A010049 -id:A010049 - cp's OEIS frontend

A001629 Self-convolution of Fibonacci numbers.

0, 0, 1, 2, 5, 10, 20, 38, 71, 130, 235, 420, 744, 1308, 2285, 3970, 6865, 11822, 20284, 34690, 59155, 100610, 170711, 289032, 488400, 823800, 1387225, 2332418, 3916061, 6566290, 10996580, 18394910, 30737759, 51310978, 85573315, 142587180, 237387960, 394905492

Offset: 0

N. J. A. Sloane

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
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
Number of 3412-avoiding involutions containing exactly one subsequence of type 321.
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
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
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
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
a(n+1) = (-i)^(n-1)*(d/dx)S(n,x)|A049310%20for%20the%20S-polynomials.%20-%20_Wolfdieter%20Lang">{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
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
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
For n > 0: a(n) = Sum_{k=1..n-1} (A039913(n-1,k)) / 2. - Reinhard Zumkeller, Oct 07 2012
The right-hand side of a binomial-coefficient identity [Gauthier]. - N. J. A. Sloane, Apr 09 2013
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
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
Also the number of maximal and maximum cliques in the (n-1)-Fibonacci cube graph. - Eric W. Weisstein, Sep 07 2017
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
From Steven Finch, Mar 22 2020: (Start)
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.
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.
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.
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)
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

			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

Donald E. Knuth, Fundamental Algorithms, Addison-Wesley, 1968, p. 83, Eq. 1.2.8--(17). - Don Knuth, Feb 26 2019
Thomas Koshy, Fibonacci and Lucas Numbers with Applications, 2001, Chapter 15, page 187, "Hosoya's Triangle", and p. 375, eq. (32.13).
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 101.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
S. Vajda, Fibonacci and Lucas numbers and the Golden Section, Ellis Horwood Ltd., Chichester, 1989, p. 183, Nr.(98).

T. D. Noe, Table of n, a(n) for n = 0..500
Marco Abrate, Stefano Barbero, Umberto Cerruti and Nadir Murru, Colored compositions, Invert operator and elegant compositions with the "black tie", Discrete Math. 335 (2014), 1--7. MR3248794.
Katharine A. Ahrens, Combinatorial Applications of the k-Fibonacci Numbers: A Cryptographically Motivated Analysis, Ph. D. thesis, North Carolina State University (2020).
Carlos Alirio Rico Acevedo and Ana Paula Chaves, Double-Recurrence Fibonacci Numbers and Generalizations, arXiv:1903.07490 [math.NT], 2019.
Marcella Anselmo, Giuseppa Castiglione, Manuela Flores, Dora Giammarresi, Maria Madonia, and Sabrina Mantaci, Hypercubes and Isometric Words based on Swap and Mismatch Distance, arXiv:2303.09898 [math.CO], 2023.
Kassie Archer and Robert P. Laudone, Pattern avoidance and the fundamental bijection, arXiv:2407.06338 [math.CO], 2024. See p. 6.
Ali Reza Ashrafi, Jernej Azarija, Khadijeh Fathalikhani, Sandi Klavžar and Marko Petkovšek, Vertex and edge orbits of Fibonacci and Lucas cubes, arXiv:1407.4962 [math.CO], 2014. See Table 2.
Ali Reza Ashrafi, Jernej Azarija, Khadijeh Fathalikhani, Sandi Klavžar, et al., Orbits of Fibonacci and Lucas cubes, dihedral transformations, and asymmetric strings, 2014.
Jean-Luc Baril, Sergey Kirgizov and Vincent Vajnovszki, Gray codes for Fibonacci q-decreasing words, arXiv:2010.09505 [cs.DM], 2020.
Daniel Birmajer, Juan Gil and Michael D. Weiner, Linear recurrence sequences and their convolutions via Bell polynomials, arXiv:1405.7727 [math.CO], 2014.
Daniel Birmajer, Juan B. Gil and Michael D. Weiner, Linear Recurrence Sequences and Their Convolutions via Bell Polynomials, Journal of Integer Sequences, 18 (2015), #15.1.2.
Matthew Blair, Rigoberto Flórez and Antara Mukherjee, Matrices in the Hosoya triangle, arXiv:1808.05278 [math.CO], 2018.
Matthew Blair, Rigoberto Flórez and Antara Mukherjee, Honeycombs in the Pascal triangle and beyond, arXiv:2203.13205 [math.HO], 2022. See p. 5.
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Giuseppa Castiglione, Antonio Restivo and Marinella Sciortino, Circular Sturmian words and Hopcroft's algorithm, Theor. Comput. Sci. 410, No. 43, 4372-4381 (2009)
Charles H. Conley and Valentin Ovsienko, Shadows of rationals and irrationals: supersymmetric continued fractions and the super modular group, arXiv:2209.10426 [math-ph], 2022.
Éva Czabarka, Rigoberto Flórez and Leandro Junes, A Discrete Convolution on the Generalized Hosoya Triangle, Journal of Integer Sequences, 18 (2015), #15.1.6.
Eric S. Egge, Restricted 3412-Avoiding Involutions, p. 16, arXiv:math/0307050 [math.CO], 2003.
Sergio Falcon, Half self-convolution of the k-Fibonacci sequence, Notes on Number Theory and Discrete Mathematics (2020) Vol. 26, No. 3, 96-106.
Luca Ferrari and Emanuele Munarini, Enumeration of edges in some lattices of paths, arXiv:1203.6792 [math.CO], 2012.
Steven Finch, Cantor-solus and Cantor-multus distributions, arXiv:2003.09458 [math.CO], 2020.
Rigoberto Flórez, Robinson Higuita and Alexander Ramírez, The resultant, the discriminant, and the derivative of generalized Fibonacci polynomials, arXiv:1808.01264 [math.NT], 2018.
Napoleon Gauthier (Proposer), Problem H-703, Fib. Quart., 50 (2012), 379-381.
Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See p. 23.
Martin Griffiths, Digit Proportions in Zeckendorf Representations, Fibonacci Quart. 48 (2010), no. 2, 168-174.
Verner E. Hoggatt, Jr. and M. Bicknell-Johnson, Fibonacci convolution sequences, Fib. Quart., 15 (1977), 117-122.
Milan Janjić, Hessenberg Matrices and Integer Sequences, J. Int. Seq. 13 (2010) # 10.7.8, section 3.
Omar Khadir, László Németh, and László Szalay, Tiling of dominoes with ranked colors, Results in Math. (2024) Vol. 79, Art. No. 253. See p. 8.
Sandi Klavžar, On median nature and enumerative properties of Fibonacci-like cubes, Disc. Math. 299 (2005), 145-153.
Sandi Klavžar, Structure of Fibonacci cubes: a survey, Institute of Mathematics, Physics and Mechanics Jadranska 19, 1000 Ljubljana, Slovenia; Preprint series Vol. 49 (2011), 1150 ISSN 2232-2094. (See Section 4.)
Toufik Mansour, Generalization of some identities involving the Fibonacci numbers, arXiv:math/0301157 [math.CO], 2003.
Toufik Mansour and Mark Shattuck, Pattern avoidance in flattened derangements, Disc. Math. Lett. (2025) Vol. 15, 67-74. See p. 74.
Pieter Moree, Convoluted convolved Fibonacci numbers, arXiv:math/0311205 [math.CO], 2003.
Pieter Moree, Convoluted Convolved Fibonacci Numbers, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.2.
László Németh, Walks on tiled boards, arXiv:2403.12159 [math.CO], 2024. See p. 2.
László Németh and László Szalay, Explicit solution of system of two higher-order recurrences, arXiv:2408.12196 [math.NT], 2024. See p. 10.
Valentin Ovsienko, Shadow sequences of integers, from Fibonacci to Markov and back, arXiv:2111.02553 [math.CO], 2021.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Mihai Prunescu and Lorenzo Sauras-Altuzarra, On the representation of C-recursive integer sequences by arithmetic terms, arXiv:2405.04083 [math.LO], 2024. See p. 18.
Jeffrey B. Remmel and J. L. B. Tiefenbruck, Q-analogues of convolutions of Fibonacci numbers, Australasian Journal of Combinatorics, Volume 64(1) (2016), Pages 166-193.
J. Riordan, Notes to N. J. A. Sloane, Jul. 1968
Eric Weisstein's World of Mathematics, Edge Count
Eric Weisstein's World of Mathematics, Fibonacci Cube Graph
Eric Weisstein's World of Mathematics, Edge Count
Eric Weisstein's World of Mathematics, Maximal Clique
Eric Weisstein's World of Mathematics, Maximum Clique
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

