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

Author

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

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A002426.

Cf. A059727.

Programs

Haskell

a011769 n = a011769_list !! n
a011769_list = 1 : zipWith (-) (map (* 3) a011769_list) a059727_list
-- Reinhard Zumkeller, 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 *)

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

Extensions

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