A136170 Triangle T, read by rows, where row n of T = row n-1 of T^fibonacci(n) with appended '1' for n>=1 starting with a single '1' in row 0.

1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 19, 9, 3, 1, 1, 310, 105, 25, 5, 1, 1, 10978, 2702, 480, 68, 8, 1, 1, 868140, 154609, 20657, 2184, 182, 13, 1, 1, 149688297, 19092682, 1906051, 152579, 9562, 483, 21, 1, 1, 57339888914, 5161046609, 378639419, 22799907

Offset: 0

Author

Paul D. Hanna, Dec 17 2007

Keywords

• nonn
• tabl

Examples

			Triangle T begins:
1;
1, 1;
1, 1, 1;
3, 2, 1, 1;
19, 9, 3, 1, 1;
310, 105, 25, 5, 1, 1;
10978, 2702, 480, 68, 8, 1, 1;
868140, 154609, 20657, 2184, 182, 13, 1, 1;
149688297, 19092682, 1906051, 152579, 9562, 483, 21, 1, 1;
57339888914, 5161046609, 378639419, 22799907, 1090125, 41480, 1275, 34, 1, 1; ...
GENERATE T FROM MATRIX POWERS OF T.
Row n of T = row n-1 of T^fibonacci(n) with appended '1'.
Examples.
Row 5 of T is given by row 4 of matrix power T^fibonacci(5) = T^5:
1;
5, 1;
15, 5, 1;
55, 20, 5, 1;
310, 105, 25, 5, 1; <== row 5 of T
3796, 1070, 215, 35, 5, 1; ...
Row 6 of T is given by row 5 of matrix power T^fibonacci(6) = T^8:
1;
8, 1;
36, 8, 1;
164, 44, 8, 1;
978, 268, 52, 8, 1;
10978, 2702, 480, 68, 8, 1; <== row 6 of T
262838, 53648, 8082, 964, 92, 8, 1; ...
ALTERNATE GENERATING METHOD.
To obtain row n, start with a '1' repeated fibonacci(n) times,
and build a table where row k+1 equals the partial sums of row k
but with the last term appearing fibonacci(n-k) times, for k=1..n-1;
listing the final terms in each row forms row n of this triangle.
Example.
To obtain row 5, start with a '1' repeated fibonacci(5)=5 times:
(1,1,1,1,1);
take partial sums, writing the last term fibonacci(4)=3 times:
1,2,3,4, (5,5,5);
take partial sums, writing the last term fibonacci(3)=2 times:
1,3,6,10,15,20, (25,25);
take partial sums, writing the last term fibonacci(2)=1 times:
1,4,10,20,35,55,80, (105);
take partial sums, writing the last term fibonacci(1)=1 times:
1,5,15,35,70,125,205, (310).
Final terms in the above partial sums forms row 5: [310,105,25,5,1];
repeating this process will generate all the rows of this triangle.
		

Crossrefs

Cf. columns: A136171, A136172, A136173; variants: A101479, A132610, A132615.

Programs

PARI

/* Generate using matrix power method: */ T(n,k)=local(A=Mat(1), B); for(m=1, n+1, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i, B[i, j]=1, B[i, j]=(A^(fibonacci(i-1)))[i-1, j]); )); A=B); return( ((A)[n+1, k+1]))

PARI

/* Generate using partial sums method (faster) */ T(n, k)=local(A=vector(n+1), p); A[1]=1; for(j=1, n-k, p=fibonacci(n+2)-fibonacci(n-j+2)-j; A=Vec((Polrev(A)+x*O(x^p))/(1-x))); A[p+1]

Formula

See example section for two different methods of generating this triangle.