A002704 Number of sets with a congruence property. (Formerly M2146 N0855)

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

Offset

0,1

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

Table of n, a(n) for n=0..15.

Alexander Rosa and Štefan Znám, A combinatorial problem in the theory of congruences (Russian with English summary), Mat.-Fys. Casopis Sloven. Akad. Vied 15 1965 49-59. [Annotated scanned copy.]

Alexander Rosa and Štefan Znám, A remark on a combinatorial problem (Russian with English summary), Mat.-Fyz. Casopis Sloven. Akad. Vied 15 1965 313-316. [Annotated scanned copy]. See Table 3 on page 316.

Formula

See Maple code!

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 *)

Crossrefs

Cf. A002703, A002705, A262570, A262583, A262584.

Sequence in context: A210672 A290688 A173103 * A015215 A363892 A158120

Adjacent sequences: A002701 A002702 A002703 * A002705 A002706 A002707

Keyword

nonn

Author

N. J. A. Sloane

Extensions

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

Status

approved