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A007598 Squared Fibonacci numbers: a(n) = F(n)^2 where F = A000045.

%I A007598 M3364 #271 May 31 2025 05:19:20
%S A007598 0,1,1,4,9,25,64,169,441,1156,3025,7921,20736,54289,142129,372100,
%T A007598 974169,2550409,6677056,17480761,45765225,119814916,313679521,
%U A007598 821223649,2149991424,5628750625,14736260449,38580030724,101003831721,264431464441,692290561600
%N A007598 Squared Fibonacci numbers: a(n) = F(n)^2 where F = A000045.
%C A007598 a(n)*(-1)^(n+1) = (2*(1-T(n,-3/2))/5), n>=0, with Chebyshev's polynomials T(n,x) of the first kind, is the r=-1 member of the r-family of sequences S_r(n) defined in A092184 where more information can be found. - _Wolfdieter Lang_, Oct 18 2004
%C A007598 From _Giorgio Balzarotti_, Mar 11 2009: (Start)
%C A007598 Determinant of power series with alternate signs of gamma matrix with determinant 1!.
%C A007598 a(n) = Determinant(A - A^2 + A^3 - A^4 + A^5 - ... - (-1)^n*A^n) where A is the submatrix A(1..2,1..2) of the matrix with factorial determinant.
%C A007598 A = [[1,1,1,1,1,1,...], [1,2,1,2,1,2,...], [1,2,3,1,2,3,...], [1,2,3,4,1,2,...], [1,2,3,4,5,1,...], [1,2,3,4,5,6,...], ...]; note: Determinant A(1..n,1..n) = (n-1)!.
%C A007598 a(n) is even with respect to signs of power of A.
%C A007598 See A158039...A158050 for sequence with matrix 2!, 3!, ... (End)
%C A007598 Equals the INVERT transform of (1, 3, 2, 2, 2, ...). Example: a(7) = 169 = (1, 1, 4, 9, 25, 64) dot (2, 2, 2, 2, 3, 1) = (2 + 2 + 8 + 18 + 75 + 64). - _Gary W. Adamson_, Apr 27 2009
%C A007598 This is a divisibility sequence.
%C A007598 a(n+1)*(-1)^n, n>=0, is the sequence of the alternating row sums of the Riordan triangle A158454. - _Wolfdieter Lang_, Dec 18 2010
%C A007598 a(n+1) is the number of tilings of a 2 X 2n rectangle with n tetrominoes of any shape, cf. A230031. - _Alois P. Heinz_, Nov 29 2013
%C A007598 This is the case P1 = 1, P2 = -6, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - _Peter Bala_, Mar 31 2014
%C A007598 Differences between successive golden rectangle numbers A001654. - _Jonathan Sondow_, Nov 05 2015
%C A007598 a(n+1) is the number of 2 X n matrices that can be obtained from a 2 X n matrix by moving each element to an adjacent position, horizontally or vertically. This is because F(n+1) is the number of domino tilings of that matrix, therefore with a checkerboard coloring and two domino tilings we can move the black element of each domino of the first tiling to the white element of the same domino and similarly move the white element of each domino of the second tiling to the black element of the same domino. - _Fabio Visonà_, May 04 2022
%C A007598 In general, squaring the terms of a second-order linear recurrence with signature (c,d) will result in a third-order linear recurrence with signature (c^2+d,(c^2+d)*d,-d^3). - _Gary Detlefs_, Jan 05 2023
%D A007598 Arthur T. Benjamin and Jennifer J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 8.
%D A007598 Ross Honsberger, Mathematical Gems III, M.A.A., 1985, p. 130.
%D A007598 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A007598 Richard P. Stanley, Enumerative Combinatorics I, Example 4.7.14, p. 251.
%H A007598 Indranil Ghosh, <a href="/A007598/b007598.txt">Table of n, a(n) for n = 0..2389</a> (terms 0..200 from T. D. Noe)
%H A007598 Mohammad K. Azarian, <a href="http://www.m-hikari.com/ijcms/ijcms-2012/37-40-2012/azarianIJCMS37-40-2012.pdf">Fibonacci Identities as Binomial Sums</a>, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 38, 2012, pp. 1871-1876.
%H A007598 Mohammad K. Azarian, <a href="http://www.m-hikari.com/ijcms/ijcms-2012/41-44-2012/azarianIJCMS41-44-2012.pdf">Fibonacci Identities as Binomial Sums II</a>, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 42, 2012, pp. 2053-2059.
%H A007598 R. C. Alperin, <a href="https://www.fq.math.ca/Papers/57-4/alperin07132019.pdf">A family of nonlinear recurrences and their linear solutions</a>, Fib. Q., 57:4 (2019), 318-321.
