cp's OEIS Frontend

A001629 Self-convolution of Fibonacci numbers.

%I A001629 M1377 N0537 #419 Aug 04 2025 01:04:46
%S A001629 0,0,1,2,5,10,20,38,71,130,235,420,744,1308,2285,3970,6865,11822,
%T A001629 20284,34690,59155,100610,170711,289032,488400,823800,1387225,2332418,
%U A001629 3916061,6566290,10996580,18394910,30737759,51310978,85573315,142587180,237387960,394905492
%N A001629 Self-convolution of Fibonacci numbers.
%C A001629 Number of elements in all subsets of {1,2,...,n-1} with no consecutive integers. Example: a(5)=10 because the subsets of {1,2,3,4} that have no consecutive elements, i.e., {}, {1}, {2}, {3}, {4}, {1,3}, {1,4}, {2,4}, the total number of elements is 10. - _Emeric Deutsch_, Dec 10 2003
%C A001629 If g is either of the real solutions to x^2-x-1=0, g'=1-g is the other one and phi is any 2 X 2-matricial solution to the same equation, not of the form gI or g'I, then Sum'_{i+j=n-1} g^i phi^j = F_n + (A001629(n) - A001629(n-1)g')*(phi-g'I), where i,j >= 0, F_n is the n-th Fibonacci number and I is the 2 X 2 identity matrix... - Michele Dondi (blazar(AT)lcm.mi.infn.it), Apr 06 2004
%C A001629 Number of 3412-avoiding involutions containing exactly one subsequence of type 321.
%C A001629 Number of binary sequences of length n with exactly one pair of consecutive 1's. - George J. Schaeffer (gschaeff(AT)andrew.cmu.edu), Sep 02 2004
%C A001629 For this sequence the n-th term is given by (nF(n+1)-F(n)+nF(n-1))/5 where F(n) is the n-th Fibonacci number. - Mrs. J. P. Shiwalkar and M. N. Deshpande (dpratap_ngp(AT)sancharnet.in), Apr 20 2005
%C A001629 If an unbiased coin is tossed n times then there are 2^n possible strings of H and T. Out of these, number of strings with exactly one 'HH' is given by a(n) where a(n) denotes n-th term of this sequence. - Mrs. J. P. Shiwalkar and M. N. Deshpande (dpratap_ngp(AT)sancharnet.in), May 04 2005
%C A001629 a(n) is half the number of horizontal dominoes in all domino tilings of a horizontally aligned 2 X n rectangle; a(n+1) = the number of vertical dominoes in all domino tilings of a horizontally aligned 2 X n rectangle; thus 2*a(n)+a(n+1)=n*F(n+1) = the number of dominoes in all domino tilings of a 2 X n rectangle, where F=A000045, the Fibonacci sequence. - _Roberto Tauraso_, May 02 2005; _Graeme McRae_, Jun 02 2006
%C A001629 a(n+1) = (-i)^(n-1)*(d/dx)S(n,x)|_{x=i}, where i is the imaginary unit, n >= 1. First derivative of Chebyshev S-polynomials evaluated at x=i multiplied by (-i)^(n-1). See A049310 for the S-polynomials. - _Wolfdieter Lang_, Apr 04 2007
%C A001629 For n >= 4, a(n) is the number of weak compositions of n-2 in which exactly one part is 0 and all other parts are either 1 or 2. - _Milan Janjic_, Jun 28 2010
%C A001629 For n greater than 1, a(n) equals the absolute value of (1 - (1/2 - i/2)*(1 + (-1)^(n + 1))) times the x-coefficient of the characteristic polynomial of the (n-1) X (n-1) tridiagonal matrix with i's along the main diagonal (i is the imaginary unit), 1's along the superdiagonal and the subdiagonal and 0's everywhere else (see Mathematica code below). - _John M. Campbell_, Jun 23 2011
%C A001629 For n > 0: a(n) = Sum_{k=1..n-1} (A039913(n-1,k)) / 2. - _Reinhard Zumkeller_, Oct 07 2012
%C A001629 The right-hand side of a binomial-coefficient identity [Gauthier]. - _N. J. A. Sloane_, Apr 09 2013
