A262568 - OEIS

%I #23 Oct 20 2023 09:50:13

%S 2,2,2,4,8,16,26,48,90,164,302,564,1058,1984,3744,7084,13440,25576,

%T 48770,93200,178482,342394,657920,1266204,2440320,4709376,9099506,

%U 17602324,34087012,66076416,128207978,248983552,483939978,941362696,1832519264,3569842948

%N a(n) = A002703(n) + 2.

%H Alexander Rosa and Štefan Znám, <a href="/A002703/a002703.pdf">A combinatorial problem in the theory of congruences. (Russian)</a>, Mat.-Fys. Casopis Sloven. Akad. Vied 15 1965 49-59. [Annotated scanned copy] This is Q(n) in Table 3.

%F See Maple code! - _N. J. A. Sloane_, Oct 21 2015

%p A178666 := proc(r,s)

%p     product( (1+x^(2*i+1)),i=0..floor((s-1)/2)) ;

%p     expand(%) ;

%p     coeftayl(%,x=0,r) ;

%p end proc:

%p kstart := proc(n,m)

%p     ceil(binomial(n+1,2)/m) ;

%p end proc:

%p kend := proc(n,m)

%p     floor(binomial(3*n+1,2)/3/m) ;

%p end proc:

%p A262568 := proc(n)

%p     local s,m,Q ,vi,k;

%p     s := 2*n-1 ;

%p     m := 2*n+1 ;

%p     Q := 0 ;

%p     for k from kstart(n,m) to kend(n,m) do

%p         vi := m*k-binomial(n+1,2) ;

%p         Q := Q+A178666(vi,s) ;

%p     end do:

%p     Q ;

%p end proc: # _R. J. Mathar_, Oct 21 2015

%t A178666[r_, s_] := SeriesCoefficient[Product[(1 + x^(2i+1)), {i, 0, Floor[ (s - 1)/2]}], {x, 0, r}];

%t kstart [n_, m_] := Ceiling[Binomial[n+1, 2]/m];

%t kend[n_, m_] := Floor[Binomial[3n+1, 2]/3/m];

%t a[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 a /@ Range[3, 38] (* _Jean-François Alcover_, Mar 24 2020, after _R. J. Mathar_ *)

%Y Cf. A002703, A262567, A262569.

%Y Tables 1 and 2 of the first Rosa-Znám 1965 paper are A053632 and A178666 respectively.

%K nonn

%O 3,1

%A _N. J. A. Sloane_, Oct 20 2015

%E More terms from _R. J. Mathar_, Oct 21 2015

%E Missing a(16) inserted by _Sean A. Irvine_, Oct 23 2015