%H A007598 Paul S. Bruckman, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Problems/elementary45-2.pdf">Problem B-1023: And a cubic as a sum of two squares</a>, Fibonacci Quarterly, Vol. 45, Number 2; May 2007; p. 186.
%H A007598 Andrej Dujella, <a href="http://dx.doi.org/10.1016/S0012-365X(98)00341-0">A bijective proof of Riordan's theorem on powers of Fibonacci numbers</a>, Discrete Math. 199 (1999), no. 1-3, 217--220. MR1675924 (99k:05016).
%H A007598 Sergio Falcon, <a href="https://ikprress.org/index.php/AJOMCOR/article/view/442">Some series of reciprocal k-Fibonacci numbers</a>, Asian Journal of Mathematics and Computer Research, Vol. 11, No. 3 (2016), pp. 184-191; <a href="https://www.researchgate.net/publication/297715665_SOME_SERIES_OF_RECIPROCAL_k-FIBONACCI_NUMBERS">ResearchGate link</a>.
%H A007598 Dominique Foata and Guo-Niu Han, <a href="https://irma.math.unistra.fr/~foata/paper/pub71.html">Nombres de Fibonacci et polynômes orthogonaux</a>, in: Marcello Morelli and Marco Tangheroni (eds.), Leonardo Fibonacci : il tempo, le opere, l'ereditá scientifica, Pisa, 23-25 Marzo 1994, Pisa, Pacini Editore, 1994, pp. 179-200.
%H A007598 Svenja Huntemann and Neil A. McKay, <a href="https://arxiv.org/abs/1909.12419">Counting Domineering Positions</a>, arXiv:1909.12419 [math.CO], 2019.
%H A007598 Jong Hyun Kim, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Kim/kim18.html">Hadamard products and tilings</a>, JIS 12 (2009) 09.7.4.
%H A007598 Toufik Mansour, <a href="https://arxiv.org/abs/math/0302015">A note on sum of k-th power of Horadam's sequence</a>, arXiv:math/0302015 [math.CO], 2003.
%H A007598 Toufik Mansour, <a href="https://arxiv.org/abs/math/0303138">Squaring the terms of an l-th order linear recurrence</a>, arXiv:math/0303138 [math.CO], 2003.
%H A007598 Hilary I. Okagbue, Muminu O. Adamu, Sheila A. Bishop, and Abiodun A. Opanuga, <a href="https://www.ripublication.com/ijaer16/ijaerv11n6_150.pdf">Digit and Iterative Digit Sum of Fibonacci numbers, their identities and powers</a>, International Journal of Applied Engineering Research ISSN 0973-4562 Volume 11, Number 6 (2016) pp 4623-4627.
%H A007598 Pantelimon Stanica, <a href="https://arxiv.org/abs/math/0010149">Generating functions, weighted and non-weighted sums of powers of second-order recurrence sequences</a>, arXiv:math/0010149 [math.CO], 2000.
%H A007598 H. C. Williams and R. K. Guy, <a href="http://dx.doi.org/10.1142/S1793042111004587">Some fourth-order linear divisibility sequences</a>, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
%H A007598 H. C. Williams and R. K. Guy, <a href="http://www.emis.de/journals/INTEGERS/papers/a17self/a17self.Abstract.html">Some Monoapparitic Fourth Order Linear Divisibility Sequences</a> Integers, Volume 12A (2012) The John Selfridge Memorial Volume.
%H A007598 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-1).
%H A007598 <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>.
%H A007598 <a href="/index/Di#divseq">Index to divisibility sequences</a>.
%F A007598 G.f.: x*(1-x)/((1+x)*(1-3*x+x^2)).
%F A007598 a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3), n > 2. a(0)=0, a(1)=1, a(2)=1.
%F A007598 a(-n) = a(n) for all n in Z.
%F A007598 a(n) = A080097(n-2) + 1.
%F A007598 L.g.f.: 1/5*log((1+3*x+x^2)/(1-6*x+x^2)) = Sum_{n>=0} a(n)/n*x^n; special case of l.g.f. given in A079291. - _Joerg Arndt_, Apr 13 2011
%F A007598 a(0) = 0, a(1) = 1; a(n) = a(n-1) + Sum(a(n-i)) + k, 0 <= i < n where k = 1 when n is odd, or k = -1 when n is even. E.g., a(2) = 1 = 1 + (1 + 1 + 0) - 1, a(3) = 4 = 1 + (1 + 1 + 0) + 1, a(4) = 9 = 4 + (4 + 1 + 1 + 0) - 1, a(5) = 25 = 9 + (9 + 4 + 1 + 1 + 0) + 1. - Sadrul Habib Chowdhury (adil040(AT)yahoo.com), Mar 02 2004
%F A007598 a(n) = (2*Fibonacci(2*n+1) - Fibonacci(2*n) - 2*(-1)^n)/5. - _Ralf Stephan_, May 14 2004
%F A007598 a(n) = F(n-1)*F(n+1) - (-1)^n = A059929(n-1) - A033999(n).
