A002705 Sets with a congruence property.

0, 4, 40, 468, 5828, 76260, 1032444, 14316584, 202116108, 2893451652, 41886157564, 611902123284, 9007199254740, 133439988963012, 1987795697598012, 29752813022112180, 447193795726343004, 6746237832670921768, 102105221251235572188

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

The values for k=1, Q(n,1) in table 1 on page 315 for n = 3,5,7,9,... are 0, 2, 6, 18, 62, 210, 728, 2570, 9198, 33288, 121574, 447394, 1657008, 6170930, 23091222, 86767016, 327235610, 1238188770, 4698767640 ... (see A262590), - R. J. Mathar, Oct 21 2015

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

• 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 2.

Crossrefs

Cf. A002703, A002704, A262585, A262590.

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:
A002705 := proc(n)
    nloc := 2+4*n ;
    Q(nloc,2) ;
end proc: # 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[4 n + 2, 2];
a /@ Range[0, 18] (* Jean-François Alcover, Mar 27 2020, after R. J. Mathar *)

Formula

See Maple code!

Extensions

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