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A103280 Array read by antidiagonals, generated by the matrix M = [1,1,1;1,N,1;1,1,1].

%I A103280 #10 Oct 05 2024 19:12:09
%S A103280 1,1,2,1,3,6,1,4,9,16,1,5,14,27,44,1,6,21,48,81,120,1,7,30,85,164,243,
%T A103280 328,1,8,41,144,341,560,729,896,1,9,54,231,684,1365,1912,2187,2448,1,
%U A103280 10,69,352,1289,3240,5461,6528,6561,6688,1,11,86,513,2276,7175,15336,21845
%N A103280 Array read by antidiagonals, generated by the matrix M = [1,1,1;1,N,1;1,1,1].
%C A103280 Consider the matrix M = [1,1,1;1,N,1;1,1,1];
%C A103280 Characteristic polynomial of M is x^3 + (-N - 2)*x^2 + (2*N - 2)*x.
%C A103280 Now (M^n)[1,2] is equivalent to the recursion a(1) = 1, a(2) = N+2, a(n) = (N+2)a(n-1)+(2N-2)a(n-2). (This also holds for negative N and fractional N.)
%C A103280 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 A103280 Columns of array follow the polynomials:
%C A103280 0
%C A103280 1
%C A103280 N + 2
%C A103280 N^2 + 2*N + 6
%C A103280 N^3 + 2*N^2 + 8*N + 16
%C A103280 N^4 + 2*N^3 + 10*N^2 + 24*N + 44
%C A103280 N^5 + 2*N^4 + 12*N^3 + 32*N^2 + 76*N + 120
%C A103280 N^6 + 2*N^5 + 14*N^4 + 40*N^3 + 112*N^2 + 232*N + 328
%C A103280 N^7 + 2*N^6 + 16*N^5 + 48*N^4 + 152*N^3 + 368*N^2 + 704*N + 896
%C A103280 N^8 + 2*N^7 + 18*N^6 + 56*N^5 + 196*N^4 + 528*N^3 + 1200*N^2 + 2112*N + 2448
%C A103280 etc.
%F A103280 T(N, 1)=1, T(N, 2)=N+2, T(N, n)=(N+2)*T(N, n-1)-(2*N-2)*T(N, n-2).
%e A103280 Array begins:
%e A103280 1,2,6,16,44,120,328,896,2448,6688,...
%e A103280 1,3,9,27,81,243,729,2187,6561,19683, ...
%e A103280 1,4,14,48,164,560,1912,6528,22288,76096,...
%e A103280 1,5,21,85,341,1365,5461,21845,87381,349525,...
%e A103280 1,6,30,144,684,3240,15336,72576,343440,1625184,...
%e A103280 1,7,41,231,1289,7175,39913,221991,1234633,6866503,...
%e A103280 ...
%o A103280 (PARI) T12(N, n) = if(n==1,1,if(n==2,N+2,(N+2)*T12(N,n-1)-(2*N-2)*T12(N,n-2)))
%o A103280 for(k=0,10,print1(k,": ");for(i=1,10,print1(T12(k,i),","));print())
%Y A103280 Cf. A103279 (for (M^n)[1, 1]), A002605 (for N=0), A000244 (for N=1), A007070 (for N=2), A002450 (for N=3), A030192 (for N=4), A152268 (for N=5), A006131 (for N=-1), A000400 (bisection for N=-2), A015443 (for N=-3), A083102 (for N=-4).
%K A103280 nonn,tabl,easy
%O A103280 0,3
%A A103280 Lambert Klasen (lambert.klasen(AT)gmx.net), Jan 27 2005