A321986 - OEIS

%I #7 Nov 23 2018 12:09:01

%S 0,0,1,3,3,5,9,14,14,19,26,34,34,42,52,63,63,74,87,101,101,115,131,

%T 148,148,165,184,204,204,224,246,269,269,292,317,343,343,369,397,426,

%U 426

%N Number of integer pairs (x,y) with x+y < 3*n/4, x-y < 3*n/4 and -x/2 < y < 2*x.

%C The Comtet formula for I(n) = round(9*n^2+18-n*b(n)/16) with b(n)=bar(7,4,1,10) with period 4, is missing divisors (32?) somewhere.

%D L. Comtet, Advanced Combinatorics (Reidel, 1974), page 122, exercise 19 sequence (2).

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,2,-2,0,0,-1,1).

%F G.f.: -x^2*(x^2 - x + 1)*(x^5 + x^4 + x^3 + 2*x^2 + 3*x + 1) / ( (1+x)^2*(x^2+1)^2*(x-1)^3 ).

%e The 3 solutions for n=3 or n=4 are (x,y)=(1,0), (1,1), (2,0).

%p A056594 := proc(n)

%p     if type (n,'odd') then

%p         0;

%p     else

%p         (-1)^(n/2) ;

%p     end if;

%p end proc:

%p A008619 := proc(n)

%p     1+iquo(n,2) ;

%p end proc:

%p A321986 := proc(n)

%p     if n =0 then

%p         0;

%p     else

%p         -11*n +35/2 +9*n^2 +9/2*(-1)^n -3*(-1)^n*n +22*A056594(n) -2*A056594(n-1) +12*(-1)^A008619(n)*A008619(n) ;

%p         %/32 ;

%p     end if;

%p end proc:

%p seq(A321986(n),n=0..30) ;

%K nonn,easy,less

%O 0,4

%A _R. J. Mathar_, Nov 23 2018