A001628 Convolved Fibonacci numbers.

1, 3, 9, 22, 51, 111, 233, 474, 942, 1836, 3522, 6666, 12473, 23109, 42447, 77378, 140109, 252177, 451441, 804228, 1426380, 2519640, 4434420, 7777860, 13599505, 23709783, 41225349, 71501422, 123723351, 213619683, 368080793, 633011454, 1086665562, 1862264196

Offset: 0

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn
• easy

Comments

a(n-2) = (((-i)^(n-2))/2)*(d^2/dx^2)S(n,x)|A049310%20for%20the%20S-polynomials.%20-%20_Wolfdieter%20Lang">{x=i}, n>=2. Second derivative of Chebyshev S-polynomials evaluated at x=i (imaginary unit) multiplied by ((-i)^(n-2))/2. See A049310 for the S-polynomials. - _Wolfdieter Lang, Apr 04 2007

a(n) = number of weak compositions of n in which exactly 2 parts are 0 and all other parts are either 1 or 2. - Milan Janjic, Jun 28 2010

Number of 4-cycles in the Fibonacci cube Gamma[n+3] (see the Klavzar reference, p. 511). - Emeric Deutsch, Apr 17 2014

Examples

			G.f. = 1 + 3*x + 9*x^2 + 22*x^3 + 51*x^4 + 111*x^5 + 233*x^6 + 474*x^7 + ...
		

References

• T. Koshy, "Fibonacci and Lucas Numbers with Applications", John Wiley and Sons, 2001, p. 375.
• 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).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..4500 (terms 0..500 from T. D. Noe)
• 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.
• Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
• Verner E. Hoggatt, Jr., Letters to N. J. A. Sloane, 1974-1975
• Verner E. Hoggatt, Jr. and Marjorie 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.
• Sandi Klavžar, Structure of Fibonacci cubes: a survey, preprint.
• Sandi Klavžar, Structure of Fibonacci cubes: a survey, J. Comb. Optim., 25, 2013, 505-522
• Toufik Mansour, Generalization of some identities involving the Fibonacci numbers, arXiv:math/0301157 [math.CO], 2003.
• 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.
• 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.
• John Riordan, Letter to N. J. A. Sloane, Oct. 1970
• Eric Weisstein's World of Mathematics, Graph Cycle
• Eric Weisstein's World of Mathematics, Fibonacci Cube Graph
• Index entries for linear recurrences with constant coefficients, signature (3,0,-5,0,3,1).

Crossrefs

a(n) = A037027(n+2,2) (Fibonacci convolution triangle).

Cf. A055243 (first differences).

Cf. A291915 (6-cycles).

Programs

Magma

[(&+[Binomial(k,n-k)*Binomial(k+2,2): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Jan 10 2018

Maple

A001628:=-1/(z**2+z-1)**3; [Simon Plouffe in his 1992 dissertation.]
a:= n-> (Matrix(6, (i, j)-> `if` (i=j-1, 1, `if` (j=1, [3, 0, -5, 0, 3, 1][i], 0)))^n)[1,1]: seq (a(n), n=0..29); # Alois P. Heinz, Aug 01 2008

Mathematica

CoefficientList[Series[1/(-z^2 - z + 1)^3, {z, 0, 100}], z] (* Vladimir Joseph Stephan Orlovsky, Jul 01 2011 *)
(* Start Eric W. Weisstein, Sep 05 2017 *)
Table[Derivative[2][Fibonacci[n + 2, #] &][1]/2, {n, 20}]
Derivative[2][Fibonacci[Range[20] + 2, #] &][1]/2
LinearRecurrence[{3, 0, -5, 0, 3, 1}, {1, 3, 9, 22, 51, 111}, 20]
Table[-I^(n + 1) Derivative[2][ChebyshevU[n + 1, -#/2] &][I]/2, {n, 20}]
(* End *)

PARI

Vec((1 - x - x^2 )^-3+O(x^99)) \\ Charles R Greathouse IV, Jul 01 2011

Formula

G.f.: 1 / (1 - x - x^2)^3.

a(n) = A037027(n+2,2) (Fibonacci convolution triangle).

a(n) = ((5*n+16)*(n+1)*F(n+2)+(5*n+17)*(n+2)*F(n+1))/50, F=A000045. - Wolfdieter Lang, Apr 12 2000 (This formula coincides with eq. (32.14) of the Koshy reference, p. 375, if there n -> n+3. - Wolfdieter Lang, Aug 03 2012)

For n>2, a(n-2) = sum(i+j+k=n, F(i)*F(j)*F(k)). - Benoit Cloitre, Nov 01 2002

a(n) = F''(n+2, 1)/2, i.e. 1/2 times the 2nd derivative of the (n+2)th Fibonacci polynomial evaluated at 1. - T. D. Noe, Jan 18 2006

a(n) = Sum_{k=0..n} C(k,n-k)*C(k+2,2). - Paul Barry, Apr 13 2008

0 = n*a(n) - (n+2)*a(n-1) - (n+4)*a(n-2), n>1. - Michael D. Weiner, Nov 18 2014

a(n) = 3*a(n-1) - 5*a(n-3) + 3*a(n-5) + a(n-6). - Eric W. Weisstein, Sep 05 2017