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

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

Author

Clark Kimberling

Keywords

• nonn
• easy

Comments

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

Links

• 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).

Crossrefs

Cf. A000045 (Fibonacci numbers).

Column 1 of triangle A063967.

Cf. A001629, A002940, A004798, A010049, A029907, A104326, A144154, A181630.

Programs

Haskell

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

Mathematica

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

PARI

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

Sage

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

Formula

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