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, 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

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.: (1-3*x-3*x^2+9*x^3+3*x^4-3*x^5) / ( (3*x-1)*(1+x)*(x^2+x-1)*(x^2-3*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