%I #7 Sep 01 2024 00:13:23

%S 1,1,3,1,3,8,1,3,9,22,1,3,10,27,60,1,3,11,34,81,164,1,3,12,43,116,243,

%T 448,1,3,13,54,171,396,729,1224,1,3,14,67,252,683,1352,2187,3344,1,3,

%U 15,82,365,1188,2731,4616,6561,9136,1,3,16,99,516,2019,5616,10923,15760

%N Array read by antidiagonals, generated by the matrix M = [1,1,1;1,N,1;1,1,1].

%C Consider the matrix M = [1,1,1;1,N,1;1,1,1]; Characteristic polynomial of M is x^3 + (-N - 2)*x^2 + (2*N - 2)*x.

%C Now (M^n)[1,1] is equivalent to the recursion a(1) = 1, a(2) = 3, a(n) = (N+2)a(n-1)+(2N-2)a(n-2). (This also holds for negative N and fractional N.)

%C a(n+1)/a(n) converges to the upper root of the characteristic polynomial ((N + 2) + sqrt((N - 2)^2 + 8))/2 for n to infinity.

%C Columns of array follow the polynomials:

%C 1,

%C 3,

%C N + 8,

%C N^2 + 4*N + 22,

%C N^3 + 4*N^2 + 16*N + 60,

%C N^4 + 4*N^3 + 18*N^2 + 56*N + 164,

%C N^5 + 4*N^4 + 20*N^3 + 68*N^2 + 188*N + 448,

%C N^6 + 4*N^5 + 22*N^4 + 80*N^3 + 248*N^2 + 608*N + 1224,

%C N^7 + 4*N^6 + 24*N^5 + 92*N^4 + 312*N^3 + 864*N^2 + 1920*N + 3344,

%C N^8 + 4*N^7 + 26*N^6 + 104*N^5 + 380*N^4 + 1152*N^3 + 2928*N^2 + 5952*N + 9136,

%C etc.

%F T(N, 1)=1, T(N, 2)=3, T(N, n)=(N+2)*T(N, n-1)-(2*N-2)*T(N, n-2).

%e Array begins:

%e 1,3,8,22,60,164,448,1224,3344,9136,...

%e 1,3,9,27,81,243,729,2187,6561,19683,...

%e 1,3,10,34,116,396,1352,4616,15760,53808,...

%e 1,3,11,43,171,683,2731,10923,43691,174763,...

%e 1,3,12,54,252,1188,5616,26568,125712,594864,...

%e ...

%o (PARI) T11(N, n) = if(n==1,1,if(n==2,3,(N+2)*r1(N,n-1)-(2*N-2)*r1(N,n-2))) for(k=0,10,print1(k,": ");for(i=1,10,print1(T11(k,i),","));print())

%Y Cf. A103280 (for (M^n)[1, 2]), A028859 (for N=0), A000244 (for N=1), A007052 (for N=2), A007583 (for N=3), A083881 (for N=4), A026581 (for N=-1), A026532 (for N=-2), A026568.

%K nonn,tabl

%O 0,3

%A Lambert Klasen (lambert.klasen(AT)gmx.net), Jan 27 2005