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A007574 Patterns in a dual ring.
(Formerly M2653)
1
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; text; internal format)
OFFSET

1,2

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).

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (4, -4, -2, 4, 0, -1).

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). [R. J. Mathar, Feb 06 2010]

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;

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] (* Vladimir Joseph Stephan Orlovsky, Feb 08 2012 *)

CoefficientList[Series[-(- 1 + x + x^2 - x^3 - x^4 + 2 x^5) / ((x - 1)^2 (x^2 + x - 1)^2), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 09 2013 *)

PROG

(MAGMA) I:=[1, 3, 7, 15, 31, 60]; [n le 6 select I[n] else 4*Self(n-1)-4*Self(n-2)-2*Self(n-3)+4*Self(n-4)-Self(n-6): n in [1..50]]; // Vincenzo Librandi, Jun 09 2013

CROSSREFS

Cf. A000045.

Sequence in context: A023424 A276647 A006778 * A034480 A218281 A057703

Adjacent sequences:  A007571 A007572 A007573 * A007575 A007576 A007577

KEYWORD

nonn,easy

AUTHOR

Simon Plouffe and N. J. A. Sloane, Robert G. Wilson v

STATUS

approved

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Last modified May 25 05:56 EDT 2017. Contains 287012 sequences.