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.

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

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