A155002 Triangle read by rows, A104762 * (A000129 * 0^(n-k)).

1, 1, 1, 2, 1, 2, 3, 2, 2, 5, 5, 3, 4, 5, 12, 8, 5, 6, 10, 12, 29, 13, 8, 10, 15, 24, 29, 70, 21, 13, 16, 25, 36, 58, 70, 169, 34, 21, 26, 40, 60, 87, 140, 169, 408, 55, 34, 42, 65, 96, 145, 210, 338, 408, 985

Offset: 1

Author

Gary W. Adamson & Roger L. Bagula, Jan 18 2009

Keywords

• eigen
• nonn
• easy
• tabl

Comments

Eigentriangle, row sums = rightmost term of next row.

Row sums = the Pell series starting with offset 1: (1, 2, 5, 12, 29, ...).

Examples

			First ten rows of the triangle T(n, m):
n \ m   1  2  3  4  5   6   7   8   9  10 ...
1:      1
2:      1  1
3:      2  1  2
4:      3  2  2  5
5:      5  3  4  5 12
6:      8  5  6 10 12  29
7:     13  8 10 15 24  29  70
8:     21 13 16 25 36  58  70 169
9:     34 21 26 40 60  87 140 169 408
10:    55 34 42 65 96 145 210 338 408 985
... reformatted by - _Wolfdieter Lang_, Apr 13 2021
Row 4 = (3, 2, 2, 5) = termwise products of (3, 2, 1, 1) and (1, 1, 2, 5).
		

Links

• M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210; arXiv:math/0205301 [math.CO], 2002. [Link to arXiv version]
• M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]

Crossrefs

Cf. A104762, A000045, A000129, A215928.

Formula

Triangle read by rows, A104762 * (A000129 * 0^(n-k)).

A104762 = Fibonacci numbers "decrescendo", (1, 1, 2, 3, 5, ...) in every column.

(A000129 * 0^(n-k)) ) = the Pell series prefaced with a 1:

(1, 1, 2, 5, 12, ...) as the main diagonal and the rest zeros

From Wolfdieter Lang, Apr 13 2021: (Start)

T(n, m) = F(n+1-m)*A215928(m), with F = A000045, for n >= m >= 1, and 0 otherwise.

The lower triangular (infinite) matrix t with elements t(n, m) = T(n+1, m+1), for n >= m >= 0, and 0 otherwise, has row polynomials R(n, x) = Sum_{m=0..n} t(n, m)*x^m with o.g.f. G(z, x) = A(z)/(1 - x*z*A(x*z)) =

(1 - x*z - (x*z)^2)/((1 - z - z^2)*(1 - 2*x*z - (x*z)^2)), with the o.g.f. A(x) of (F_{n+1})_{n>=0}, where F = A000045.

The infinite dimensional lower triangular Riordan matrix TB := (1/(1 - x - x^2), x) (a Toeplitz matrix) with nonzero elements A104762(n+1, m+1) has sequence (A215928(m)){m >=0} as 'L-eigen-sequence' (cf. the Bernstein-Sloane link for 'eigen-sequence'). This means that (TB - L)*vec(B) = 0-matrix, where L has elements L(i, j) = delta{i, j-1} (first upper diagonal with 1s, otherwise 0), and the infinite vector vec(B) has the elements of A215928.

Thanks to Gary W. Adamson for motivating me to look at such triangles and sequences. (End)