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A002703 Sets with a congruence property.

%I A002703 M1598 N0624 #63 Oct 20 2023 12:34:26
%S A002703 0,0,0,2,6,14,24,46,88,162,300,562,1056,1982,3742,7082,13438,25574,
%T A002703 48768,93198,178480,342392,657918,1266202,2440318,4709374,9099504,
%U A002703 17602322,34087010,66076414,128207976,248983550,483939976,941362694,1832519262,3569842946,6958934352
%N A002703 Sets with a congruence property.
%C A002703 a(n) is the sequence k(n) in Table 3 of the first 1965 paper. - _N. J. A. Sloane_, Oct 20 2015
%C A002703 See English summary at the end of the first 1965 paper, which is repeated in the Zentralblatt review. - _Jonathan Sondow_, Nov 02 2013
%D A002703 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002703 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002703 Alexander Rosa and Štefan Znám, <a href="http://dml.cz/dmlcz/126392">A combinatorial problem in the theory of congruences (Russian with English summary)</a>, Mat.-Fys. Casopis Sloven. Akad. Vied 15 (1965) 49-59. [<a href="/A002703/a002703.pdf">Annotated scanned copy</a>.] This is the sequence k(n) in Table 3. Tables 1 and 2 are A053632 and A178666 respectively.
%H A002703 Alexander Rosa and Štefan Znám, <a href="http://dml.cz/dmlcz/126444">A remark on a combinatorial problem (Russian with English summary)</a>, Mat.-Fyz. Casopis Sloven. Akad. Vied 15 (1965) 313-316. [<a href="/A002703/a002703_1.pdf">Annotated scanned copy</a>]
%H A002703 Zentralblatt, <a href="http://zbmath.org/scans/128/268@2x.gif">Review of Rosa and Znám, A combinatorial problem in the theory of congruences</a>.
%p A002703 A002703 := proc(n)
%p A002703     A262568(n)-2 ;
%p A002703 end proc: # _R. J. Mathar_, Oct 21 2015
%t A002703 A178666[r_, s_] := SeriesCoefficient[Product[ (1 + x^(2i+1)), {i, 0, Floor[(s-1)/2]}], {x, 0, r}];
%t A002703 kstart[n_, m_] := Ceiling[Binomial[n+1, 2]/m];
%t A002703 kend[n_, m_] := Floor[Binomial[3n+1, 2]/3/m];
%t A002703 A262568[n_] := Module[{s = 2n-1, m = 2n+1, Q=0, vi, k}, For[k = kstart[n, m], k <= kend[n, m], k++, vi = m k - Binomial[n+1, 2]; Q += A178666[vi, s] ]; Q];
%t A002703 a[n_] := A262568[n] - 2;
%t A002703 a /@ Range[3, 39] (* _Jean-François Alcover_, Mar 24 2020, after _R. J. Mathar_ in A262568 *)
%Y A002703 Cf. A002704, A002705.
%Y A002703 See A262567, A262568, A262569 for other versions.
%Y A002703 Tables 1 and 2 of the first Rosa-Znám 1965 paper are A053632 and A178666 respectively.
%K A002703 nonn
%O A002703 3,4
%A A002703 _N. J. A. Sloane_
%E A002703 More terms from _R. J. Mathar_, Oct 21 2015