|
%I M2146 N0855 #47 Oct 19 2023 08:48:54
%S 2,26,938,42800,2130458,111557594,6041272682,335089258634,
%T 18922687509962,1083572842675610,62744027461625648,
%U 3666433604712457466,215879610645469496234,12792865816027823374874,762278349313657804740842,45638342462133835019322554
%N Number of sets with a congruence property.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Alexander Rosa and Štefan Znám, <a href="/A002703/a002703.pdf">A combinatorial problem in the theory of congruences (Russian with English summary)</a>, Mat.-Fys. Casopis Sloven. Akad. Vied 15 1965 49-59. [Annotated scanned copy.]
%H Alexander Rosa and Štefan Znám, <a href="/A002703/a002703_1.pdf">A remark on a combinatorial problem (Russian with English summary)</a>, Mat.-Fyz. Casopis Sloven. Akad. Vied 15 1965 313-316. [Annotated scanned copy]. See Table 3 on page 316.
%F See Maple code!
%p p := proc(r,s,k)
%p option remember;
%p if r = 0 then
%p 1;
%p elif r < 0 then
%p 0;
%p elif s < 0 then
%p 0;
%p elif igcd(s,2*k+1) > 1 then
%p procname(r,s-1,k) ;
%p else
%p procname(r,s-1,k)+procname(r-s,s-1,k) ;
%p end if;
%p end proc:
%p Q := proc(n,k)
%p local q,knrat,alpha,m ;
%p q := 0 ;
%p knrat := (2*k*n^2+n^2+k^2)/4/k ;
%p if type(knrat,'integer') then
%p for alpha from 0 to knrat do
%p m := 2*n+n/k ;
%p if modp(2*alpha,m) = modp(knrat,m) then
%p q := q+p(alpha,n+(n-k)/2/k,k) ;
%p end if;
%p end do:
%p end if;
%p q ;
%p end proc:
%p A002704 := proc(n)
%p nloc := 3+6*n ;
%p Q(nloc,3) ;
%p end proc:
%p seq(A002704(n),n=0..15) ; # _R. J. Mathar_, Oct 21 2015
%t 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]];
%t 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];
%t a[n_] := Q[6 n + 3, 3];
%t a /@ Range[0, 15] (* _Jean-François Alcover_, Mar 27 2020, after _R. J. Mathar_ *)
%Y Cf. A002703, A002705, A262570, A262583, A262584.
%K nonn
%O 0,1
%A _N. J. A. Sloane_
%E More terms from _R. J. Mathar_, Oct 21 2015
|