Row sums of triangles A058071, A134510, A134836.
First differences of A006478.
Cf. A000032, A000045, A001628, A010049, A037027, A055244, A214178.

List([0..40],n->Sum([0..n],k->Fibonacci(k)*Fibonacci(n-k))); # Muniru A Asiru, Jun 24 2018

a001629 n = a001629_list !! (n-1)
a001629_list = f [] $ tail a000045_list where
   f us (v:vs) = (sum $ zipWith (*) us a000045_list) : f (v:us) vs
-- Reinhard Zumkeller, Jan 18 2014, Oct 16 2011

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

a:= n-> (<<2|1|0|0>, <1|0|1|0>, <-2|0|0|1>, <-1|0|0|0>>^n)[1,3]:
seq(a(n), n=0..40); # Alois P. Heinz, Aug 01 2008
# Alternative:
A001629 := n -> `if`(n<2, 0, (n-1)*hypergeom([1-n/2, (3-n)/2], [1-n], -4)):
seq(simplify(A001629(n)), n=0..37); # Peter Luschny, Apr 10 2018

Table[Sum[Binomial[n-i, i] i, {i, 0, n}], {n, 0, 34}] (* Geoffrey Critzer, May 04 2009 *)
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 *)
LinearRecurrence[{2,1,-2,-1}, {0,0,1,2}, 40] (* Harvey P. Dale, Aug 26 2013 *)
CoefficientList[Series[x^2/(1-x-x^2)^2, {x, 0, 40}], x] (* Vincenzo Librandi, Nov 19 2014 *)
Table[(2nFibonacci[n-1] + (n-1)Fibonacci[n])/5, {n, 0, 40}] (* Vladimir Reshetnikov, May 08 2016 *)
Table[With[{fibs=Fibonacci[Range[n]]},ListConvolve[fibs,fibs]],{n,-1,40}]//Flatten (* Harvey P. Dale, Aug 19 2018 *)

Vec(1/(1-x-x^2)^2+O(x^99)) \\ Charles R Greathouse IV, Feb 03 2014

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

def A001629(n): return (1/5)*(n*lucas_number2(n, 1, -1) - fibonacci(n))
[A001629(n) for n in (0..40)] # G. C. Greubel, Apr 06 2022

