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.

A262585 a(0)=0; thereafter a(n) = A002705(n) - 2.

Original entry on oeis.org

0, 2, 38, 466, 5826, 76258, 1032442, 14316582, 202116106, 2893451650, 41886157562, 611902123282, 9007199254738, 133439988963010, 1987795697598010, 29752813022112178, 447193795726343002, 6746237832670921766, 102105221251235572186
Offset: 0

Views

Author

N. J. A. Sloane, Oct 20 2015

Keywords

Links

Crossrefs

Cf. A002705.

Extensions

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

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

A002704 Number of sets with a congruence property.

Original entry on oeis.org

2, 26, 938, 42800, 2130458, 111557594, 6041272682, 335089258634, 18922687509962, 1083572842675610, 62744027461625648, 3666433604712457466, 215879610645469496234, 12792865816027823374874, 762278349313657804740842, 45638342462133835019322554
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

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. A002703, A002705, A262570, A262583, A262584.

Programs

  • Maple
    p := proc(r,s,k)
    option remember;
    if r = 0 then
        1;
    elif r < 0 then
        0;
    elif s < 0 then
        0;
    elif igcd(s,2*k+1) > 1 then
        procname(r,s-1,k) ;
    else
        procname(r,s-1,k)+procname(r-s,s-1,k) ;
    end if;
end proc:
Q := proc(n,k)
    local q,knrat,alpha,m ;
    q := 0 ;
    knrat := (2*k*n^2+n^2+k^2)/4/k ;
    if type(knrat,'integer') then
        for alpha from 0 to knrat do
            m := 2*n+n/k ;
            if modp(2*alpha,m) = modp(knrat,m) then
                q := q+p(alpha,n+(n-k)/2/k,k) ;
            end if;
        end do:
    end if;
    q ;
end proc:
A002704 := proc(n)
    nloc := 3+6*n ;
    Q(nloc,3) ;
end proc:
seq(A002704(n),n=0..15) ; # R. J. Mathar, Oct 21 2015
  • Mathematica
    p[r_, s_, k_] := p[r, s, k] = Which[r == 0, 1, r < 0, 0, s < 0, 0, GCD[s, 2 k + 1] > 1, p[r, s - 1, k], True, p[r, s - 1, k] + p[r - s, s - 1, k]];
Q[n_, k_] := Module[{q = 0, knrat, alpha, m}, knrat = (2 k n^2 + n^2 + k^2)/4/k; If[IntegerQ[knrat], For[alpha = 0, alpha <= knrat, alpha++, m = 2 n + n/k; If[Mod[2 alpha, m] == Mod[knrat, m], q += p[alpha, n + (n - k)/2/k, k]]]]; q];
a[n_] := Q[6 n + 3, 3];
a /@ Range[0, 15] (* Jean-François Alcover, Mar 27 2020, after R. J. Mathar *)

Formula

See Maple code!

Extensions

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

A262590 Sets with a congruence property.

Original entry on oeis.org

0, 2, 6, 18, 62, 210, 728, 2570, 9198, 33288, 121574, 447394, 1657008, 6170930, 23091222, 86767016, 327235610, 1238188770, 4698767640
Offset: 3

Views

Author

R. J. Mathar, Oct 21 2015

Keywords

Links

Formula

See Maple code in A002705.

A262591 Sets with a congruence property.

Original entry on oeis.org

0, 0, 4, 16, 60, 208, 726, 2568, 9196, 33286, 121572, 447392, 1657006, 6170928, 23091220, 86767014, 327235608, 1238188768, 4698767638
Offset: 3

Views

Author

R. J. Mathar, Oct 21 2015

Keywords

Links

Formula

See Maple code in A002705.
Showing 1-5 of 5 results.

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

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