%C A001629 a(n) is the number of edges in the Fibonacci cube Gamma(n-1) (see the Klavzar 2005 reference, p. 149). Example: a(3)=2; indeed, the Fibonacci cube Gamma(2) is the path P(3) having 2 edges. - _Emeric Deutsch_, Aug 10 2014
%C A001629 a(n) is the number of c(i)'s, including repetitions, in p(n), where p(n)/q(n) is the n-th convergent p(n)/q(n) of the formal infinite continued fraction [c(0), c(1), ...]; e.g., the number of c(i)'s in p(3) = c(0)*c(1)*c(2)*c(3) + c(0)*c(1) + c(0)*c(3) + c(2)*c(3) + 1 is a(5) = 10. - _Clark Kimberling_, Dec 23 2015
%C A001629 Also the number of maximal and maximum cliques in the (n-1)-Fibonacci cube graph. - _Eric W. Weisstein_, Sep 07 2017
%C A001629 a(n+1) is the total number of fixed points in all permutations p on 1, 2, ..., n such that |k-p(k)| <= 1 for 1 <= k <= n. - _Katharine Ahrens_, Sep 03 2019
%C A001629 From _Steven Finch_, Mar 22 2020: (Start)
%C A001629 a(n+1) is the total binary weight (cf. A000120) of all A000045(n+2) binary sequences of length n not containing any adjacent 1's.
%C A001629 The only three 2-bitstrings without adjacent 1's are 00, 01 and 10. The bitsums of these are 0, 1 and 1. Adding these give a(3)=2.
%C A001629 The only five 3-bitstrings without adjacent 1's are 000, 001, 010, 100 and 101. The bitsums of these are 0, 1, 1, 1 and 2. Adding these give a(4)=5.
%C A001629 The only eight 4-bitstrings without adjacent 1's are 0000, 0001, 0010, 0100, 1000, 0101, 1010 and 1001. The bitsums of these are 0, 1, 1, 1, 1, 2, 2, and 2. Adding these give a(5)=10. (End)
%C A001629 Number of tilings of a 1 X n strip with monominoes (1 X 1 squares) and at least one domino (1 X 2 rectangles), where exactly one of the dominoes is colored gold. - _Greg Dresden_ and Jiachen Weng, Jul 31 2025
%D A001629 Donald E. Knuth, Fundamental Algorithms, Addison-Wesley, 1968, p. 83, Eq. 1.2.8--(17). - _Don Knuth_, Feb 26 2019
%D A001629 Thomas Koshy, Fibonacci and Lucas Numbers with Applications, 2001, Chapter 15, page 187, "Hosoya's Triangle", and p. 375, eq. (32.13).
%D A001629 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 101.
%D A001629 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001629 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A001629 S. Vajda, Fibonacci and Lucas numbers and the Golden Section, Ellis Horwood Ltd., Chichester, 1989, p. 183, Nr.(98).
%H A001629 T. D. Noe, <a href="/A001629/b001629.txt">Table of n, a(n) for n = 0..500</a>
%H A001629 Marco Abrate, Stefano Barbero, Umberto Cerruti and Nadir Murru, <a href="http://dx.doi.org/10.1016/j.disc.2014.06.026">Colored compositions, Invert operator and elegant compositions with the "black tie"</a>, Discrete Math. 335 (2014), 1--7. MR3248794.
%H A001629 Katharine A. Ahrens, <a href="https://www.lib.ncsu.edu/resolver/1840.20/37364">Combinatorial Applications of the k-Fibonacci Numbers: A Cryptographically Motivated Analysis</a>, Ph. D. thesis, North Carolina State University (2020).
%H A001629 Carlos Alirio Rico Acevedo and Ana Paula Chaves, <a href="https://arxiv.org/abs/1903.07490">Double-Recurrence Fibonacci Numbers and Generalizations</a>, arXiv:1903.07490 [math.NT], 2019.
%H A001629 Marcella Anselmo, Giuseppa Castiglione, Manuela Flores, Dora Giammarresi, Maria Madonia, and Sabrina Mantaci, <a href="https://arxiv.org/abs/2303.09898">Hypercubes and Isometric Words based on Swap and Mismatch Distance</a>, arXiv:2303.09898 [math.CO], 2023.