G.f.: x^2/(1 - x - x^2)^2. - Simon Plouffe in his 1992 dissertation
a(n) = A037027(n-1, 1), n >= 1 (Fibonacci convolution triangle).
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4), n > 3.
a(n) = Sum_{k=0..n} A000045(k)*A000045(n-k).
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
a(n) = Sum_{k=0..floor(n/2)-1} (k+1)*binomial(n-k-1, k+1). - Emeric Deutsch, Nov 15 2001
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
a(n) = a(n-1) + A010049(n-1) for n > 0. - Emeric Deutsch, Dec 10 2003
a(n) = Sum_{k=0..floor((n-2)/2)} (n-k-1)*binomial(n-k-2, k). - Paul Barry, Jan 25 2005
a(n) = ((n-1)*F(n) + 2*n*F(n-1))/5, F(n)=A000045(n) (see Vajda and Koshy reference).
F'(n, 1), the first derivative of the n-th Fibonacci polynomial evaluated at 1. - T. D. Noe, Jan 18 2006
a(n) = a(n-1) + a(n-2) + F(n-1), where F=A000045, the Fibonacci sequence. - Graeme McRae, Jun 02 2006
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
a(n) = A055244(n-1) - F(n-2). Example: a(6) = 20 = A055244(5) - F(3) = (23 - 3). - Gary W. Adamson, Jul 27 2007
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
a(n) = A214178(n,1) for n > 0. - Reinhard Zumkeller, Jul 08 2012
a(n) = ((n+1)*F(n-1) + (n-1)*F(n+1))/5. - Richard R. Forberg, Aug 04 2014
(n-2)*a(n) - (n-1)*a(n-1) - n*a(n-2) = 0, n > 1. - Michael D. Weiner, Nov 18 2014
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
a(n) = (n*Lucas(n) - Fibonacci(n))/5, where Lucas = A000032, Fibonacci = A000045. - Vladimir Reshetnikov, Sep 27 2016
a(n) = (n-1)*hypergeom([1-n/2, (3-n)/2], [1-n], -4) for n >= 2. - Peter Luschny, Apr 10 2018
a(n) = -(-1)^n a(-n) for all n in Z. - Michael Somos, Jun 24 2018
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
From Peter Bala, Jan 14 2025: (Start)
a(2*n+1) is even and a(2*n) has the same parity as Fibonacci(n).
For n >= 1, a(n) = (2/n)*Sum_{k = 0..n} k*Fibonacci(k)*Fibonacci(n-k). (End)

A029907 a(n+1) = a(n) + a(n-1) + Fibonacci(n), with a(0) = 0 and a(1) = 1.

Original entry on oeis.org

0, 1, 2, 4, 8, 15, 28, 51, 92, 164, 290, 509, 888, 1541, 2662, 4580, 7852, 13419, 22868, 38871, 65920, 111556, 188422, 317689, 534768, 898825, 1508618, 2528836, 4233872, 7080519, 11828620, 19741179, 32916068, 54835556, 91276202, 151814645, 252318312

Offset: 0

N. J. A. Sloane

Number of matchings of the fan graph on n vertices, n>0 (a fan is the join of the path graph with one extra vertex).
a(n+1) gives row sums of A054450. - Paul Barry, Oct 23 2004
Number of parts in all compositions of n into odd parts. Example: a(5)=15 because the compositions 5, 311, 131, 113, and 11111 have a total of 1+3+3+3+5=15 parts.
a(n-1) is the number of compositions of n that contain one even part; for example, a(5-1)=a(4)=8 counts the compositions 1112, 1121, 1211, 14, 2111, 23, 32, 41. - Joerg Arndt, May 21 2013

			a(4)=8 because matchings of fan graph with edges {OA,OB,OC,AB,AC} are: {},{OA},{OB},{OC},{AB},{AC},{OA,BC},{OC,AB}.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Jean-Luc Baril and Nathanaël Hassler, Intervals in a family of Fibonacci lattices, Univ. de Bourgogne (France, 2024). See p. 5.
David Broadhurst, Zig-zag resistance and OEIS sequence A029907, Apr 11 2024
Jia Huang, Compositions with restricted parts, arXiv:1812.11010 [math.CO], 2018.
Mengmeng Liu and Andrew Yezhou Wang, The Number of Designated Parts in Compositions with Restricted Parts, J. Int. Seq., Vol. 23 (2020), Article 20.1.8.
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

Cf. A000032, A000045, A001629, A010049, A054450, A152163, A240847.

a029907 n = a029907_list !! n
a029907_list = 0 : 1 : zipWith (+) (tail a000045_list)
                      (zipWith (+) (tail a029907_list) a029907_list)
-- Reinhard Zumkeller, Nov 01 2013

[((n+4)*Fibonacci(n)+2*n*Fibonacci(n-1))/5: n in [0..40]]; // Vincenzo Librandi, Feb 25 2018

with(combinat); A029907 := proc(n) options remember; if n <= 1 then n else procname(n-1)+procname(n-2)+fibonacci(n-1); fi; end;

CoefficientList[Series[x(1-x^2)/(1-x-x^2)^2, {x, 0, 37}], x] (* or *)
a[n_]:= a[n]= a[n-1] +a[n-2] +Fibonacci[n-1]; a[0]=0; a[1]=1; Array[a, 37] (* or *)
LinearRecurrence[{2,1,-2,-1}, {0,1,2,4}, 38] (* Robert G. Wilson v, Jun 22 2014 *)

alias(F,fibonacci); a(n)=((n+4)*F(n)+2*n*F(n-1))/5;

def A029907(n): return (1/5)*(n*lucas_number2(n, 1, -1) + 4*fibonacci(n))
[A029907(n) for n in (0..40)] # G. C. Greubel, Apr 06 2022

G.f.: x*(1-x^2)/(1-x-x^2)^2.
a(n) = ((n+4)*Fibonacci(n) + 2*n*Fibonacci(n-1))/5.
a(n+1) = Sum_{k=0..n} Sum_{j=0..floor(k/2)} binomial(n-j, j). - Paul Barry, Oct 23 2004
a(n) = A010049(n+1) + A152163(n+1). - R. J. Mathar, Dec 10 2011
a(n) = F(n) + Sum_{k=1..n-1} F(k)*F(n-k), where F=Fibonacci. - Reinhard Zumkeller, Nov 01 2013
a(n) = (1/5)*(n*A000032(n) + 4*A000045(n)). - G. C. Greubel, Apr 06 2022
a(n) = A001629(n+1) - A001629(n-1), where A001629 is the first convolution of the Fibonacci numbers. - Gregory L. Simay, Aug 30 2022
E.g.f.: exp(x/2)*(5*x*cosh(sqrt(5)*x/2) + sqrt(5)*(5*x + 8)*sinh(sqrt(5)*x/2))/25. - Stefano Spezia, Dec 04 2023

Additional formula from Wolfdieter Lang, May 02 2000

Additional comments from Michael Somos, Jul 23 2002

A023610 Convolution of Fibonacci numbers and {F(2), F(3), F(4), ...}.

