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A006491 Generalized Lucas numbers.
(Formerly M3258)
3
1, 0, 4, 5, 15, 28, 60, 117, 230, 440, 834, 1560, 2891, 5310, 9680, 17527, 31545, 56468, 100590, 178395, 315106, 554530, 972564, 1700400, 2964325, 5153868, 8938300, 15465497, 26700915, 46004620, 79112304, 135801105, 232715006, 398151740 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

For n>2 note that (n+1)|a(n) unless n is prime, in which case (n+1)|2*a(n). This sequence is not the better-known generalized Lucas numbers V(n,a,b) defined for fixed integers a and b such that D = a^2 + 4*b is nonnegative, V(0) = 2, V(1) = a and for n>1 the recurrence V(n) = V(n-1) + V(n-2). The a = b = 1 case gives the Lucas Numbers. - Jonathan Vos Post, Mar 16 2005

Number of circular binary words of length n+1 having exactly two occurrences of 00. Example: a(4)=5 because we have 00011, 10001, 11000, 00110 and 01100. Column 2 of A119458. - Emeric Deutsch, May 20 2006

REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1000

L. Carlitz and R. Scoville, Zero-one sequences and Fibonacci numbers, Fibonacci Quarterly, 15 (1977), 246-254.

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.

Index entries for linear recurrences with constant coefficients, signature (3,0,-5,0,3,1).

FORMULA

G.f.: x*(1-x)*(1-2*x+2*x^2)/(1-x-x^2)^3. - Ralf Stephan, Apr 23 2004, corrected Feb 08 2006

a(n) = a(n-1) + a(n-2) + n*Fibonacci(n-2) - (n-1)*Fibonacci(n-3) for n >= 3; a(1)=1, a(2)=0. - Emeric Deutsch, May 20 2006

a(n) = 3*a(n-1) - 5*a(n-3) + 3*a(n-5) + a(n-6). - G. C. Greubel, Jan 01 2018

MAPLE

G:=x*(1-x)*(1-2*x+2*x^2)/(1-x-x^2)^3: Gser:=series(G, x=0, 45): seq(coeff(Gser, x^n), n=1..40); # Emeric Deutsch, Feb 07 2006

with(combinat): a[1]:=1: a[2]:=0: for n from 3 to 40 do a[n]:=a[n-1]+a[n-2]+n*fibonacci(n-2)-(n-1)*fibonacci(n-3) od: seq(a[n], n=1..40); # Emeric Deutsch, May 20 2006

A006491:=(z-1)*(1-2*z+2*z**2)/(z**2+z-1)**3; # conjectured by Simon Plouffe in his 1992 dissertation

MATHEMATICA

LinearRecurrence[{3, 0, -5, 0, 3, 1}, {1, 0, 4, 5, 15, 28}, 50] (* G. C. Greubel, Jan 01 2018 *)

PROG

(PARI) x='x+O('x^30); Vec(x*(1-x)*(1-2*x+2*x^2)/(1-x-x^2)^3) \\ G. C. Greubel, Jan 01 2018

(MAGMA) I:=[1, 0, 4, 5, 15, 28]; [n le 6 select I[n] else 3*Self(n-1) -5*Self(n-3) +3*Self(n-5)+Self(n-6): n in [1..30]]; // G. C. Greubel, Jan 01 2018

CROSSREFS

Cf. A006490, A119458.

Sequence in context: A026634 A026656 A184244 * A304921 A051721 A050226

Adjacent sequences:  A006488 A006489 A006490 * A006492 A006493 A006494

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Emeric Deutsch, Feb 07 2006

STATUS

approved

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Last modified October 16 12:35 EDT 2018. Contains 316263 sequences. (Running on oeis4.)