cp's OEIS Frontend

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

%I A023610 #157 May 13 2025 09:55:03
%S A023610 1,3,7,15,30,58,109,201,365,655,1164,2052,3593,6255,10835,18687,32106,
%T A023610 54974,93845,159765,271321,459743,777432,1312200,2211025,3719643,
%U A023610 6248479,10482351,17562870,29391490,49132669,82048737,136884293,228160495,379975140,632293452
%N A023610 Convolution of Fibonacci numbers and {F(2), F(3), F(4), ...}.
%C A023610 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
%C A023610 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
%C A023610 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
%C A023610 Partial sums of A029907. First differences of A002940. - _Peter Bala_, Oct 24 2007
%C A023610 Equals row sums of triangle A144154. - _Gary W. Adamson_, Sep 12 2008
%C A023610 Equals the number of 1's in Fibonacci Maximal notation for subsets of
%C A023610 (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
%C A023610 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
%C A023610 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
%H A023610 Reinhard Zumkeller, <a href="/A023610/b023610.txt">Table of n, a(n) for n = 0..1000</a>
%H A023610 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 A023610 Kassie Archer and Noel Bourne, <a href="https://arxiv.org/abs/2505.05218">Pattern avoidance in compositions and powers of permutations</a>, arXiv:2505.05218 [math.CO], 2025. See p. 6.
%H A023610 Jean-Luc Baril and José Luis Ramírez, <a href="https://arxiv.org/abs/2302.12741">Descent distribution on Catalan words avoiding ordered pairs of Relations</a>, arXiv:2302.12741 [math.CO], 2023.
%H A023610 Jean-Luc Baril and José L. Ramírez, <a href="http://jl.baril.u-bourgogne.fr/pathwall.pdf">Fibonacci and Catalan paths in a wall</a>, 2023.
%H A023610 Giuseppa Castiglione and Marinella Sciortino, <a href="https://doi.org/10.1016/j.tcs.2015.07.023">Standard Sturmian words and automata minimization algorithms</a>, Theoretical Computer Science, Volume 601, 11 October 2015, Pages 58-66 ("WORDS 2013").
%H A023610 Tomislav Došlic and Luka Podrug, <a href="https://arxiv.org/abs/2203.11761">Tilings of a Honeycomb Strip and Higher Order Fibonacci Numbers</a>, arXiv:2203.11761 [math.CO], 2022.
%H A023610 Dmitry Efimov, <a href="https://arxiv.org/abs/2101.09722">Hafnian of two-parameter matrices</a>, arXiv:2101.09722 [math.CO], 2021.
%H A023610 Eric S. Egge, <a href="https://arxiv.org/abs/math/0307050">Restricted 3412-Avoiding Involutions: Continued Fractions, Chebyshev Polynomials and Enumerations</a>, arXiv:math/0307050 [math.CO], 2003. Sec. 8.
%H A023610 Neven Elezović, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Elezovic/elezovic5.html">Asymptotic Expansions of Central Binomial Coefficients and Catalan Numbers</a>, J. Int. Seq. 17 (2014) # 14.2.1.
%H A023610 Martin Griffiths, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Griffiths/griffiths16.html">A Restricted Random Walk defined via a Fibonacci Process</a>, Journal of Integer Sequences, Vol. 14 (2011), #11.5.4.
%H A023610 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 A023610 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-2,-1).
%F A023610 O.g.f.: (x+1)/(1-x-x^2)^2. - _Len Smiley_, Dec 11 2001
%F A023610 a(n) = (1/5)*((n+2)*F(n+4) + (n-1)*F(n+2)), with F(n)=A000045(n). - _Ralf Stephan_, Jul 06 2003
%F A023610 a(n) = Sum_{k=0..n+1} (n-k+1)*binomial(n-k+1, k). - _Paul Barry_, Nov 05 2005
%F A023610 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
%F A023610 a(n) = (n+1)*F(n+2) - A001629(n+1) where F(n) is the n-th Fibonacci number. - _Geoffrey Critzer_, Apr 07 2008
%F A023610 a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4), n >= 4. - _L. Edson Jeffery_, Mar 29 2013
%F A023610 a(n+1) = A004798(n) + A000045(n+2) for n >= 0. - _John Molokach_, Jul 04 2013
%F A023610 a(n) = A001629(n+1) + A001629(n+2). - _Philippe Deléham_, Oct 30 2013
%F A023610 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
%t A023610 Table[Sum[Binomial[n - i, i]*(n - i), {i, 0, n}], {n, 1, 33}] (* _Geoffrey Critzer_, May 04 2009 *)
%o A023610 (Sage)
%o A023610 def A023610():
%o A023610     a, b, c, d = 1, 3, 7, 15
%o A023610     while True:
%o A023610         yield a
%o A023610         a, b, c, d = b, c, d, 2*(d-b)+c-a
%o A023610 a = A023610(); [next(a) for i in range(33)]  # _Peter Luschny_, Nov 20 2013
%o A023610 (Haskell)
%o A023610 a023610 n = a023610_list !! n
%o A023610 a023610_list = f [1] $ drop 3 a000045_list where
%o A023610    f us (v:vs) = (sum $ zipWith (*) us $ tail a000045_list) : f (v:us) vs
%o A023610 -- _Reinhard Zumkeller_, Jan 18 2014
%o A023610 (PARI) a(n)=(n+2)*fibonacci(n+4)/5+(n-1)*fibonacci(n+2)/5 \\ _Charles R Greathouse IV_, Jun 11 2015
%Y A023610 Cf. A000045 (Fibonacci numbers).
%Y A023610 Column 1 of triangle A063967.
%Y A023610 Cf. A001629, A002940, A004798, A010049, A029907, A104326, A144154, A181630.
%K A023610 nonn,easy
%O A023610 0,2
%A A023610 _Clark Kimberling_