Original entry on oeis.org

1, 3, 7, 15, 30, 58, 109, 201, 365, 655, 1164, 2052, 3593, 6255, 10835, 18687, 32106, 54974, 93845, 159765, 271321, 459743, 777432, 1312200, 2211025, 3719643, 6248479, 10482351, 17562870, 29391490, 49132669, 82048737, 136884293, 228160495, 379975140, 632293452

Offset: 0

Clark Kimberling

a(n-2) + 1 is the number of (3412,1243)-, (3412,2134)- and (3412,1324)-avoiding involutions in S_n, n>1. - Ralf Stephan, Jul 06 2003
The number of terms in all ordered partitions of (n+1) using only ones and twos. For example, a(3)=15 because there are 15 terms in 1+1+1+1;2+1+1;1+2+1;1+1+2;2+2 - Geoffrey Critzer, Apr 07 2008
a(n) is the number of n-matchings in the graph obtained by a zig-zag triangulation of a convex (2n+1)-gon. Example: a(2)=7 because in the triangulation of the convex pentagon ABCDEA with diagonals AD and AC we have 7 2-matchings: {AB,CD},{AB,DE},{BC,AD},{BC,DE},{BC,EA},{CD,EA} and {DE,AC}. - Emeric Deutsch, Dec 25 2004
Partial sums of A029907. First differences of A002940. - Peter Bala, Oct 24 2007
Equals row sums of triangle A144154. - Gary W. Adamson, Sep 12 2008
Equals the number of 1's in Fibonacci Maximal notation for subsets of
(1, 2, 3, 5, 8, 13, ...) terms. For example (cf. A181630): 4, 5, and 6 are the 3 terms 101, 110, and 111 in Fibonacci Maximal. Total number of 1's for those terms = 7 = a(2). - Gary W. Adamson, Nov 02 2010
a(n) is half the number of strokes needed to draw all the domino tilings of a 2 X (n+2) rectangle. - Roberto Tauraso, Mar 15 2014
a(n) is the total number of 1's in all (n+1)-bit dual Zeckendorf representations of integers (A104326). For example, a(2) = 7 counts the 1's in 101, 110, 111. - Shenghui Yang, Feb 09 2025

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Marcella Anselmo, Giuseppa Castiglione, Manuela Flores, Dora Giammarresi, Maria Madonia, and Sabrina Mantaci, Hypercubes and Isometric Words based on Swap and Mismatch Distance, arXiv:2303.09898 [math.CO], 2023.
Kassie Archer and Noel Bourne, Pattern avoidance in compositions and powers of permutations, arXiv:2505.05218 [math.CO], 2025. See p. 6.
Jean-Luc Baril and José Luis Ramírez, Descent distribution on Catalan words avoiding ordered pairs of Relations, arXiv:2302.12741 [math.CO], 2023.
Jean-Luc Baril and José L. Ramírez, Fibonacci and Catalan paths in a wall, 2023.
Giuseppa Castiglione and Marinella Sciortino, Standard Sturmian words and automata minimization algorithms, Theoretical Computer Science, Volume 601, 11 October 2015, Pages 58-66 ("WORDS 2013").
Tomislav Došlic and Luka Podrug, Tilings of a Honeycomb Strip and Higher Order Fibonacci Numbers, arXiv:2203.11761 [math.CO], 2022.
Dmitry Efimov, Hafnian of two-parameter matrices, arXiv:2101.09722 [math.CO], 2021.
Eric S. Egge, Restricted 3412-Avoiding Involutions: Continued Fractions, Chebyshev Polynomials and Enumerations, arXiv:math/0307050 [math.CO], 2003. Sec. 8.
Neven Elezović, Asymptotic Expansions of Central Binomial Coefficients and Catalan Numbers, J. Int. Seq. 17 (2014) # 14.2.1.
Martin Griffiths, A Restricted Random Walk defined via a Fibonacci Process, Journal of Integer Sequences, Vol. 14 (2011), #11.5.4.
László Németh and László Szalay, Explicit solution of system of two higher-order recurrences, arXiv:2408.12196 [math.NT], 2024. See p. 10.
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

Cf. A000045 (Fibonacci numbers).
Column 1 of triangle A063967.
Cf. A001629, A002940, A004798, A010049, A029907, A104326, A144154, A181630.

a023610 n = a023610_list !! n
a023610_list = f [1] $ drop 3 a000045_list where
   f us (v:vs) = (sum $ zipWith (*) us $ tail a000045_list) : f (v:us) vs
-- Reinhard Zumkeller, Jan 18 2014

Table[Sum[Binomial[n - i, i]*(n - i), {i, 0, n}], {n, 1, 33}] (* Geoffrey Critzer, May 04 2009 *)

a(n)=(n+2)*fibonacci(n+4)/5+(n-1)*fibonacci(n+2)/5 \\ Charles R Greathouse IV, Jun 11 2015

def A023610():
    a, b, c, d = 1, 3, 7, 15
    while True:
        yield a
        a, b, c, d = b, c, d, 2*(d-b)+c-a
a = A023610(); [next(a) for i in range(33)]  # Peter Luschny, Nov 20 2013

