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Search: id:A106196
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| A106196 |
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Triangle read by rows, generated from Pascal's triangle. |
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+0
1
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| 1, 1, 1, 2, 2, 1, 3, 5, 3, 1, 5, 10, 8, 4, 1, 8, 20, 17, 11, 5, 1, 13, 38, 35, 24, 14, 6, 1, 21, 71, 68, 50, 31, 17, 7, 1
(list; table; graph; listen)
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OFFSET
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0,4
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COMMENT
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The array P =
1, 0, 0, 0, 0, 0,...
0, 1, 0, 0, 0, 0,...
0, 1, 1, 0, 0, 0,...
0, 0, 2, 1, 0, 0,...
0, 0, 1, 3, 1, 0,...
0, 0, 0, 3, 4, 1,...
...
...as shown on page 107 of "A Primer for the Fibonacci Numbers".
The array A is composed of arithmetic sequences, as a matrix.
1, 1, 1, 1, 1,...
1, 2, 3, 4, 5,...
1, 3, 5, 7, 9,...
1, 4, 7, 10, 13...
1, 5, 9, 13, 17...
...
Leftmost column = Fibonacci numbers, next column (1, 2, 5, 10, 20...) = Fibonacci numbers convolved with themselves.
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REFERENCES
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V. E. Hoggatt, Jr., editor; "A Primer for the Fibonacci Numbers", 1963, p. 107.
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FORMULA
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Let P = an array with columns comprised of Pascal's Triangle rows, offset, spaces filled in with zeros; A = an array composed of arithmetic sequences.(n, k). Perform P * A and extract antidiagonals which become the rows of A106196.
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EXAMPLE
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The operation P * A generates the array:
1, 1, 1, 1, 1,...
1, 2, 3, 4, 5...
2, 5, 8, 11, 14...
3, 10, 17, 24, 31...
5, 20, 35, 50, 65...
...; from which we extract antidiagonals, read by rows, become triangle A106196:
1;
1, 1;
2, 2, 1;
3, 5, 3, 1;
5, 10, 8, 4, 1;
8, 20, 17, 11, 5, 1;
13, 38, 35, 24, 14, 6, 1;
21, 71, 68, 50, 31, 17, 7, 1;
...
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CROSSREFS
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Cf. A052996, A007678, A106196.
Sequence in context: A106179 A081572 A144287 this_sequence A037027 A139375 A106198
Adjacent sequences: A106193 A106194 A106195 this_sequence A106197 A106198 A106199
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KEYWORD
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nonn,tabl
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AUTHOR
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Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 24 2005
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