%F A007598 Sum_{j=0..2*n} binomial(2*n,j)*a(j) = 5^(n-1)*A005248(n+1) for n >= 1 [P. Stanica]. Sum_{j=0..2*n+1} binomial(2*n+1,j)*a(j) = 5^n*A001519(n+1) [P. Stanica]. - _R. J. Mathar_, Oct 16 2006
%F A007598 a(n) = (A005248(n) - 2*(-1)^n)/5. - _R. J. Mathar_, Sep 12 2010
%F A007598 a(n) = (-1)^k*(Fibonacci(n+k)^2-Fibonacci(k)*Fibonacci(2*n+k)), for any k. - _Gary Detlefs_, Dec 13 2010
%F A007598 a(n) = 3*a(n-1) - a(n-2) + 2*(-1)^(n+1), n > 1. - _Gary Detlefs_, Dec 20 2010
%F A007598 a(n) = Fibonacci(2*n-2) + a(n-2). - _Gary Detlefs_, Dec 20 2010
%F A007598 a(n) = (Fibonacci(3*n) - 3*(-1)^n*Fibonacci(n))/(5*Fibonacci(n)), n > 0. - _Gary Detlefs_, Dec 20 2010
%F A007598 a(n) = (Fibonacci(n)*Fibonacci(n+4) - 3*Fibonacci(n)*Fibonacci(n+1))/2. - _Gary Detlefs_, Jan 17 2011
%F A007598 a(n) = (((3+sqrt(5))/2)^n + ((3-sqrt(5))/2)^n - 2*(-1)^n)/5; without leading zero we would have a(n) = ((3+sqrt(5))*((3+sqrt(5))/2)^n + (3-sqrt(5))*((3-sqrt(5))/2)^n + 4*(-1)^n)/10. - _Tim Monahan_, Jul 17 2011
%F A007598 E.g.f.: (exp((phi+1)*x) + exp((2-phi)*x) - 2*exp(-x))/5, with the golden section phi:=(1+sqrt(5))/2. From the Binet-de Moivre formula for F(n). - _Wolfdieter Lang_, Jan 13 2012
%F A007598 Starting with "1" = triangle A059260 * the Fibonacci sequence as a vector. - _Gary W. Adamson_, Mar 06 2012
%F A007598 a(0) = 0, a(1) = 1; a(n+1) = (a(n)^(1/2) + a(n-1)^(1/2))^2. - _Thomas Ordowski_, Jan 06 2013
%F A007598 a(n) + a(n-1) = A001519(n), n > 0. - _R. J. Mathar_, Mar 19 2014
%F A007598 From _Peter Bala_, Mar 31 2014: (Start)
%F A007598 a(n) = ( T(n,alpha) - T(n,beta) )/(alpha - beta), where alpha = 3/2 and beta = -1 and T(n,x) denotes the Chebyshev polynomial of the first kind.
%F A007598 a(n) = the bottom left entry of the 2 X 2 matrix T(n, M), where M is the 2 X 2 matrix [0, 3/2; 1, 1/2].
%F A007598 a(n) = U(n-1,i/2)*U(n-1,-i/2), where U(n,x) denotes the Chebyshev polynomial of the second kind.