%H A001629 Kassie Archer and Robert P. Laudone, <a href="https://arxiv.org/abs/2407.06338">Pattern avoidance and the fundamental bijection</a>, arXiv:2407.06338 [math.CO], 2024. See p. 6.
%H A001629 Ali Reza Ashrafi, Jernej Azarija, Khadijeh Fathalikhani, Sandi Klavžar and Marko Petkovšek, <a href="http://arxiv.org/abs/1407.4962">Vertex and edge orbits of Fibonacci and Lucas cubes</a>, arXiv:1407.4962 [math.CO], 2014. See Table 2.
%H A001629 Ali Reza Ashrafi, Jernej Azarija, Khadijeh Fathalikhani, Sandi Klavžar, et al., <a href="http://www.fmf.uni-lj.si/~klavzar/preprints/Fib-Luc-orbits-August-11-2014.pdf">Orbits of Fibonacci and Lucas cubes, dihedral transformations, and asymmetric strings</a>, 2014.
%H A001629 Jean-Luc Baril, Sergey Kirgizov and Vincent Vajnovszki, <a href="https://arxiv.org/abs/2010.09505">Gray codes for Fibonacci q-decreasing words</a>, arXiv:2010.09505 [cs.DM], 2020.
%H A001629 Daniel Birmajer, Juan Gil and Michael D. Weiner, <a href="http://arxiv.org/abs/1405.7727">Linear recurrence sequences and their convolutions via Bell polynomials</a>, arXiv:1405.7727 [math.CO], 2014.
%H A001629 Daniel Birmajer, Juan B. Gil and Michael D. Weiner, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Gil/gil3.html">Linear Recurrence Sequences and Their Convolutions via Bell Polynomials</a>, Journal of Integer Sequences, 18 (2015), #15.1.2.
%H A001629 Matthew Blair, Rigoberto Flórez and Antara Mukherjee, <a href="https://arxiv.org/abs/1808.05278">Matrices in the Hosoya triangle</a>, arXiv:1808.05278 [math.CO], 2018.
%H A001629 Matthew Blair, Rigoberto Flórez and Antara Mukherjee, <a href="https://arxiv.org/abs/2203.13205">Honeycombs in the Pascal triangle and beyond</a>, arXiv:2203.13205 [math.HO], 2022. See p. 5.
%H A001629 Peter J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/groups.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A001629 Giuseppa Castiglione, Antonio Restivo and Marinella Sciortino, <a href="https://doi.org/10.1016/j.tcs.2009.07.018">Circular Sturmian words and Hopcroft's algorithm</a>, Theor. Comput. Sci. 410, No. 43, 4372-4381 (2009)
%H A001629 Charles H. Conley and Valentin Ovsienko, <a href="https://arxiv.org/abs/2209.10426">Shadows of rationals and irrationals: supersymmetric continued fractions and the super modular group</a>, arXiv:2209.10426 [math-ph], 2022.
%H A001629 Éva Czabarka, Rigoberto Flórez and Leandro Junes, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Florez/florez12.html">A Discrete Convolution on the Generalized Hosoya Triangle</a>, Journal of Integer Sequences, 18 (2015), #15.1.6.
%H A001629 Eric S. Egge, <a href="https://arxiv.org/abs/math/0307050">Restricted 3412-Avoiding Involutions</a>, p. 16, arXiv:math/0307050 [math.CO], 2003.
%H A001629 Sergio Falcon, <a href="https://doi.org/10.7546/nntdm.2020.26.3.96-106">Half self-convolution of the k-Fibonacci sequence</a>, Notes on Number Theory and Discrete Mathematics (2020) Vol. 26, No. 3, 96-106.
%H A001629 Luca Ferrari and Emanuele Munarini, <a href="http://arxiv.org/abs/1203.6792">Enumeration of edges in some lattices of paths</a>, arXiv:1203.6792 [math.CO], 2012.
%H A001629 Steven Finch, <a href="https://arxiv.org/abs/2003.09458">Cantor-solus and Cantor-multus distributions</a>, arXiv:2003.09458 [math.CO], 2020.
%H A001629 Rigoberto Flórez, Robinson Higuita and Alexander Ramírez, <a href="https://arxiv.org/abs/1808.01264">The resultant, the discriminant, and the derivative of generalized Fibonacci polynomials</a>, arXiv:1808.01264 [math.NT], 2018.
