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A113350
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Triangle Q, read by rows, such that Q^2 transforms column k of Q^2 into column k+1 of Q^2, so that column k of Q^2 equals column 0 of Q^(2*k+2), where Q^2 denotes the matrix square of Q.
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28
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1, 2, 1, 5, 4, 1, 19, 22, 6, 1, 113, 166, 51, 8, 1, 966, 1671, 561, 92, 10, 1, 10958, 21510, 7726, 1324, 145, 12, 1, 156700, 341463, 129406, 23010, 2575, 210, 14, 1, 2727794, 6496923, 2572892, 471724, 53935, 4434, 287, 16, 1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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FORMULA
| Let [Q^m]_k denote column k of matrix power Q^m,
so that triangular matrix Q may be defined by
[Q]_k = [P^(2*k+2)]_0, for k>=0, where
the dual triangular matrix P = A113340 is defined by
[P]_k = [P^(2*k+1)]_0, for k>=0.
Then, amazingly, powers of P and Q satisfy:
[P^(2*j+1)]_k = [P^(2*k+1)]_j,
[P^(2*j+2)]_k = [Q^(2*k+1)]_j,
[Q^(2*j+2)]_k = [Q^(2*k+2)]_j,
for all j>=0, k>=0.
Also, we have the column transformations:
P^2 * [P]_k = [P]_{k+1},
P^2 * [Q]_k = [Q]_{k+1},
Q^2 * [P^2]_k = [P^2]_{k+1},
Q^2 * [Q^2]_k = [Q^2]_{k+1},
for all k>=0.
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EXAMPLE
| Triangle Q begins:
1;
2,1;
5,4,1;
19,22,6,1;
113,166,51,8,1;
966,1671,561,92,10,1;
10958,21510,7726,1324,145,12,1;
156700,341463,129406,23010,2575,210,14,1;
2727794,6496923,2572892,471724,53935,4434,287,16,1;
56306696,144856710,59525136,11198006,1305070,108593,7021,376,18,1;
Matrix square Q^2 begins:
1;
4,1;
18,8,1;
112,68,12,1;
965,712,150,16,1;
10957,9270,2184,264,20,1; ...
where Q^2 transforms column k of Q^2 into column k+1:
at k=0, [Q^2]*[1,4,18,112,965,...] = [1,8,68,712,9270,...];
at k=1, [Q^2]*[1,8,68,712,9270,...] =
[1,12,150,2184,37523,...].
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PROG
| (PARI) {Q(n, k)=local(A, B); A=matrix(1, 1); A[1, 1]=1; for(m=2, n+1, B=matrix(m, m); for(i=1, m, for(j=1, i, if(i<3|j==i|j>m-1, B[i, j]=1, if(j==1, B[i, 1]=1, B[i, j]=(A^(2*j-1))[i-j+1, 1])); )); A=B); (A^(2*k+2))[n-k+1, 1]}
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CROSSREFS
| Cf. A113351 (column 1), A113352 (column 2), A113353 (column 3), A113354 (column 4); A113355 (Q^2), A113365 (Q^3), A113340 (P), A113345 (P^2), A113360 (P^3).
Sequence in context: A110271 A073107 A103718 * A164678 A164679 A185023
Adjacent sequences: A113347 A113348 A113349 * A113351 A113352 A113353
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KEYWORD
| nonn,tabl
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Nov 08 2005
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