O.g.f.: (x+1)/(1-x-x^2)^2. - Len Smiley, Dec 11 2001
a(n) = (1/5)*((n+2)*F(n+4) + (n-1)*F(n+2)), with F(n)=A000045(n). - Ralf Stephan, Jul 06 2003
a(n) = Sum_{k=0..n+1} (n-k+1)*binomial(n-k+1, k). - Paul Barry, Nov 05 2005
Recurrence: a(n+2) = a(n+1) + a(n) + Fib(n+4), n >= 0. For n >= 2, a(n-2) = (-1)^n*((-2n+3)*Fib(-n) - (-n)*Fib(-n-1))/5 = (-1)^n*A010049(-n), the second-order Fibonacci numbers of negative index, where Fib(-n) = (-1)^(n+1)*Fib(n). - Peter Bala, Oct 24 2007
a(n) = (n+1)*F(n+2) - A001629(n+1) where F(n) is the n-th Fibonacci number. - Geoffrey Critzer, Apr 07 2008
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4), n >= 4. - L. Edson Jeffery, Mar 29 2013
a(n+1) = A004798(n) + A000045(n+2) for n >= 0. - John Molokach, Jul 04 2013
a(n) = A001629(n+1) + A001629(n+2). - Philippe Deléham, Oct 30 2013
E.g.f.: exp(x/2)*(5*(5 + 7*x)*cosh(sqrt(5)*x/2) + sqrt(5)*(11 + 15*x)*sinh(sqrt(5)*x/2))/25. - Stefano Spezia, Dec 04 2023

A099920 a(n) = (n+1)*F(n), F(n) = Fibonacci numbers A000045.

Original entry on oeis.org

0, 2, 3, 8, 15, 30, 56, 104, 189, 340, 605, 1068, 1872, 3262, 5655, 9760, 16779, 28746, 49096, 83620, 142065, 240812, 407353, 687768, 1159200, 1950650, 3277611, 5499704, 9216519, 15426870, 25793240, 43080608, 71884197, 119835652

Offset: 0

Paul Barry and Ralf Stephan, Oct 15 2004

A Fibonacci-Lucas convolution.
The number of edges in the Lucas cube L_(n+1) [Klavzar]. - R. J. Mathar, Nov 05 2008
Sums of rows of the triangle in A108037. - Reinhard Zumkeller, Oct 07 2012
a(n-1) is the total binary weight of all bimultus bitstrings of length n. A bitstring is bimultus if each of its 1's possess at least one neighboring 1 and each of its 0's possess at least one neighboring 0. - Steven Finch, May 26 2020

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 35.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
S. Klavzar, On median nature and enumerative properties of Fibonacci-like cubes, Discr. Math. 299 (2005), 145-153.
Franck Ramaharo, A one-variable bracket polynomial for some Turk's head knots, arXiv:1807.05256 [math.CO], 2018.
Steven Finch, Variance of longest run duration in a random bitstring, arXiv:2005.12185 [math.CO], 2020.
Tamás Szakács, Convolution of second order linear recursive sequences. II. Commun. Math. 25, No. 2, 137-148 (2017), remark 2.
Tamás Szakács, Linear recursive sequences and factorials, Ph. D. Thesis, Univ. Debrecen (Hungary, 2024). See p. 35.
Eric Weisstein's World of Mathematics, Edge Count.
Eric Weisstein's World of Mathematics, Lucas Cube Graph.
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1)

Equals A010049(n) + A001629(n+1).

Cf. A000045, A000032, A045925, A023607.

a099920 n = a099920_list !! n
a099920_list = zipWith (*) [1..] a000045_list
-- Reinhard Zumkeller, Oct 07 2012

[(n+1)*Fibonacci(n): n in [0..60]]; // Vincenzo Librandi, Apr 23 2011

Table[(n + 1) Fibonacci[n], {n, 0, 40}] (* Harvey P. Dale, Jan 18 2012 *)
LinearRecurrence[{2, 1, -2, -1}, {0, 2, 3, 8}, 40] (* Harvey P. Dale, Jan 18 2012 *)
CoefficientList[Series[(2 - x) x/(-1 + x + x^2)^2, {x, 0, 20}], x] (* Eric W. Weisstein, Jul 28 2023 *)

a(n)=(n+1)*fibonacci(n) \\ Charles R Greathouse IV, Jun 11 2015

G.f.: x*(2-x)/(1-x-x^2)^2;
a(n) = Sum_{k=0..n} F(n-k)*(L(k-1) + 0^k).
a(n) = Sum_{k=0..n+1} F(n-k)*binomial(n-k+1, k)*binomial(1, (k+1)/2)*(1-(-1)^k)/2.
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4); a(0)=0, a(1)=2, a(2)=3, a(3)=8. - Harvey P. Dale, Jan 18 2012
a(n) = a(n-1) + a(n-2) + A000032(n-1) (Lucas numbers). - Bob Selcoe, Aug 19 2015
a(n) = 2*A001629(n+1) - A001629(n). - R. J. Mathar, Feb 04 2022

Entry revised by N. J. A. Sloane, Jan 23 2006. The offset changed, so some of the formulas may now be slightly off.

A105809 Riordan array (1/(1 - x - x^2), x/(1 - x)).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 3, 4, 3, 1, 5, 7, 7, 4, 1, 8, 12, 14, 11, 5, 1, 13, 20, 26, 25, 16, 6, 1, 21, 33, 46, 51, 41, 22, 7, 1, 34, 54, 79, 97, 92, 63, 29, 8, 1, 55, 88, 133, 176, 189, 155, 92, 37, 9, 1, 89, 143, 221, 309, 365, 344, 247, 129, 46, 10, 1, 144, 232, 364, 530, 674, 709, 591

Offset: 0

Paul Barry, May 04 2005

Previous name was: A Fibonacci-Pascal matrix.
From Wolfdieter Lang, Oct 04 2014: (Start)
In the column k of this triangle (without leading zeros) is the k-fold iterated partial sums of the Fibonacci numbers, starting with 1. A000045(n+1), A000071(n+3), A001924(n+1), A014162(n+1), A014166(n+1), ..., n >= 0. See the Riordan property.
For a combinatorial interpretation of these iterated partial sums see the H. Belbachir and A. Belkhir link. There table 1 shows in the rows these columns. In their notation (with r = k) f^(k)(n) = T(k, n+k).
The A-sequence of this Riordan triangle is [1, 1] (see the recurrence for T(n, k), k >= 1, given in the formula section). The Z-sequence is A165326 = [1, repeat(1, -1)]. See the W. Lang link under A006232 for Riordan A- and Z-sequences. (End)

