A067331 Convolution of Fibonacci F(n+1), n >= 0, with F(n+3), n >= 0.

2, 5, 12, 25, 50, 96, 180, 331, 600, 1075, 1908, 3360, 5878, 10225, 17700, 30509, 52390, 89664, 153000, 260375, 442032, 748775, 1265832, 2136000, 3598250, 6052061, 10164540, 17048641, 28559450, 47786400, 79870428, 133359715, 222457608, 370747675, 617363100

Offset: 0

Author

Wolfdieter Lang, Feb 15 2002

Keywords

• nonn
• easy
• nice

Comments

Third diagonal of A067330. Third column of A067418.

From Emeric Deutsch, Jun 15 2010: (Start)

a(n) is the external path length of the Fibonacci tree of order n+3. A Fibonacci tree of order n (n >= 2) is a complete binary tree whose left subtree is the Fibonacci tree of order n-1 and whose right subtree is the Fibonacci tree of order n-2; each of the Fibonacci trees of order 0 and 1 is defined as a single node. The external path length of a tree is the sum of the levels of its external nodes (i.e., leaves).

a(n) = Sum_{k>=0} k*A178524(n+2,k).

(End)

a(n) equals the penultimate immanant of the (n+3) X (n+3) tridiagonal matrix with ones along the main diagonal, the superdiagonal, and the subdiagonal. - John M. Campbell, Jan 01 2016

a(n) is the sum of the eccentricities of the vertices of the Fibonacci cube G(n+1). Example: a(1)=5; indeed, the Fibonacci cube G(2) is the path graph P(3), the vertices of which have eccentricities 2, 1, 2. - Emeric Deutsch, May 28 2017

Examples

			From _John M. Campbell_, Jan 03 2016: (Start)
Letting n=2, the external path length of the Fibonacci tree T(5) of order n+3=5 illustrated below is 12 = a(2) = F(1)*F(5) + F(2)*F(4) + F(3)*F(3).
     .
    / \
   /\ /\
  /\
(End)
		

References

• D. E. Knuth, The Art of Computer Programming, Vol. 3, 2nd edition, Addison-Wesley, Reading, MA, 1998, p. 417.

Links

• Robert Israel, Table of n, a(n) for n = 0..4720
• Matthew Blair, Rigoberto Flórez, Antara Mukherjee, and José L. Ramírez, Matrices in the determinant Hosoya triangle, Fibonacci Quart. 58 (2020), no. 5, 34-54.
• Matthew Blair, Rigoberto Flórez, and Antara Mukherjee, Geometric Patterns in The Determinant Hosoya Triangle, INTEGERS, A90, 2021.
• J. Bodeen, S. Butler, T. Kim, X. Sun, and S. Wang, Tiling a strip with triangles, Electron. J. Combin. 21 (1) (2014), P1.7.
• John M. Campbell, On the external path length of a Fibonacci tree.
• Y. Horibe, An entropy view of Fibonacci trees, Fibonacci Quarterly, 20(2) (1982), 168-178.
• S. Klavzar and M. Mollard, Asymptotic properties of Fibonacci cubes and Lucas cubes, HAL Id: hal-00836788, 2013.
• S. Klavzar and M. Mollard, Asymptotic properties of Fibonacci cubes and Lucas cubes, Ann. Comb. 18 (2014), 447-457.
• Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

Crossrefs

Row sums of A108038.

Cf. A000045, A004070, A067330, A067418, A178523, A178524.

Programs

Magma

[((7*n+10)*Fibonacci(n+1)+4*(n+1)*Fibonacci(n))/5: n in [0..40]]; // Vincenzo Librandi, Jan 02 2016

Maple

f:= gfun:-rectoproc({a(n) = 2*a(n-1)+a(n-2) - 2*a(n-3)-a(n-4),a(0)=2,a(1)=5,a(2)=12,a(3)=25},a(n),remember):
map(f, [$0..50]); # Robert Israel, Jan 06 2016

Mathematica

LinearRecurrence[{2, 1, -2, -1}, {2, 5, 12, 25}, 70] (* Vincenzo Librandi, Jan 02 2016 *)
Table[SeriesCoefficient[(2 + x)/(1 - x - x^2)^2, {x, 0, n}], {n, 0, 34}] (* Michael De Vlieger, Jan 02 2016 *)
Print[Table[Sum[Binomial[n + 3 - i, i]*(n + 2 - 2*i), {i, 0, Floor[(n + 3)/2]}], {n, 0, 100}]] (* John M. Campbell, Jan 04 2016 *)
Module[{nn=40,fibs},fibs=Fibonacci[Range[nn]];Table[ListConvolve[Take[ fibs,n],Take[fibs,{2,n+2}]],{n,nn-2}]][[All,2]] (* Harvey P. Dale, Aug 03 2019 *)

PARI

Vec((2+x)/(1-x-x^2)^2 + O(x^100)) \\ Altug Alkan, Jan 04 2016

Formula

a(n) = A067330(n+2, n) = A067418(n+2, 2) = Sum_{k=0..n} F(k+1)*F(n+3-k), n >= 0.

a(n) = ((7*n + 10)*F(n + 1) + 4*(n + 1)*F(n))/5, with F(n) = A000045(n) (Fibonacci).

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

a(n) = Sum_{i=0..floor((n+3)/2)} binomial(n+3-i, i)*(n + 2 - 2*i). - John M. Campbell, Jan 04 2016

E.g.f.: exp(x/2)*((50 + 55*x)*cosh(sqrt(5)*x/2) + sqrt(5)*(18 + 25*x)*sinh(sqrt(5)*x/2))/25. - Stefano Spezia, Dec 04 2023