A323943 Trapezoidal matrix T(n,k) (n>=1, 1<=k<=n+2) read by rows, arising in enumeration of unbranched k-4-catafusenes.

1, 2, 1, 3, 7, 5, 1, 9, 24, 22, 8, 1, 27, 81, 90, 46, 11, 1, 81, 270, 351, 228, 79, 14, 1, 243, 891, 1323, 1035, 465, 121, 17, 1, 729, 2916, 4860, 4428, 2430, 828, 172, 20, 1, 2187, 9477, 17496, 18144, 11718, 4914, 1344, 232, 23, 1, 6561, 30618, 61965, 71928, 53298, 26460, 8946, 2040, 301, 26, 1, 19683

Offset

1,2

Comments

Rows sums are powers of 4.

Links

Table of n, a(n) for n=1..64.

S. J. Cyvin, B. N. Cyvin and J. Brunvoll, Isomer enumeration of some polygonal systems representing polycyclic conjugated hydrocarbons, Journal of molecular structure 376.1-3 (1996): 495-505. See Section 7.3.

Formula

T(1,1)=T(1,3)=1, T(1,2)=2; thereafter T(n+1,k) = 3*T(n,k)+T(n,k-1).

Example

Matrix begins:
1, 2, 1,
3, 7, 5, 1,
9, 24, 22, 8, 1,
27, 81, 90, 46, 11, 1,
81, 270, 351, 228, 79, 14, 1,
...

Maple

A323943 := proc(n, k)
    option remember;
    if n = 1 then
        if k>=1 and k<=3 then
            op(k, [1, 2, 1]) ;
        else
            0;
        end if;
    else
        3*procname(n-1, k)+procname(n-1, k-1) ;
    end if;
end proc:
seq(seq(A323943(n, k), k=1..n+2), n=1..12) ; # R. J. Mathar, May 08 2019

Mathematica

T[n_, k_] := T[n, k] = If[n == 1, If[k >= 1 && k <= 3, {1, 2, 1}[[k]], 0], 3*T[n - 1, k] + T[n - 1, k - 1]];
Table[Table[T[n, k], {k, 1, n + 2}], {n, 1, 12}] // Flatten (* Jean-François Alcover, Nov 08 2023, after R. J. Mathar *)

Crossrefs

Cf. A024462 (reversed rows).

Cf. A000244 (column 1), A038765 (column 2), A081892 (column 3)

Sequence in context: A238206 A127896 A010757 * A286616 A019320 A201615

Adjacent sequences: A323940 A323941 A323942 * A323944 A323945 A323946

Keyword

nonn,tabf,easy

Author

N. J. A. Sloane, Feb 09 2019

Status

approved