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A002705 Sets with a congruence property.
(Formerly M3673 N1497)
6
0, 4, 40, 468, 5828, 76260, 1032444, 14316584, 202116108, 2893451652, 41886157564, 611902123284, 9007199254740, 133439988963012, 1987795697598012, 29752813022112180, 447193795726343004, 6746237832670921768, 102105221251235572188 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A264112 A304447 A370444 * A235372 A034385 A249927
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from R. J. Mathar, Oct 21 2015
STATUS
approved

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Last modified March 28 16:12 EDT 2024. Contains 371254 sequences. (Running on oeis4.)