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A129100 Triangle T, read by rows, where column n of T = column 0 of T^(2^n) for n>0, such that column 0 (A129092) equals the row sums of the prior row, starting with T(0,0)=1. 8
1, 1, 1, 2, 2, 1, 5, 6, 4, 1, 16, 24, 20, 8, 1, 69, 136, 136, 72, 16, 1, 430, 1162, 1360, 880, 272, 32, 1, 4137, 15702, 21204, 16032, 6240, 1056, 64, 1, 64436, 346768, 537748, 461992, 214336, 46784, 4160, 128, 1, 1676353, 12836904, 22891448, 21944520 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

T(n,0) = A129092(n) = A030067(2^n - 1) for n>0 where A030067 is the Semi-Fibonacci numbers.

LINKS

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

FORMULA

Row k = row 0 of matrix power A129104^k, where A129104 equals triangle A129100 with an additional leftmost column of all 1's.

EXAMPLE

Column 0 of row n equals A129092(n) = A030067(2^n-1) for n>=1,

where A030067 is the semi-Fibonacci numbers:

[(1), 1, (2), 1, 3, 2, (5), 1, 6, 3, 9, 2, 11, 5, (16), 1, ...],

which obey the recurrence:

A030067(n) = A030067(n/2) when n is even; and

A030067(n) = A030067(n-1) + A030067(n-2) when n is odd.

Triangle begins:

1;

1, 1;

2, 2, 1;

5, 6, 4, 1;

16, 24, 20, 8, 1;

69, 136, 136, 72, 16, 1;

430, 1162, 1360, 880, 272, 32, 1;

4137, 15702, 21204, 16032, 6240, 1056, 64, 1;

64436, 346768, 537748, 461992, 214336, 46784, 4160, 128, 1;

1676353, 12836904, 22891448, 21944520, 11720016, 3107456, 361856, 16512, 256, 1; ...

where columns shift left under matrix square, A129100^2, which starts:

1;

2, 1;

6, 4, 1;

24, 20, 8, 1;

136, 136, 72, 16, 1;

1162, 1360, 880, 272, 32, 1; ...

Inserting a left column of all 1's, yields matrix A129104:

1, 1;

1, 1, 1;

1, 2, 2, 1;

1, 5, 6, 4, 1;

1, 16, 24, 20, 8, 1;

1, 69, 136, 136, 72, 16, 1; ...

where row 0 of matrix power A129104^k forms row k of A129100,

as illustrated below.

For row 2: A129104^2 begins:

2, 2, 1;

3, 4, 3, 1;

6, 12, 12, 6, 1;

17, 54, 65, 42, 12, 1;

70, 362, 512, 400, 156, 24, 1;

431, 3708, 6223, 5656, 2744, 600, 48, 1; ...

and row 0 of A129104^2 equals row 2 of A129100: [2, 2, 1].

For row 3: A129104^3 begins:

5, 6, 4, 1;

11, 18, 16, 7, 1;

37, 88, 96, 56, 14, 1;

191, 672, 860, 609, 210, 28, 1;

1525, 8038, 11956, 9856, 4256, 812, 56, 1; ...

and row 0 of A129104^3 equals row 3 of A129100: [5, 6, 4, 1].

For row 4: A129104^4 begins:

16, 24, 20, 8, 1;

53, 112, 116, 64, 15, 1;

292, 890, 1088, 736, 240, 30, 1;

2571, 11350, 16056, 12664, 5185, 930, 60, 1; ...

and row 0 of A129104^4 equals row 4 of A129100: [16, 24, 20, 8, 1].

PROG

(PARI) T(n, k)=local(A=Mat(1), B); for(m=1, n+1, B=matrix(m, m); for(r=1, m, for(c=1, r, if(r==c || r==1 || r==2, B[r, c]=1, if(c==1, B[r, 1]=sum(i=1, r-1, A[r-1, i]), B[r, c]=(A^(2^(c-1)))[r-c+1, 1])); )); A=B); return(A[n+1, k+1])

CROSSREFS

Cf. A030067 (Semi-Fibonacci); A129092 (row sums=column 0), A129101 (column 1), A129102 (column 2), A129103 (column 3); variant: A129104.

Sequence in context: A259691 A056857 A175579 * A309991 A162382 A325580

Adjacent sequences:  A129097 A129098 A129099 * A129101 A129102 A129103

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Mar 29 2007

STATUS

approved

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Last modified January 25 21:23 EST 2021. Contains 340427 sequences. (Running on oeis4.)