A108038 -id:A108038 - cp's OEIS frontend

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

Wolfdieter Lang, Feb 15 2002

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

			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)

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

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).

Row sums of A108038.

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

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

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

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 *)

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

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

A360932 Primes of the form H(m,k) = F(k+1)F(m-k+2) - F(k)F(m-k+1), where F(m) is the m-th Fibonacci number and m >= 0, 0 <= k <= m.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 23, 29, 37, 41, 47, 89, 97, 103, 107, 157, 167, 173, 199, 233, 443, 521, 733, 1597, 1741, 1867, 1871, 1877, 2207, 3037, 3571, 7841, 7919, 7951, 9349, 11933, 12823, 28657, 33503, 50549, 54277, 54287, 54293, 54319, 54497, 55717, 142099

Offset: 1

Rigoberto Florez, Feb 25 2023

nonn

This sequence appears in the triangle A108038 in this order (reading by rows): 3, 2, 7, 5, 11, 13, 29, 23, 47, 37, 41, 97, 107, 103, 89, 199, 157, 173, 167.
Are there infinitely many primes of the form H(m,k)?
This sequence appears within the determinant Hosoya triangle.

			29 is a term because it is prime and A108038(8,2) = H(8,2) = 29.  Also A108038(8,7) = H(8,7) = 29.

Robert Israel, Table of n, a(n) for n = 1..9313
Hsin-Yun Ching, Rigoberto Florez, F. Luca, Antara Mukherjee, and J. C. Saunders, Primes and composites in the determinant Hosoya triangle, Fibonacci Quarterly, 2023.

Cf. A000040, A000045, A005478 (subsequence), A108038, A153892, A067331.

Res:= {}:
M:= 50: # for terms <= F(M)
fmax:= combinat:-fibonacci(M):
T[1]:= [1,1]:
T[2]:= [1,3,1]:
for i from 3 to M do
  t1:= [op(T[i-1][1..i-1] + T[i-2][1..i-1]),T[i-1][i],0];
  t2:= ListTools:-Reverse(t1);
  T[i]:= zip(max,t1,t2);
  Res:= Res union convert(select(t -> t <= fmax and isprime(t), T[i][1..ceil((i+1)/2)]),set)
od:
sort(convert(Res,list)); # Robert Israel, Mar 14 2024

H[r_, k_] := Det[{{Fibonacci[r-k+2], Fibonacci[r-k+1]}, {Fibonacci[k], Fibonacci[k+1]}}];  DeterminantPrimes[t_, m_] :=  Table[If[PrimeQ[H[r,k]],H[r,k], Unevaluated[Sequence[]]], {r,t,m}, {k,1,Ceiling[r/2]}]; ListOfPrimes[t_,m_]:=  Sort[DeleteDuplicates[Flatten[DeterminantPrimes[t, m]]]]; ListOfPrimes[2, 100]
Select[Union[Flatten[Table[Fibonacci[k+1]Fibonacci[m-k+2]-Fibonacci[k]Fibonacci[m-k+1],{m,0,40},{k,0,m}]]],PrimeQ] (* Harvey P. Dale, Aug 14 2025 *)

Intersection of A000040 and A108038.

cp's OEIS Frontend

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

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A360932 Primes of the form H(m,k) = F(k+1)F(m-k+2) - F(k)F(m-k+1), where F(m) is the m-th Fibonacci number and m >= 0, 0 <= k <= m.

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cp's OEIS Frontend

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

Views

Author

Keywords

Comments

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References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A360932 Primes of the form H(m,k) = F(k+1)*F(m-k+2) - F(k)*F(m-k+1), where F(m) is the m-th Fibonacci number and m >= 0, 0 <= k <= m.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A360932 Primes of the form H(m,k) = F(k+1)F(m-k+2) - F(k)F(m-k+1), where F(m) is the m-th Fibonacci number and m >= 0, 0 <= k <= m.