%F A007598 See the remarks in A100047 for the general connection between Chebyshev polynomials and 4th-order linear divisibility sequences. (End)
%F A007598 a(n) = (F(n+2)*F(n+3) - L(n)*L(n+1))/3 for F = A000045 and L = A000032. - _J. M. Bergot_, Jun 02 2014
%F A007598 0 = a(n)*(+a(n) - 2*a(n+1) - 2*a(n+2)) + a(n+1)*(+a(n+1) - 2*a(n+2)) + a(n+2)*(+a(n+2)) for all n in Z. - _Michael Somos_, Jun 03 2014
%F A007598 (F(n)*b(n+2))^2 + (F(n+1)*b(n-1))^2 = F(2*n+1)^3 = A001519(n+1)^3, with b(n) = a(n) + 2*(-1)^n and F(n) = A000045(n) (see Bruckman link). - _Michel Marcus_, Jan 24 2015
%F A007598 a(n) = 1/4*( a(n-2) - a(n-1) - a(n+1) + a(n+2) ). The same recurrence holds for A001254. - _Peter Bala_, Aug 18 2015
%F A007598 a(n) = F(n)*F(n+1) - F(n-1)*F(n). - _Jonathan Sondow_, Nov 05 2015
%F A007598 For n>2, a(n) = F(n-2)*(3*F(n-1) + F(n-3)) + F(2*n-5). Also, for n>2 a(n)=2*F(n-3)*F(n) + F(2*n-3) -(2)*(-1)^n. - _J. M. Bergot_, Nov 05 2015
%F A007598 a(n) = (F(n+2)^2 + L(n+1)^2) - 2*F(n+2)*L(n+1). - _J. M. Bergot_, Nov 08 2015
%F A007598 a(n) = F(n+3)^2 - 4*F(n+1)*F(n+2). - _J. M. Bergot_, Mar 17 2016
%F A007598 a(n) = (F(n-2)*F(n+2) + F(n-1)*F(n+1))/2. - _J. M. Bergot_, May 25 2017
%F A007598 4*a(n) = L(n+1)*L(n-1) - F(n+2)*F(n-2), where L = A000032. - _Bruno Berselli_, Sep 27 2017
%F A007598 a(n) = F(n+k)*F(n-k) + (-1)^(n+k)*a(k), for every integer k >= 0. - _Federico Provvedi_, Dec 10 2018
%F A007598 From _Peter Bala_, Nov 19 2019: (Start)
%F A007598 Sum_{n >= 3} 1/(a(n) - 1/a(n)) = 4/9.
%F A007598 Sum_{n >= 3} (-1)^n/(a(n) - 1/a(n)) =  (10 - 3*sqrt(5))/18.
%F A007598 Conjecture: Sum_{n >= 1, n != 2*k+1} 1/(a(n) + (-1)^n*a(2*k+1)) = 1/a(4*k+2) for k = 0,1,2,.... (End)
%F A007598 Sum_{n>=1} 1/a(n) = A105393. - _Amiram Eldar_, Oct 22 2020
%F A007598 Product_{n>=2} (1 + (-1)^n/a(n)) = phi (A001622) (Falcon, 2016, p. 189, eq. (3.1)). - _Amiram Eldar_, Dec 03 2024
%e A007598 G.f. = x + x^2 + 4*x^3 + 9*x^4 + 25*x^5 + 64*x^6 + 169*x^7 + 441*x^8 + ...
%p A007598 with(combinat): seq(fibonacci(n)^2, n=0..27); # _Zerinvary Lajos_, Sep 21 2007
%t A007598 f[n_] := Fibonacci[n]^2; Array[f, 4!, 0] (* _Vladimir Joseph Stephan Orlovsky_, Oct 25 2009 *)
%t A007598 LinearRecurrence[{2,2,-1},{0,1,1},41] (* _Harvey P. Dale_, May 18 2011 *)
%o A007598 (PARI) {a(n) = fibonacci(n)^2};
%o A007598 (PARI) concat(0, Vec(x*(1-x)/((1+x)*(1-3*x+x^2)) + O(x^30))) \\ _Altug Alkan_, Nov 06 2015
%o A007598 (Sage) [(fibonacci(n))^2 for n in range(0, 28)]# _Zerinvary Lajos_, May 15 2009
%o A007598 (Magma) [Fibonacci(n)^2: n in [0..30]]; // _Vincenzo Librandi_, Apr 14 2011
%o A007598 (Haskell)
%o A007598 a007598 = (^ 2) . a000045  -- _Reinhard Zumkeller_, Sep 01 2013
%o A007598 (Sage) [fibonacci(n)^2 for n in range(30)] # _G. C. Greubel_, Dec 10 2018
%o A007598 (GAP) List([0..30], n -> Fibonacci(n)^2); # _G. C. Greubel_, Dec 10 2018
%o A007598 (Python)
%o A007598 from sympy import fibonacci
%o A007598 def A007598(n): return fibonacci(n)**2 # _Chai Wah Wu_, Apr 14 2025
%Y A007598 Cf. A000032, A000045, A001254, A001622, A001654, A047946, A056570, A059260, A061646, A065885, A079291, A080097, A105393.
%Y A007598 Bisection of A006498 and A074677. First differences of A001654.
%Y A007598 Second row of array A103323.
%Y A007598 Half of A175395.
%K A007598 nonn,easy,nice
%O A007598 0,4
%A A007598 _N. J. A. Sloane_, _Robert G. Wilson v_