%H A001629 Napoleon Gauthier (Proposer), <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Problems/November2012advanced.pdf">Problem H-703</a>, Fib. Quart., 50 (2012), 379-381.
%H A001629 Ricardo Gómez Aíza, <a href="https://arxiv.org/abs/2402.16111">Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis</a>, arXiv:2402.16111 [math.CO], 2024. See p. 23.
%H A001629 Martin Griffiths, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/48-2/Griffiths.pdf">Digit Proportions in Zeckendorf Representations</a>, Fibonacci Quart. 48 (2010), no. 2, 168-174.
%H A001629 Verner E. Hoggatt, Jr. and M. Bicknell-Johnson, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/15-2/hoggatt1.pdf">Fibonacci convolution sequences</a>, Fib. Quart., 15 (1977), 117-122.
%H A001629 Milan Janjić, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Janjic/janjic33.html">Hessenberg Matrices and Integer Sequences</a>, J. Int. Seq. 13 (2010) # 10.7.8, section 3.
%H A001629 Omar Khadir, László Németh, and László Szalay, <a href="https://doi.org/10.1007/s00025-024-02284-3">Tiling of dominoes with ranked colors</a>, Results in Math. (2024) Vol. 79, Art. No. 253. See p. 8.
%H A001629 Sandi Klavžar, <a href="https://doi.org/10.1016/j.disc.2004.02.023">On median nature and enumerative properties of Fibonacci-like cubes</a>, Disc. Math. 299 (2005), 145-153.
%H A001629 Sandi Klavžar, <a href="http://www.imfm.si/preprinti/PDF/01150.pdf">Structure of Fibonacci cubes: a survey</a>, Institute of Mathematics, Physics and Mechanics Jadranska 19, 1000 Ljubljana, Slovenia; Preprint series Vol. 49 (2011), 1150 ISSN 2232-2094. (See Section 4.)
%H A001629 Toufik Mansour, <a href="https://arxiv.org/abs/math/0301157">Generalization of some identities involving the Fibonacci numbers</a>, arXiv:math/0301157 [math.CO], 2003.
%H A001629 Toufik Mansour and Mark Shattuck, <a href="https://doi.org/10.47443/dml.2025.060">Pattern avoidance in flattened derangements</a>, Disc. Math. Lett. (2025) Vol. 15, 67-74. See p. 74.
%H A001629 Pieter Moree, <a href="https://arxiv.org/abs/math/0311205">Convoluted convolved Fibonacci numbers</a>, arXiv:math/0311205 [math.CO], 2003.
%H A001629 Pieter Moree, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL7/Moree/moree12.htm">Convoluted Convolved Fibonacci Numbers</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.2.
%H A001629 László Németh, <a href="https://arxiv.org/abs/2403.12159">Walks on tiled boards</a>, arXiv:2403.12159 [math.CO], 2024. See p. 2.
%H A001629 László Németh and László Szalay, <a href="https://arxiv.org/abs/2408.12196">Explicit solution of system of two higher-order recurrences</a>, arXiv:2408.12196 [math.NT], 2024. See p. 10.
%H A001629 Valentin Ovsienko, <a href="https://arxiv.org/abs/2111.02553">Shadow sequences of integers, from Fibonacci to Markov and back</a>, arXiv:2111.02553 [math.CO], 2021.
%H A001629 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H A001629 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H A001629 Mihai Prunescu and Lorenzo Sauras-Altuzarra, <a href="https://arxiv.org/abs/2405.04083">On the representation of C-recursive integer sequences by arithmetic terms</a>, arXiv:2405.04083 [math.LO], 2024. See p. 18.
%H A001629 Jeffrey B. Remmel and J. L. B. Tiefenbruck, <a href="https://ajc.maths.uq.edu.au/pdf/64/ajc_v64_p166.pdf">Q-analogues of convolutions of Fibonacci numbers</a>, Australasian Journal of Combinatorics, Volume 64(1) (2016), Pages 166-193.