			The triangle T(n,k) begins:
n\k   0   1   2    3    4    5    6    7    8   9  10 11 12 13 ...
0:    1
1:    1   1
2:    2   2   1
3:    3   4   3    1
4:    5   7   7    4    1
5:    8  12  14   11    5    1
6:   13  20  26   25   16    6    1
7:   21  33  46   51   41   22    7    1
8:   34  54  79   97   92   63   29    8    1
9:   55  88 133  176  189  155   92   37    9   1
10:  89 143 221  309  365  344  247  129   46  10   1
11: 144 232 364  530  674  709  591  376  175  56  11  1
12: 233 376 596  894 1204 1383 1300  967  551 231  67 12  1
13: 377 609 972 1490 2098 2587 2683 2267 1518 782 298 79 13  1
... reformatted and extended - _Wolfdieter Lang_, Oct 03 2014
------------------------------------------------------------------
Recurrence from Z-sequence (see a comment above): 8 = T(0,5) = (+1)*5 + (+1)*7 + (-1)*7 + (+1)*4 + (-1)*1 = 8. - _Wolfdieter Lang_, Oct 04 2014

Reinhard Zumkeller, Rows n = 0..120 of table, flattened
H. Belbachir and A. Belkhir, Combinatorial Expressions Involving Fibonacci, Hyperfibonacci, and Incomplete Fibonacci Numbers, Journal of Integer Sequences, Vol. 17 (2014), Article 14.4.3.
Hung Viet Chu, Partial Sums of the Fibonacci Sequence, arXiv:2106.03659 [math.CO], 2021.
Anton Vladimirovich Shutov, On some analogue of the Gelfond problem for Zeckendorf representations, Chebyshevskii Sbornik (2024) Vol. 25, No. 5, 195-215. See p. 215. (In Russian)
Index entries for triangles and arrays related to Pascal's triangle

Cf. A165326 (Z-sequence), A027934 (row sums), A010049(n+1) (antidiagonal sums), A212804 (alternating row sums), inverse is A105810.

Some other Fibonacci-Pascal triangles: A027926, A036355, A037027, A074829, A109906, A111006, A114197, A162741, A228074.

a105809 n k = a105809_tabl !! n !! k
a105809_row n = a105809_tabl !! n
a105809_tabl = map fst $ iterate
   (\(u:_, vs) -> (vs, zipWith (+) ([u] ++ vs) (vs ++ [0]))) ([1], [1,1])
-- Reinhard Zumkeller, Aug 15 2013

T := (n,k) -> `if`(n=0,1,binomial(n,k)*hypergeom([1,k/2-n/2,k/2-n/2+1/2], [k+1,-n], -4)); for n from 0 to 13 do seq(simplify(T(n,k)),k=0..n) od; # Peter Luschny, Oct 10 2014

T[n_, k_] := Sum[Binomial[n-j, k+j], {j, 0, n}]; Table[T[n, k], {n, 0, 11}, {k, 0, n}] (* Jean-François Alcover, Jun 11 2019 *)

Riordan array (1/(1-x-x^2), x/(1-x)).
Triangle T(n, k) = Sum_{j=0..n} binomial(n-j, k+j); T(n, 0) = A000045(n+1);
T(n, m) = T(n-1, m-1) + T(n-1, m).
T(n, k) = Sum_{j=0..n} binomial(j, n+k-j). - Paul Barry, Oct 23 2006
G.f. of row polynomials Sum_{k=0..n} T(n, k)*x^k is (1 - z)/((1 - z - z^2)*(1 - (1 + x)*z)) (Riordan property). - Wolfdieter Lang, Oct 04 2014
T(n, k) = binomial(n, k)*hypergeom([1, k/2 - n/2, k/2 - n/2 + 1/2],[k + 1, -n], -4) for n > 0. - Peter Luschny, Oct 10 2014
From Wolfdieter Lang, Feb 13 2025: (Start)
Array A(k, n) = Sum_{j=0..n} F(j+1)*binomial(k-1+n-j, k-1), k >= 0, n >= 0, with F = A000045, (from Riordan triangle k-th convolution in columns without leading 0s).
A(k, n) = F(n+1+2*k) - Sum_{j=0..k-1} F(2*(k-j)-1) * binomial(n+1+j, j), (from iteration of partial sums).
Triangle T(n, k) = A(k, n-k) = Sum_{j=k..n} F(n-j+1) * binomial(j-1, k-1), 0 <= k <= n.
T(n, k) = F(n+1+k) - Sum_{j=0..k-1} F(2*(k-j)-1) * binomial(n - (k-1-j), j). (End)
T(n, k) =  A027926(n, n+k), for 0 <= k <= n. - Wolfdieter Lang, Mar 08 2025

Use first formula as a more descriptive name, Joerg Arndt, Jun 08 2021

A053538 Triangle: a(n,m) = ways to place p balls in n slots with m in the rightmost p slots, 0<=p<=n, 0<=m<=n, summed over p, a(n,m)= Sum_{k=0..n} binomial(k,m)*binomial(n-k,k-m), (see program line).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 5, 4, 1, 1, 8, 10, 7, 5, 1, 1, 13, 18, 16, 9, 6, 1, 1, 21, 33, 31, 23, 11, 7, 1, 1, 34, 59, 62, 47, 31, 13, 8, 1, 1, 55, 105, 119, 101, 66, 40, 15, 9, 1, 1, 89, 185, 227, 205, 151, 88, 50, 17, 10, 1, 1, 144, 324, 426, 414, 321, 213, 113, 61, 19, 11, 1, 1

