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A006478 a(n) = a(n-1) + a(n-2) + F(n) - 1, a(0) = a(1) = 0, where F() = Fibonacci numbers A000045.
(Formerly M2733)
12
1, 3, 8, 18, 38, 76, 147, 277, 512, 932, 1676, 2984, 5269, 9239, 16104, 27926, 48210, 82900, 142055, 242665, 413376, 702408, 1190808, 2014608, 3401833, 5734251, 9650312, 16216602, 27213182, 45608092, 76345851, 127656829, 213230144, 355817324, 593205284 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,2

COMMENTS

Partial sums of A001629.

Number of edges in the Fibonacci hypercube FQ(n-2) (defined in the Rispoli and Cosares reference). - Emeric Deutsch, Oct 06 2014

Circuit rank (cyclomatic number) of the n-Fibonacci cube graph. - Eric W. Weisstein, Sep 05 2017

REFERENCES

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 = 3..1000

K. J. Overholt, Efficiency of the Fibonacci search method, Nordisk Tidskr. Informationsbehandling (BIT) 13 (1973), 92-96.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

F. J. Rispoli and S. Cosares, The Fibonacci hypercube, Australasian J. Combinatorics, 40, 2008, 187-196.

Eric Weisstein's World of Mathematics, Circuit Rank

Eric Weisstein's World of Mathematics, Fibonacci Cube Graph

Index entries for linear recurrences with constant coefficients, signature (3,-1,-3,1,1).

FORMULA

a(n) = 1+((n-2)*F(n+2)+(3*n+1)*F(n+3))/5 (with offset 0).

G.f.: x^3/((1-x)*(1-x-x^2)^2).

a(n) = sum(k=0, n-1, sum(i=0, k, F(i)*F(k-i))). - Benoit Cloitre, Jan 26 2003

a(n) = A175722(-2-n). - Michael Somos, Mar 11 2014

a(n) = 3*a(n-1) - a(n-2) - 3*a(n-3) + a(n-4) + a(n-5). - Eric W. Weisstein, Sep 05 2017

EXAMPLE

G.f. = x^3 + 3*x^4 + 8*x^5 + 18*x^6 + 38*x^7 + 76*x^8 + 147*x^9 + 277*x^10 + ...

MAPLE

A006478:=-1/(z-1)/(z**2+z-1)**2; # Simon Plouffe in his 1992 dissertation with offset 0.

MATHEMATICA

CoefficientList[Series[1/((1 - x) (1 - x - x^2)^2), {x, 0, 50}], x] (* Vincenzo Librandi, Mar 13 2014 *)

LinearRecurrence[{3, -1, -3, 1, 1}, {0, 0, 1, 3, 8}, 20] (* Eric W. Weisstein, Sep 05 2017 *)

Table[1 + (2 (n + 1) Fibonacci[n] + n Fibonacci[n + 1])/5 - Fibonacci[n + 2], {n, 20}] (* Eric W. Weisstein, Sep 05 2017 *)

PROG

(PARI) {a(n) = if( n<0, polcoeff( x^2 / ((1 - x) * (1 + x - x^2)^2) + x * O(x^-n), -n), polcoeff( x^3 / ((1 - x) * (1 - x - x^2)^2) + x * O(x^n), n))}; /* Michael Somos, Mar 11 2014 */

(Haskell)

a006478 n = a006478_list !! (n-3)

a006478_list = scanl1 (+) $ drop 2 a001629_list

-- Reinhard Zumkeller, Sep 12 2015

CROSSREFS

Cf. A006479, A175722.

Cf. A001629.

Sequence in context: A078409 A036642 A000235 * A104187 A051633 A131051

Adjacent sequences:  A006475 A006476 A006477 * A006479 A006480 A006481

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified October 17 10:43 EDT 2017. Contains 293469 sequences.