++, vi = m k - Binomial[n+1, 2]; Q += A178666[vi, s]]; Q];
a /@ Range[3, 38] (* Jean-François Alcover, Mar 24 2020, after R. J. Mathar *)

Formula

See Maple code! - N. J. A. Sloane, Oct 21 2015

Extensions

More terms from R. J. Mathar, Oct 21 2015
Missing a(16) inserted by Sean A. Irvine, Oct 23 2015

A262567 a(n) = A002703(n)/2.

Original entry on oeis.org

0, 0, 0, 1, 3, 7, 12, 23, 44, 81, 150, 281, 528, 991, 1871, 3541, 6719, 12787, 24384, 46599, 89240, 171196, 328959, 633101, 1220159, 2354687, 4549752, 8801161, 17043505, 33038207, 64103988, 124491775, 241969988, 470681347, 916259631, 1784921473, 3479467176, 6787108712, 13247128044, 25870861823
Offset: 3

Views

Author

N. J. A. Sloane, Oct 20 2015

Keywords

Comments

A002703 is somewhat mysterious. Having four versions (A002703, this sequence, A262568, A262569) instead of one increases the chance that one of them will be found in a different context.

Links

Crossrefs

Cf. A002703, A262568, A262569.
Tables 1 and 2 of the first Rosa-Znám 1965 paper are A053632 and A178666 respectively.

Programs

Extensions

More terms from R. J. Mathar, Oct 21 2015
Showing 1-3 of 3 results.

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

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