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%I #41 Jan 07 2019 11:30:33
%S 1,3,7,19,51,141,393,1107,3139,8955,25675,73945,213825,620595,1807263,
%T 5279283,15465139,45420261,133708777,394446691,1165855131,3451793403,
%U 10235554347,30392965809,90357645121,268922897571,801139867063,2388683219347,7127469430899
%N a(0) = 1, a(n+1) = 3 * a(n) - F(n)*(F(n) + 1), where F(n) = A000045(n) is n-th Fibonacci number.
%D L. Euler, (E326) Observationes analyticae, reprinted in: Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 15, p. 59.
%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 575.
%D P. Henrici, Applied and Computational Complex Analysis. Wiley, NY, 3 vols., 1974-1986. (Vol. 1, p. 42.)
%D J. Riordan, Combinatorial Identities, Wiley, 1968, p. 74.
%D See also the references mentioned under A002426.
%H Reinhard Zumkeller, <a href="/A011769/b011769.txt">Table of n, a(n) for n = 0..1000</a>
%H R. K. Guy, <a href="http://www.jstor.org/stable/2691503">The Second Strong Law of Small Numbers</a>, Math. Mag, 63 (1990) 3-20, esp. 18-19.
%H V. E. Hoggatt, Jr. and M. Bicknell, <a href="http://www.fq.math.ca/Scanned/7-4/hoggatt-a.pdf">Diagonal sums of generalized Pascal triangles</a>, Fib. Quart., 7 (1969), 341-358, 393.
%H L. W. Shapiro et al., <a href="http://dx.doi.org/10.1016/0166-218X(91)90088-E">The Riordan group</a>, Discrete Applied Math., 34 (1991), 229-239.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-8,-8,14,4,-3).
%F 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]
%F 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
%F a(n+1) = (1/10) * (3^n + 2*Lucas(2n) + Lucas(n) + (-1)^n ). - _Ralf Stephan_, Aug 10 2013
%F 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
%p 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:
%p seq(A011769(n),n=0..40) ;
%t 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 *)
%o (Haskell)
%o a011769 n = a011769_list !! n
%o a011769_list = 1 : zipWith (-) (map (* 3) a011769_list) a059727_list
%o -- _Reinhard Zumkeller_, Dec 17 2011
%Y Cf. A002426.
%Y Cf. A059727.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, _R. K. Guy_
%E Values at n>=18 corrected by _R. J. Mathar_, Sep 04 2010