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A007574
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Patterns in a dual ring.
(Formerly M2653)
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1
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1, 3, 7, 15, 31, 60, 113, 207, 373, 663, 1167, 2038, 3537, 6107, 10499, 17983, 30703, 52272, 88769, 150407, 254321, 429223, 723167, 1216490, 2043361, 3427635, 5742463, 9609327, 16062463, 26821668, 44744657, 74576703, 124192237, 206650167, 343594479
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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REFERENCES
| C. A. Church, Jr., Lattice paths and Fibonacci and Lucas numbers, Fibonacci Quarterly 12(4) (1974) 336-338.
W. Dotson, F. Norwood and C. Taylor, Fiber optics and Fibonacci, Math. Mag., 66 (1993), 167-174.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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FORMULA
| a(n)= 4*a(n-1) -4*a(n-2) -2*a(n-3) +4*a(n-4) -a(n-6). G.f.: -x*(-1+x+x^2-x^3-x^4+2*x^5)/ ((x-1)^2 * (x^2+x-1)^2). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 06 2010]
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MAPLE
| with(combinat): A007574 := proc(n) local k; if n=1 then RETURN(1) fi; if n=2 then RETURN(3) fi; if n=3 then RETURN(7) fi; if n>3 then RETURN( fibonacci(n)+2*fibonacci(n-1)+n*sum(fibonacci(n-k), k=2..n-1)) fi; end;
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MATHEMATICA
| Table[ Fibonacci[n] + 2 Fibonacci[n - 1] + n*Sum[Fibonacci[n - k], {k, 2, n - 1}], {n, 1, 35} ]
LinearRecurrence[{4, -4, -2, 4, 0, -1}, {1, 3, 7, 15, 31, 60}, 60] (* From Vladimir Joseph Stephan Orlovsky, Feb 08 2012 *)
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CROSSREFS
| Cf. A000045.
Sequence in context: A151338 A023424 A006778 * A034480 A057703 A006739
Adjacent sequences: A007571 A007572 A007573 * A007575 A007576 A007577
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KEYWORD
| nonn,changed
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AUTHOR
| Simon Plouffe, N. J. A. Sloane (njas(AT)research.att.com), Robert G. Wilson v (rgwv(AT)rgwv.com)
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