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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. 3
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,7

LINKS

Table of n, a(n) for n=0..48.

FORMULA

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

EXAMPLE

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.

PROG

(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]

CROSSREFS

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

Sequence in context: A204124 A316674 A101479 * A245188 A137241 A331539

Adjacent sequences:  A136167 A136168 A136169 * A136171 A136172 A136173

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Dec 17 2007

STATUS

approved

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Last modified May 12 02:06 EDT 2021. Contains 343808 sequences. (Running on oeis4.)