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Search: id:A083857
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| A083857 |
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Square array of binomial transforms of generalized Fibonacci numbers, read by antidiagonals. |
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+0
5
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| 0, 0, 1, 0, 1, 3, 0, 1, 3, 7, 0, 1, 3, 8, 15, 0, 1, 3, 9, 21, 31, 0, 1, 3, 10, 27, 55, 63, 0, 1, 3, 11, 33, 81, 144, 127, 0, 1, 3, 12, 39, 109, 243, 377, 255, 0, 1, 3, 13, 45, 139, 360, 729, 987, 511, 0, 1, 3, 14, 51, 171, 495, 1189, 2187, 2584, 1023, 0, 1, 3, 15, 57, 205, 648
(list; table; graph; listen)
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OFFSET
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0,6
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COMMENT
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Array gives solutions to the recurrences a(n)=3a(n-1)+ka(n-2), a(0)=0, a(1)=1, k>=-2. These are the binomial transforms of the generalized Fibonacci numbers A083856. Rows include A000225, A001906, A000244, A006190, A007482, A030195, A015521.
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FORMULA
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T(n, k)=((3+sqrt(4k+9))/2)^n/sqrt(4k+9)-((3-sqrt(4k+9))/2)^n/sqrt(4k+9), k>=-2
O.g.f. row k: -x/(-1+3*x+k*x^2) . - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 23 2007
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EXAMPLE
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Rows begin
0 1 3 7 15 31 ...
0 1 3 8 21 55 ...
0 1 3 9 27 81 ...
0 1 3 10 33 109 ...
0 1 3 11 39 139 ...
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CROSSREFS
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Sequence in context: A127158 A112367 A035623 this_sequence A115142 A048963 A119458
Adjacent sequences: A083854 A083855 A083856 this_sequence A083858 A083859 A083860
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KEYWORD
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easy,nonn,tabl
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AUTHOR
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Paul Barry (pbarry(AT)wit.ie), May 06 2003
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