Offset: 0

Wouter Meeussen, May 23 2001

Riordan array (1/(1-x-x^2), x(1-x)/(1-x-x^2)). Row sums are A000079. Diagonal sums are A006053(n+2). - Paul Barry, Nov 01 2006
Subtriangle of the triangle given by (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 05 2012
Mirror image of triangle in A208342. - Philippe Deléham, Mar 05 2012
A053538 is jointly generated with A076791 as an array of coefficients of polynomials u(n,x):  initially, u(1,x)=v(1,x)=1, for n>1, u(n,x) = x*u(n-1,x) + v(n-1,x) and v(n,x) = u(n-1,x) + v(n-1,x).  See the Mathematica section at A076791. - Clark Kimberling, Mar 08 2012
The matrix inverse starts
    1;
   -1,   1;
   -1,  -1,   1;
    1,  -2,  -1,  1;
    3,   1,  -3, -1,  1;
    1,   6,   1, -4, -1,  1;
   -7,   4,  10,  1, -5, -1,  1;
  -13, -13,   8, 15,  1, -6, -1,  1;
    3, -31, -23, 13, 21,  1, -7, -1, 1; - R. J. Mathar, Mar 15 2013
Also appears to be the number of subsets of {1..n} containing n with k maximal anti-runs of consecutive elements increasing by more than 1. For example, the subset {1,3,6,7,11,12} has maximal anti-runs ((1,3,6),(7,11),(12)) so is counted under a(12,3). For runs instead of anti-runs we get A202064. - Gus Wiseman, Jun 26 2025

			n=4; Table[binomial[k, j]binomial[n-k, k-j], {k, 0, n}, {j, 0, n}] splits {1, 4, 6, 4, 1} into {{1, 0, 0, 0, 0}, {3, 1, 0, 0, 0}, {1, 4, 1, 0, 0}, {0, 0, 3, 1, 0}, {0, 0, 0, 0, 1}} and this gives summed by columns {5, 5, 4, 1, 1}
Triangle begins :
   1;
   1,  1;
   2,  1,  1;
   3,  3,  1, 1;
   5,  5,  4, 1, 1;
   8, 10,  7, 5, 1, 1;
  13, 18, 16, 9, 6, 1, 1;
...
(0, 1, 1, -1, 0, 0, 0, ...) DELTA (1, 0, -1, 1, 0, 0, 0, ...) begins :
  1;
  0,  1;
  0,  1,  1;
  0,  2,  1,  1;
  0,  3,  3,  1, 1;
  0,  5,  5,  4, 1, 1;
  0,  8, 10,  7, 5, 1, 1;
  0, 13, 18, 16, 9, 6, 1, 1;

Alois P. Heinz, Rows n = 0..140, flattened
R. P. Grimaldi, Extraordinary subsets: a generalization, Fib. Quart., 55 (No. 3, 2017), 114-122. See Table 1.

Column k = 1 is A000045.
Row sums are A000079.
Column k = 2 is A010049.
For runs instead of anti-runs we have A202064.
For integer partitions see A268193, strict A384905, runs A116674.
A034839 counts subsets by number of maximal runs.
A384175 counts subsets with all distinct lengths of maximal runs, complement A384176.
A384877 gives lengths of maximal anti-runs in binary indices, firsts A384878.
A384893 counts subsets by number of maximal anti-runs.
Cf. A000071, A001629, A010027, A208342, A384177, A384879, A384889, A384890.

Flat(List([0..12], n-> List([0..n], k-> Sum([0..n], j->  Binomial(j,k)*Binomial(n-j,j-k)) ))); # G. C. Greubel, May 16 2019

[[(&+[Binomial(j,k)*Binomial(n-j,j-k): j in [0..n]]): k in [0..n]]: n in [0..12]]; // G. C. Greubel, May 16 2019

a:= (n, m)-> add(binomial(k, m)*binomial(n-k, k-m), k=0..n):
seq(seq(a(n,m), m=0..n), n=0..12);  # Alois P. Heinz, Sep 19 2013

Table[Sum[Binomial[k, m]*Binomial[n-k, k-m], {k,0,n}], {n,0,12}, {m,0,n}]

{T(n,k) = sum(j=0,n, binomial(j,k)*binomial(n-j,j-k))}; \\ G. C. Greubel, May 16 2019

[[sum(binomial(j,k)*binomial(n-j,j-k) for j in (0..n)) for k in (0..n)] for n in (0..12)] # G. C. Greubel, May 16 2019

From Philippe Deléham, Mar 05 2012: (Start)
T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) - T(n-2,k-1), T(0,0) = T(1,0) = T(1,1) = 1 and T(n,k) = 0 if k<0 or if k>n.
G.f.: 1/(1-(1+y)*x-(1-y)*x^2).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A077957(n), A000045(n+1), A000079(n), A001906(n+1), A007070(n), A116415(n), A084326(n+1), A190974(n+1), A190978(n+1), A190984(n+1), A190990(n+1), A190872(n+1) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 respectively. (End)

A006367 Number of binary vectors of length n+1 beginning with 0 and containing just 1 singleton.

Original entry on oeis.org

1, 0, 2, 2, 5, 8, 15, 26, 46, 80, 139, 240, 413, 708, 1210, 2062, 3505, 5944, 10059, 16990, 28646, 48220, 81047, 136032, 228025, 381768, 638450, 1066586, 1780061, 2968040, 4944519, 8230370, 13689118, 22751528, 37786915, 62716752, 104028245

