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A011769 a(0) = 1, a(n+1) = 3 * a(n) - F(n)*(F(n) + 1), where F(n) = A000045(n) is n-th Fibonacci number. 1
1, 3, 7, 19, 51, 141, 393, 1107, 3139, 8955, 25675, 73945, 213825, 620595, 1807263, 5279283, 15465139, 45420261, 133708777, 394446691, 1165855131, 3451793403, 10235554347, 30392965809, 90357645121, 268922897571, 801139867063, 2388683219347, 7127469430899 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

REFERENCES

L. Euler, (E326) Observationes analyticae, reprinted in: Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 15, p. 59.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 575.

P. Henrici, Applied and Computational Complex Analysis. Wiley, NY, 3 vols., 1974-1986. (Vol. 1, p. 42.)

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 74.

See also the references mentioned under A002426.

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990) 3-20, esp. 18-19.

V. E. Hoggatt, Jr. and M. Bicknell, Diagonal sums of generalized Pascal triangles, Fib. Quart., 7 (1969), 341-358, 393.

L. W. Shapiro et al., The Riordan group, Discrete Applied Math., 34 (1991), 229-239.

Index entries for linear recurrences with constant coefficients, signature (6,-8,-8,14,4,-3).

FORMULA

a(n) = +6*a(n-1) -8*a(n-2) -8*a(n-3) +14*a(n-4) +4*a(n-5) -3*a(n-6). [R. J. Mathar, Sep 04 2010]

G.f.: S(x)=-(3*x^5-3*x^4-9*x^3+3*x^2+3*x-1)/(3*x^6-4*x^5-14*x^4+8*x^3+8*x^2-6*x+1). - Sergei N. Gladkovskii, Dec 16 2011

a(n+1) = (1/10) * (3^n + 2*Lucas(2n) + Lucas(n) + (-1)^n ). - Ralf Stephan, Aug 10 2013

a(k) = 3^(k+1)*x^k/10 + (-1)^(k+1)*x^k/10 + p^(k+1)*x^k/5 + (-q)^(k+1)*x^k/5 + p^(2*k+2)*x^k/5 + q^(2*k+2)*x^k/5 ; p=(sqrt(5)+1)/2 , q=(sqrt(5)-1)/2 . - Sergei N. Gladkovskii, Dec 17 2011

MAPLE

A011769 := proc(n) if n = 0 then 1; else 3*procname(n-1)-combinat[fibonacci](n-1)*(1+combinat[fibonacci](n-1)) ; end if; end proc:

seq(A011769(n), n=0..40) ;

MATHEMATICA

nxt[{n_, a_}]:=Module[{fib=Fibonacci[n]}, {n+1, 3a-fib(fib+1)}]; Transpose[ [ nxt, {0, 1}, 30]][[2]] (* or *) LinearRecurrence[{6, -8, -8, 14, 4, -3}, {1, 3, 7, 19, 51, 141}, 30] (* Harvey P. Dale, Jun 05 2015 *)

PROG

(Haskell)

a011769 n = a011769_list !! n

a011769_list = 1 : zipWith (-) (map (* 3) a011769_list) a059727_list

-- Reinhard Zumkeller, Dec 17 2011

CROSSREFS

Cf. A002426.

Cf. A059727.

Sequence in context: A052948 A026325 A002426 * A087432 A135052 A198305

Adjacent sequences:  A011766 A011767 A011768 * A011770 A011771 A011772

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, R. K. Guy

EXTENSIONS

Values at n>=18 corrected by R. J. Mathar, Sep 04 2010

STATUS

approved

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Last modified October 17 17:08 EDT 2018. Contains 316290 sequences. (Running on oeis4.)