%H A001629 J. Riordan, <a href="/A001820/a001820.pdf">Notes to N. J. A. Sloane, Jul. 1968</a>
%H A001629 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EdgeCount.html">Edge Count</a>
%H A001629 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FibonacciCubeGraph.html">Fibonacci Cube Graph</a>
%H A001629 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EdgeCount.html">Edge Count</a>
%H A001629 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MaximalClique.html">Maximal Clique</a>
%H A001629 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MaximumClique.html">Maximum Clique</a>
%H A001629 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-2,-1).
%F A001629 G.f.: x^2/(1 - x - x^2)^2. - _Simon Plouffe_ in his 1992 dissertation
%F A001629 a(n) = A037027(n-1, 1), n >= 1 (Fibonacci convolution triangle).
%F A001629 a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4), n > 3.
%F A001629 a(n) = Sum_{k=0..n} A000045(k)*A000045(n-k).
%F A001629 a(n+1) = Sum_{i=0..F(n)} A007895(i), where F = A000045, the Fibonacci sequence. - Claude Lenormand (claude.lenormand(AT)free.fr), Feb 04 2001
%F A001629 a(n) = Sum_{k=0..floor(n/2)-1} (k+1)*binomial(n-k-1, k+1). - _Emeric Deutsch_, Nov 15 2001
%F A001629 a(n) = floor( (1/5)*(n - 1/sqrt(5))*phi^n + 1/2 ) where phi=(1+sqrt(5))/2 is the golden ratio. - _Benoit Cloitre_, Jan 05 2003
%F A001629 a(n) = a(n-1) + A010049(n-1) for n > 0. - _Emeric Deutsch_, Dec 10 2003
%F A001629 a(n) = Sum_{k=0..floor((n-2)/2)} (n-k-1)*binomial(n-k-2, k). - _Paul Barry_, Jan 25 2005
%F A001629 a(n) = ((n-1)*F(n) + 2*n*F(n-1))/5, F(n)=A000045(n) (see Vajda and Koshy reference).
%F A001629 F'(n, 1), the first derivative of the n-th Fibonacci polynomial evaluated at 1. - _T. D. Noe_, Jan 18 2006
%F A001629 a(n) = a(n-1) + a(n-2) + F(n-1), where F=A000045, the Fibonacci sequence. - _Graeme McRae_, Jun 02 2006
%F A001629 a(n) = (1/5)*(n-1/sqrt(5))*((1+sqrt(5))/2)^n + (1/5)*(n+1/sqrt(5))*((1-sqrt(5))/2)^n. - _Graeme McRae_, Jun 02 2006
%F A001629 a(n) = A055244(n-1) - F(n-2). Example: a(6) = 20 = A055244(5) - F(3) = (23 - 3). - _Gary W. Adamson_, Jul 27 2007
%F A001629 a(n) = term (1,3) in the 4 X 4 matrix [2,1,0,0; 1,0,1,0; -2,0,0,1; -1,0,0,0]^n. - _Alois P. Heinz_, Aug 01 2008
%F A001629 a(n) = A214178(n,1) for n > 0. - _Reinhard Zumkeller_, Jul 08 2012
%F A001629 a(n) = ((n+1)*F(n-1) + (n-1)*F(n+1))/5. - _Richard R. Forberg_, Aug 04 2014
%F A001629 (n-2)*a(n) - (n-1)*a(n-1) - n*a(n-2) = 0, n > 1. - _Michael D. Weiner_, Nov 18 2014
%F A001629 a(n) = Sum_{i=0..n-1} Sum_{j=0..i} F(j-1)*F(i-j), where F(n) = A000045 Fibonacci Numbers. - _Carlos A. Rico A._, Jul 14 2016
%F A001629 a(n) = (n*Lucas(n) - Fibonacci(n))/5, where Lucas = A000032, Fibonacci = A000045. - _Vladimir Reshetnikov_, Sep 27 2016
%F A001629 a(n) = (n-1)*hypergeom([1-n/2, (3-n)/2], [1-n], -4) for n >= 2. - _Peter Luschny_, Apr 10 2018
%F A001629 a(n) = -(-1)^n a(-n) for all n in Z. - _Michael Somos_, Jun 24 2018
%F A001629 E.g.f.: (1/50)*exp(-2*x/(1+sqrt(5)))*(2*sqrt(5)-5*(-1+sqrt(5))*x+exp(sqrt(5)*x)*(-2*sqrt(5)+5*(1+sqrt(5))*x)). - _Stefano Spezia_, Sep 03 2019
%F A001629 From _Peter Bala_, Jan 14 2025: (Start)
%F A001629 a(2*n+1) is even and a(2*n) has the same parity as Fibonacci(n).
