A103279 Array read by antidiagonals, generated by the matrix M = [1,1,1;1,N,1;1,1,1].

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, 448, 1, 3, 13, 54, 171, 396, 729, 1224, 1, 3, 14, 67, 252, 683, 1352, 2187, 3344, 1, 3, 15, 82, 365, 1188, 2731, 4616, 6561, 9136, 1, 3, 16, 99, 516, 2019, 5616, 10923, 15760

Offset

0,3

Comments

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)[1,1] 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.)

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:

1,

3,

N + 8,

N^2 + 4*N + 22,

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

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

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

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

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

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

etc.

Links

Table of n, a(n) for n=0..63.

Formula

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

Example

Array begins:
1,3,8,22,60,164,448,1224,3344,9136,...
1,3,9,27,81,243,729,2187,6561,19683,...
1,3,10,34,116,396,1352,4616,15760,53808,...
1,3,11,43,171,683,2731,10923,43691,174763,...
1,3,12,54,252,1188,5616,26568,125712,594864,...
...

Prog

(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())

Crossrefs

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.

Sequence in context: A171843 A132476 A328807 * A208910 A209760 A046544

Adjacent sequences: A103276 A103277 A103278 * A103280 A103281 A103282

Keyword

nonn,tabl

Author

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

Status

approved