A361961 Total semiperimeter of 3-Fuss-Catalan polyominoes of length 3n.

2, 18, 150, 1275, 11033, 96768, 857440, 7658001, 68827440, 621769016, 5640718746, 51355222113, 468976190634, 4293892636600, 39403880112240, 362321464909965, 3337465898598408, 30791007409655928, 284475382593582680, 2631594710532743340, 24372218297220901965, 225958143637966827240

Offset: 1

Author

R. J. Mathar, Mar 31 2023

Keywords

• nonn
• easy

Links

• Toufik Mansour, I. L. Ramirez, Enumerations of polyominoes determined by Fuss-Catalan words, Australas. J. Combin. 81 (3) (2021) 447-457, Table 2.

Crossrefs

Cf. A024482 (1-Fuss-Catalan), A078999 (total area), A361960 (2-Fuss-Catalan).

Programs

Maple

Per := proc(s,p,n)
    local i,j,a ;
    a := 0 ;
    for i from 0 to n-1 do
    for j from 0 to n-1-i do
        a := a+ (-1)^j*p^(n+1+i+(s+1)*j) *binomial(n-1+i,i)*binomial(n,j)*binomial(n+s*j,n-1-i-j)/(1-p)^(i+j) ;
    end do:
    end do:
    expand(a/n) ;
    factor(%) ;
end proc:
Per1std := proc(s,n)
    local p;
    Per(s,p,n) ;
    diff(%,p) ;
    factor(%) ;
    subs(p=1,%) ;
end proc:
seq(Per1std(3,n),n=1..30) ;

Formula

Conjecture: D-finite with recurrence 3*n*(396221*n -410120) *(3*n-1) *(3*n+1) *a(n) +4*(-86981513*n^4 [phone]*n^3 -996839467*n^2 [phone]*n -279161658) *a(n-1) +32*(2*n-5) *(4*n-9) *(4*n-7) *(2282347*n -1795413)*a(n-2)=0.