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Search: id:A083861
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| A083861 |
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Square array of binomial transforms of generalized Fibonacci numbers, read by antidiagonals. |
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+0
2
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| 0, 0, 1, 0, 1, 5, 0, 1, 5, 19, 0, 1, 5, 20, 65, 0, 1, 5, 21, 75, 211, 0, 1, 5, 22, 85, 275, 665, 0, 1, 5, 23, 95, 341, 1000, 2059, 0, 1, 5, 24, 105, 409, 1365, 3625, 6305, 0, 1, 5, 25, 115, 479, 1760, 5461, 13125, 19171, 0, 1, 5, 26, 125, 551, 2185, 7573, 21845, 47500, 58025
(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)=5a(n-1)+ka(n-2),a(0)=0,a(1)=1,k>=-6. The rows are the binomial transforms of the rows of array A083857. The rows are the second binomial transforms of the generalized Fibonacci numbers in array A083856. Rows include A002450, A004254, A000351, A052918, A015535, A015536, A015537. The main diagonal is A083863.
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FORMULA
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T(n, k)=(((5+sqrt(4k+1))/2)^n-((5-sqrt(4k+1))/2)^n)/sqrt(4k+1)
O.g.f. for row k>=-6: -x/(-1+5*x+k*x^2) . - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Dec 02 2007
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EXAMPLE
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Rows begin
0 1 5 19 65 ...
0 1 5 20 75 ...
0 1 5 21 85 ...
0 1 5 22 95 ...
0 1 5 23 105 ...
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CROSSREFS
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Cf. A082297.
Sequence in context: A054672 A021670 A060081 this_sequence A097591 A164652 A127557
Adjacent sequences: A083858 A083859 A083860 this_sequence A083862 A083863 A083864
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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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