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A103280
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Array read by antidiagonals, generated by the matrix M = [1,1,1;1,N,1;1,1,1];.
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1
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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, 328, 1, 8, 41, 144, 341, 560, 729, 896, 1, 9, 54, 231, 684, 1365, 1912, 2187, 2448, 1, 10, 69, 352, 1289, 3240, 5461, 6528, 6561, 6688, 1, 11, 86, 513, 2276, 7175, 15336, 21845
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OFFSET
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0,3
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COMMENTS
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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.
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.)
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.
Columns of array follow the polynomials:
0
1
N + 2
N^2 + 2*N + 6
N^3 + 2*N^2 + 8*N + 16
N^4 + 2*N^3 + 10*N^2 + 24*N + 44
N^5 + 2*N^4 + 12*N^3 + 32*N^2 + 76*N + 120
N^6 + 2*N^5 + 14*N^4 + 40*N^3 + 112*N^2 + 232*N + 328
N^7 + 2*N^6 + 16*N^5 + 48*N^4 + 152*N^3 + 368*N^2 + 704*N + 896
N^8 + 2*N^7 + 18*N^6 + 56*N^5 + 196*N^4 + 528*N^3 + 1200*N^2 + 2112*N + 2448
etc.
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LINKS
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FORMULA
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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)))
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EXAMPLE
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Array begins:
1,2,6,16,44,120,328,896,2448,6688,...
1,3,9,27,81,243,729,2187,6561,19683, ...
1,4,14,48,164,560,1912,6528,22288,76096,...
1,5,21,85,341,1365,5461,21845,87381,349525,...
1,6,30,144,684,3240,15336,72576,343440,1625184,...
1,7,41,231,1289,7175,39913,221991,1234633,6866503,...
...
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PROG
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(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))) for(k=0, 10, print1(k, ": "); for(i=1, 10, print1(T12(k, i), ", ")); print())
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CROSSREFS
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Cf. A103279 (for (M^n)[1, 1]), A002605 (for N=0), A000244 (for N=1), A007070 (for N=2), A002450 (for N=3), A002450 (for N=4), A030192 (for N=5) A006131 (for N=-1), A000400 (bisection for N=-2), A015443 (for N=-3), A083102 (for N=-4).
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KEYWORD
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AUTHOR
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Lambert Klasen (lambert.klasen(AT)gmx.net), Jan 27 2005
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STATUS
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approved
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