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Search: id:A098703
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| A098703 |
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a(n) = (3^n + phi^(n-1) + (-phi)^(1-n)) / 5, where phi denotes the golden ratio [A001622] (1 + sqrt(5))/2. |
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+0
5
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| 0, 1, 2, 6, 17, 50, 148, 441, 1318, 3946, 11825, 35454, 106328, 318929, 956698, 2869950, 8609617, 25828474, 77484812, 232453449, 697358750, 2092073666, 6276216817, 18828643686, 56485920112, 169457742625, 508373199218, 1525119551286
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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a(n) = sums of antidiagonals of A090888; a(n) = partial sums of A099159; a(n) = A000045(n) + A094688(n-1); for n > 2, a(n) = 3a(n-1) - A000045(n-3); for n > 3, a(n) = 3^2a(n-2) - A000285(n-4); for n > 4, a(n) = 3^3a(n-3) - A022383(n-5); lim n -> inf a(n) / a(n-1) = 3.
a(n) = A101220(1,3,n) - Ross La Haye (rlahaye(AT)new.rr.com), Dec 15 2004
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LINKS
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Eric Weisstein, Golden Ratio
Eric Weisstein, Lucas Number
Eric Weisstein, Fibonacci Number
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FORMULA
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a(n) = (((1 + SQRT(5))^n - (1 - SQRT(5))^n) / (2^nSQRT(5))) + ((3^n - (((1 + SQRT(5)) / 2)^(n+1) + ((1 - SQRT(5)) / 2)^(n+1))) / 5); a(n) = (3^n + (((1 + SQRT(5)) / 2)^(n-1) + ((1 - SQRT(5)) / 2)^(n-1))) / 5.
Let Luc(n) denote the n-th Lucas number [A000032] and Fib(n) denote the n-th Fibonacci number [A000045]. Then a(n) = (3^n + Luc(n-1)) / 5; a(n) = Fib(n) + ((3^n - Luc(n+1)) / 5); a(n) = (3^n + Fib(n) + Fib(n-2)) / 5; a(n) = (3^n + 4Fib(n) - Fib(n+2)) / 5; a(n) = Sum[(Fib(k)3^(n-k-1)) - (Fib(k-2)2^(n-k-1)), {k=0...n-1}]... and so on.
a(n) = 4a(n-1) - 2a(n-2) - 3a(n-3).
Binomial transform of unsigned A084178. Binomial transform of signed A084178 gives the Fibonacci oblongs [A001654]. - Ross La Haye (rlahaye(AT)new.rr.com), Dec 21 2004
G.f. = (x(1-2x))/((-1+3x)(-1+x+x^2)). - Ross La Haye (rlahaye(AT)new.rr.com), Aug 09 2005
a(0) = 0, a(1) = 1, a(n) = a(n-1) + a(n-2) + 3^(n-2) for n > 1. - Ross La Haye (rlahaye(AT)new.rr.com), Aug 20 2005
Binomial transform of A052964 beginning 0,1,0,3,1,10... - Ross La Haye (rlahaye(AT)new.rr.com), May 31 2006
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EXAMPLE
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a(2) = 2 because 3^2 = 9, Luc(1) = 1 and (9 + 1) / 5 = 2.
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MATHEMATICA
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f[n_] := (3^n + Fibonacci[n] + Fibonacci[n - 2])/5; Table[ f[n], {n, 0, 27}] (from Robert G. Wilson v Nov 04 2004)
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CROSSREFS
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Cf. A001622, A000032, A000045, A090888, A099159, A094688, A000285, A022383, A000244.
Sequence in context: A148445 A148446 A027914 this_sequence A025272 A148447 A148448
Adjacent sequences: A098700 A098701 A098702 this_sequence A098704 A098705 A098706
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KEYWORD
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nonn
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AUTHOR
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Ross La Haye (rlahaye(AT)new.rr.com), Oct 27 2004
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EXTENSIONS
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More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Nov 04 2004
More terms from Ross La Haye (rlahaye(AT)new.rr.com), Dec 21 2004
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