Offset: 0

David M. Bloom

Number of compositions of n+1 containing exactly one 1. - Emeric Deutsch, Mar 08 2002
Number of permutations with one fixed point avoiding 231 and 321.
A singleton is a run of length 1. - Michael Somos, Nov 29 2014
Second column of A105422. - Michael Somos, Nov 29 2014
Number of weak compositions of n with one 0 and no 1's. Example: Combine one 0 with the compositions of 5 without 1 to get a(5) = 8 weak compositions: 0,5; 5,0; 0,2,3; 0,3,2; 2,0,3; 3,0,2; 2,3,0; 3,2,0. - Gregory L. Simay, Mar 21 2018

			a(4) = 5 because among the 2^4 compositions of 5 only 4+1,1+4,2+2+1,2+1+2,1+2+2 contain exactly one 1.
a(4) = 5 because the binary vectors of length 4+1 beginning with 0 and with exactly one singleton are: 00001, 00100, 00110, 01100, 01111. - _Michael Somos_, Nov 29 2014
G.f. = 1 + 2*x^2 + 2*x^3 + 5*x^4 + 8*x^5 + 15*x^6 + 26*x^7 + 46*x^8 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Ricardo Gómez Aíza, Symbolic dynamical scales: modes, orbitals, and transversals, arXiv:2009.02669 [math.DS], 2020.
Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 4, 11.
Mengmeng Liu and Andrew Yezhou Wang, The Number of Designated Parts in Compositions with Restricted Parts, J. Int. Seq., Vol. 23 (2020), Article 20.1.8.
J. J. Madden, A generating function for the distribution of runs in binary words, arXiv:1707.04351 [math.CO], 2017, Theorem 1.1, r=k=1.
T. Mansour and A. Robertson, Refined restricted permutations avoiding subsets of patterns of length three, arXiv:math/0204005 [math.CO], 2002.
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

Cf. A000032, A000045, A004798, A006355, A010049, A105422, A139821, A212804.

I:=[1,0]; [n le 2 select I[n] else Self(n-1)+Self(n-2)+Fibonacci(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 20 2014

nn=36; CoefficientList[Series[1/(1 -x/(1-x) +x)^2, {x, 0, nn}], x] (* Geoffrey Critzer, Feb 18 2014 *)
a[n_]:= If[ n<0, SeriesCoefficient[((1-x)/(1+x-x^2))^2, {x, 0, -2-n}], SeriesCoefficient[((1-x)/(1-x-x^2))^2, {x, 0, n}]]; (* Michael Somos, Nov 29 2014 *)

Vec( (1-x)^2/(1-x-x^2)^2 + O(x^66) ) \\ Joerg Arndt, Feb 20 2014

{a(n) = if( n<0, n = -2-n; polcoeff( (1 - x)^2 / (1 + x - x^2)^2 + x * O(x^n), n), polcoeff( (1 - x)^2 / (1 - x - x^2)^2 + x * O(x^n), n))}; /* Michael Somos, Nov 29 2014 */

from sympy import fibonacci
from sympy.core.cache import cacheit
@cacheit
def a(n): return 1 if n==0 else 0 if n==1 else a(n - 1) + a(n - 2) + fibonacci(n - 3)
print([a(n) for n in range(51)]) # Indranil Ghosh, Jul 20 2017

def A006367(n): return (1/5)*(n*lucas_number2(n-2, 1, -1) + fibonacci(n+1) + 4*fibonacci(n-1))
[A006367(n) for n in (0..40)] # G. C. Greubel, Apr 06 2022

a(n) = a(n-1) + a(n-2) + Fibonacci(n-3).
G.f.: (1-x)^2/(1-x-x^2)^2. - Emeric Deutsch, Mar 08 2002
a(n) = A010049(n+1) - A010049(n). - R. J. Mathar, May 30 2014
Convolution square of A212804. - Michael Somos, Nov 29 2014
a(n) = -(-1)^n * A004798(-1-n) for all n in Z. - Michael Somos, Nov 29 2014
0 = a(n)*(-2*a(n) - 7*a(n+1) + 2*a(n+2) + a(n+3)) + a(n+1)*(-4*a(n+1) + 10*a(n+2) - 2*a(n+3)) + a(n+2)*(+4*a(n+2) - 7*a(n+3)) + a(n+3)*(+2*a(n+3)) for all n in Z. - Michael Somos, Nov 29 2014
a(n) = (n*Lucas(n-2) + Fibonacci(n))/5 + Fibonacci(n-1). - Ehren Metcalfe, Jul 29 2017

Conjecture: a(n) is the number of compositions of n if all the 1's are constrained to be in a single run; for example, a(7) counts the compositions 4,1,1,1 and 1,1,1,4 but not the compositions 1,4,1,1 and 1,1,4,1. - Gregory L. Simay, Sep 29 2018

This also gives the number of ordered partitions of n into parts of sizes 1, 2, and 3 with at most one 3. - Jerrold Grossman, Apr 03 2024

To compute a(n): replace every other bit with zero (starting with the second bit) in each run of consecutive 1's in the binary expansion of n.
From Gus Wiseman, Jul 11 2025: (Start)
This is the greatest binary rank of a sparse subset of the binary indices of n, where:
1. The binary indices of a nonnegative integer are the positions of 1 in its reversed binary expansion.
2. A set is sparse iff 1 is not a first difference.
3. The binary rank of a set {S_1,S_2,...} is Sum_i 2^(S_i-1).
(End)

2*a(n) = C_{n+3} of Turban reference eq.(2.17), C_{1}= 0 = C_{2}.

Number of binary sequences of length n+3 such that the sequence has exactly two pairs (which may overlap) of consecutive 1's. - George J. Schaeffer (gschaeff(AT)andrew.cmu.edu), Sep 07 2004

cp's OEIS Frontend

A001629 Self-convolution of Fibonacci numbers.

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A029907 a(n+1) = a(n) + a(n-1) + Fibonacci(n), with a(0) = 0 and a(1) = 1.

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A023610 Convolution of Fibonacci numbers and {F(2), F(3), F(4), ...}.

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A099920 a(n) = (n+1)*F(n), F(n) = Fibonacci numbers A000045.

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A105809 Riordan array (1/(1 - x - x^2), x/(1 - x)).

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A053538 Triangle: a(n,m) = ways to place p balls in n slots with m in the rightmost p slots, 0<=p<=n, 0<=m<=n, summed over p, a(n,m)= Sum_{k=0..n} binomial(k,m)*binomial(n-k,k-m), (see program line).