0,3

Generally, n-th term of = sequences of the form N*a(n-1) + n = right term of M^n * [1,0,0] given the sequences begin (1, N+2,...). By diagonals N=0,1,2,... of the triangle, N=0: (1, 2, 3,...); N=1: (A000217: 1, 3, 6,...); N=2: (A000295: 1, 4, 11,...); N=3: (A000340: 1, 5, 18,...); N=4: (A014825: 1, 6, 27,...); N=5: (A014827: 1, 7, 38,...); N=6: (A014829: 1, 8, 51,...); N=7: (A014830: 1, 9, 66,...); N=8: (A014831: 1, 10, 83,...); ...

Row sums = A134195: (1, 3, 7, 15, 32, 72, 178, 494, 1543,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 12 2007

Construct an array of sequences of the form N*a(n-1) + n, N=0,1,2...; then take antidiagonals. n-th term in N-th row of the array (n=1,2,3...) = right term in M^n * [1,0,0], where M = the 3 X 3 matrix [1,0,0; 1,1,0; 1,1,N].

First few rows of the array are:

1, 2, 3, 4, 5,...;

1, 3, 6, 10, 15,...;

1, 4, 11, 26, 57,...;

1, 5, 18, 58, 179,...;

1, 6, 27, 112, 453,...;

...

First few rows of the triangle are:

1;

1, 2;

1, 3, 3;

1, 4, 6, 4;

1, 5, 11, 10, 5;

1, 6, 18, 26, 15, 6;

1, 7, 27, 58, 57, 21, 7;

1, 8, 38, 112, 179, 120, 28, 8;

1, 9, 51, 194, 453, 543, 247, 36, 9;

1, 10, 66, 310, 975, 1818, 1636, 502, 45, 10;

1, 11, 83, 466, 1865, 4881, 7279, 4916, 1013, 55, 11;

...

Examples: row N=4, (A014825: 1, 6, 27, 112...) is generated from 6*A014825(n) + n.

The 4-th term in Row N (112) = right term in M^4 * [1,0,0], where M = the 3 X 3 matrix [1,0,0; 1,1,0; 1,1,4]. M^4 * [1,0,0] = [1, 4, 112].

Cf. A000217, A000295, A000340, A014825, A014827, A014829, A014830, A014831.

Cf. A134195.

Sequence in context: A135278 A034356 A075195 this_sequence A130305 A144400 A143328

Adjacent sequences: A126882 A126883 A126884 this_sequence A126886 A126887 A126888

nonn,tabl,uned

Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 30 2006