A321986 Number of integer pairs (x,y) with x+y < 3*n/4, x-y < 3*n/4 and -x/2 < y < 2*x.

0, 0, 1, 3, 3, 5, 9, 14, 14, 19, 26, 34, 34, 42, 52, 63, 63, 74, 87, 101, 101, 115, 131, 148, 148, 165, 184, 204, 204, 224, 246, 269, 269, 292, 317, 343, 343, 369, 397, 426, 426, 455, 486, 518, 518, 550, 584, 619, 619, 654, 691, 729, 729, 767, 807, 848, 848, 889, 932, 976, 976

Offset

0,4

Comments

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.

References

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

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (1,0,0,2,-2,0,0,-1,1).

Formula

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 ).

Example

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

Maple

A056594 := proc(n)
    if type (n, 'odd') then
        0;
    else
        (-1)^(n/2) ;
    end if;
end proc:
A008619 := proc(n)
    1+iquo(n, 2) ;
end proc:
A321986 := proc(n)
    if n =0 then
        0;
    else
        -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) ;
        %/32 ;
    end if;
end proc:
seq(A321986(n), n=0..30) ;

Crossrefs

Sequence in context: A350393 A325849 A104220 * A325187 A209083 A137202

Adjacent sequences: A321983 A321984 A321985 * A321987 A321988 A321989

Keyword

nonn,easy,less

Author

R. J. Mathar, Nov 23 2018

Extensions

a(41) onward from Andrew Howroyd, Nov 09 2025

Status

approved