A002705 Sets with a congruence property. (Formerly M3673 N1497)

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

Offset

0,2

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

Table of n, a(n) for n=0..18.

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.

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:
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 *)

Crossrefs

Cf. A002703, A002704, A262585, A262590.

Sequence in context: A372460 A304447 A370444 * A235372 A034385 A249927

Adjacent sequences: A002702 A002703 A002704 * A002706 A002707 A002708

Keyword

nonn

Author

N. J. A. Sloane

Extensions

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

Status

approved