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A144152
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Triangle read by rows: A128174 * X; X = an infinite lower triangular matrix with a shifted Fibonacci sequence: (1, 1, 1, 2, 3, 5, 8, ...) in the main diagonal and the rest zeros.
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1
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1, 0, 1, 1, 0, 1, 0, 1, 0, 2, 1, 0, 1, 0, 3, 0, 1, 0, 2, 0, 5, 1, 0, 1, 0, 3, 0, 8, 0, 1, 0, 2, 0, 5, 0, 13, 1, 0, 1, 0, 3, 0, 8, 0, 21, 0, 1, 0, 2, 0, 0, 5, 0, 13, 0, 34, 1, 0, 1, 0, 3, 0, 8, 0, 21, 0, 55
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OFFSET
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1,10
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COMMENTS
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The original definition was: Eigentriangle, row sums = Fibonacci numbers.
Even n rows are composed of odd-indexed Fibonacci numbers interpolated with zeros.
Odd n rows are composed of even-indexed Fibonacci numbers with alternate zeros.
Sum of n-th row terms = rightmost term of next row, = F(n-1). Row sums = F(n).
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LINKS
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FORMULA
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A128174 = the matrix: (1; 0,1; 1,0,1; 0,1,0,1; ...). These operations are equivalent to termwise products of n terms of A128174 matrix row terms and an equal number of terms in (1, 1, 1, 2, 3, 5, 8, ...).
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EXAMPLE
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First few rows of the triangle =
1;
0, 1;
1, 0, 1;
0, 1, 0, 2;
1, 0, 1, 0, 3
0, 1, 0, 2, 0, 5;
1, 0, 1, 0, 3, 0, 8;
0, 1, 0, 2, 0, 5, 0, 13;
1, 0, 1, 0, 3, 0, 8, 0, 21;
...
Row 5 = (1, 0, 1, 0, 3) = termwise products of (1, 0, 1, 0, 1) and (1, 1, 1, 2, 3).
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PROG
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(PARI) MT(n, k) = (1+(-1)^(n-k))/2;
MF(n, k) = n--; k--; if (n==k, if (n==0, 1, fibonacci(n)), 0);
tabl(nn) = {my(T=matrix(nn, nn, n, k, MT(n, k))); my(F=matrix(nn, nn, n, k, MF(n, k))); my(P=T*F); matrix(nn, nn, n, k, if (n>=k, P[n, k], 0)); } \\ Michel Marcus, Mar 08 2021
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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Moved a comment to the Name section. - Omar E. Pol, Mar 08 2021
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STATUS
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approved
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