%F A001629 For n >= 1, a(n) = (2/n)*Sum_{k = 0..n} k*Fibonacci(k)*Fibonacci(n-k). (End)
%e A001629 G.f. = x^2 + 2*x^3 + 5*x^4 + 10*x^5 + 20*x^6 + 38*x^7 + 71*x^8 + 130*x^9 + ... - _Michael Somos_, Jun 24 2018
%p A001629 a:= n-> (<<2|1|0|0>, <1|0|1|0>, <-2|0|0|1>, <-1|0|0|0>>^n)[1,3]:
%p A001629 seq(a(n), n=0..40); # _Alois P. Heinz_, Aug 01 2008
%p A001629 # Alternative:
%p A001629 A001629 := n -> `if`(n<2, 0, (n-1)*hypergeom([1-n/2, (3-n)/2], [1-n], -4)):
%p A001629 seq(simplify(A001629(n)), n=0..37); # _Peter Luschny_, Apr 10 2018
%t A001629 Table[Sum[Binomial[n-i, i] i, {i, 0, n}], {n, 0, 34}] (* _Geoffrey Critzer_, May 04 2009 *)
%t A001629 Table[Abs[(1 -(1/2 -I/2)(1 - (-1)^n))*Coefficient[CharacteristicPolynomial[ Array[KroneckerDelta[#1, #2] I + KroneckerDelta[#1 + 1, #2] + KroneckerDelta[#1 -1, #2] &, {n-1, n-1}], x], x]], {n,2,50}] (* _John M. Campbell_, Jun 23 2011 *)
%t A001629 LinearRecurrence[{2,1,-2,-1}, {0,0,1,2}, 40] (* _Harvey P. Dale_, Aug 26 2013 *)
%t A001629 CoefficientList[Series[x^2/(1-x-x^2)^2, {x, 0, 40}], x] (* _Vincenzo Librandi_, Nov 19 2014 *)
%t A001629 Table[(2nFibonacci[n-1] + (n-1)Fibonacci[n])/5, {n, 0, 40}] (* _Vladimir Reshetnikov_, May 08 2016 *)
%t A001629 Table[With[{fibs=Fibonacci[Range[n]]},ListConvolve[fibs,fibs]],{n,-1,40}]//Flatten (* _Harvey P. Dale_, Aug 19 2018 *)
%o A001629 (Haskell)
%o A001629 a001629 n = a001629_list !! (n-1)
%o A001629 a001629_list = f [] $ tail a000045_list where
%o A001629    f us (v:vs) = (sum $ zipWith (*) us a000045_list) : f (v:us) vs
%o A001629 -- _Reinhard Zumkeller_, Jan 18 2014, Oct 16 2011
%o A001629 (PARI) Vec(1/(1-x-x^2)^2+O(x^99)) \\ _Charles R Greathouse IV_, Feb 03 2014
%o A001629 (PARI) a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -1,-2,1,2]^n)[2,4] \\ _Charles R Greathouse IV_, Jul 20 2016
%o A001629 (Magma) I:=[0,0,1,2]; [n le 4 select I[n] else 2*Self(n-1)+Self(n-2)-2*Self(n-3)-Self(n-4): n in [1..40]]; // _Vincenzo Librandi_, Nov 19 2014
%o A001629 (GAP) List([0..40],n->Sum([0..n],k->Fibonacci(k)*Fibonacci(n-k))); # _Muniru A Asiru_, Jun 24 2018
%o A001629 (SageMath)
%o A001629 def A001629(n): return (1/5)*(n*lucas_number2(n, 1, -1) - fibonacci(n))
%o A001629 [A001629(n) for n in (0..40)] # _G. C. Greubel_, Apr 06 2022
%Y A001629 Row sums of triangles A058071, A134510, A134836.
%Y A001629 First differences of A006478.
%Y A001629 Cf. A000032, A000045, A001628, A010049, A037027, A055244, A214178.
%K A001629 nonn,easy,nice
%O A001629 0,4
%A A001629 _N. J. A. Sloane_