A108037 -id:A108037 - cp's OEIS frontend

A099920 a(n) = (n+1)*F(n), F(n) = Fibonacci numbers A000045.

0, 2, 3, 8, 15, 30, 56, 104, 189, 340, 605, 1068, 1872, 3262, 5655, 9760, 16779, 28746, 49096, 83620, 142065, 240812, 407353, 687768, 1159200, 1950650, 3277611, 5499704, 9216519, 15426870, 25793240, 43080608, 71884197, 119835652

Offset: 0

Paul Barry and Ralf Stephan, Oct 15 2004

A Fibonacci-Lucas convolution.
The number of edges in the Lucas cube L_(n+1) [Klavzar]. - R. J. Mathar, Nov 05 2008
Sums of rows of the triangle in A108037. - Reinhard Zumkeller, Oct 07 2012
a(n-1) is the total binary weight of all bimultus bitstrings of length n. A bitstring is bimultus if each of its 1's possess at least one neighboring 1 and each of its 0's possess at least one neighboring 0. - Steven Finch, May 26 2020

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 35.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
S. Klavzar, On median nature and enumerative properties of Fibonacci-like cubes, Discr. Math. 299 (2005), 145-153.
Franck Ramaharo, A one-variable bracket polynomial for some Turk's head knots, arXiv:1807.05256 [math.CO], 2018.
Steven Finch, Variance of longest run duration in a random bitstring, arXiv:2005.12185 [math.CO], 2020.
Tamás Szakács, Convolution of second order linear recursive sequences. II. Commun. Math. 25, No. 2, 137-148 (2017), remark 2.
Tamás Szakács, Linear recursive sequences and factorials, Ph. D. Thesis, Univ. Debrecen (Hungary, 2024). See p. 35.
Eric Weisstein's World of Mathematics, Edge Count.
Eric Weisstein's World of Mathematics, Lucas Cube Graph.
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1)

Equals A010049(n) + A001629(n+1).

Cf. A000045, A000032, A045925, A023607.

a099920 n = a099920_list !! n
a099920_list = zipWith (*) [1..] a000045_list
-- Reinhard Zumkeller, Oct 07 2012

[(n+1)*Fibonacci(n): n in [0..60]]; // Vincenzo Librandi, Apr 23 2011

Table[(n + 1) Fibonacci[n], {n, 0, 40}] (* Harvey P. Dale, Jan 18 2012 *)
LinearRecurrence[{2, 1, -2, -1}, {0, 2, 3, 8}, 40] (* Harvey P. Dale, Jan 18 2012 *)
CoefficientList[Series[(2 - x) x/(-1 + x + x^2)^2, {x, 0, 20}], x] (* Eric W. Weisstein, Jul 28 2023 *)

a(n)=(n+1)*fibonacci(n) \\ Charles R Greathouse IV, Jun 11 2015

G.f.: x*(2-x)/(1-x-x^2)^2;
a(n) = Sum_{k=0..n} F(n-k)*(L(k-1) + 0^k).
a(n) = Sum_{k=0..n+1} F(n-k)*binomial(n-k+1, k)*binomial(1, (k+1)/2)*(1-(-1)^k)/2.
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4); a(0)=0, a(1)=2, a(2)=3, a(3)=8. - Harvey P. Dale, Jan 18 2012
a(n) = a(n-1) + a(n-2) + A000032(n-1) (Lucas numbers). - Bob Selcoe, Aug 19 2015
a(n) = 2*A001629(n+1) - A001629(n). - R. J. Mathar, Feb 04 2022

Entry revised by N. J. A. Sloane, Jan 23 2006. The offset changed, so some of the formulas may now be slightly off.

A108617 Triangle read by rows: T(n,k) = T(n-1,k-1) + T(n-1,k) for 0 < k < n, T(n,0) = T(n,n) = n-th Fibonacci number.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 3, 3, 2, 3, 5, 6, 5, 3, 5, 8, 11, 11, 8, 5, 8, 13, 19, 22, 19, 13, 8, 13, 21, 32, 41, 41, 32, 21, 13, 21, 34, 53, 73, 82, 73, 53, 34, 21, 34, 55, 87, 126, 155, 155, 126, 87, 55, 34, 55, 89, 142, 213, 281, 310, 281, 213, 142, 89, 55, 89, 144, 231, 355, 494, 591, 591, 494, 355, 231, 144, 89

Offset: 0

Reinhard Zumkeller, Jun 12 2005

Sum of n-th row = 2*A027934(n). - Reinhard Zumkeller, Oct 07 2012

			Triangle begins:
   0;
   1,   1;
   1,   2,   1;
   2,   3,   3,   2;
   3,   5,   6,   5,   3;
   5,   8,  11,  11,   8,   5;
   8,  13,  19,  22,  19,  13,   8;
  13,  21,  32,  41,  41,  32,  21,  13;
  21,  34,  53,  73,  82,  73,  53,  34,  21;
  34,  55,  87, 126, 155, 155, 126,  87,  55,  34;
  55,  89, 142, 213, 281, 310, 281, 213, 142,  89,  55;

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
Hacéne Belbachir and László Szalay, On the Arithmetic Triangles, Šiauliai Mathematical Seminar, Vol. 9 (17), 2014. See Fig. 1 p. 18.
Eric Weisstein's World of Mathematics, Fibonacci Number.
Eric Weisstein's World of Mathematics, Pascal's Triangle.
Wikipedia, Fibonacci number.
Wikipedia, Pascal's triangle.
Index entries for triangles and arrays related to Pascal's triangle

Cf. A007318, A074829, A108037.

T(2n,n) gives 2*A176085(n).

a108617 n k = a108617_tabl !! n !! k
a108617_row n = a108617_tabl !! n
a108617_tabl = [0] : iterate f [1,1] where
   f row@(u:v:_) = zipWith (+) ([v - u] ++ row) (row ++ [v - u])
-- Reinhard Zumkeller, Oct 07 2012

function T(n,k) // T = A108617
  if k eq 0 or k eq n then return Fibonacci(n);
  else return T(n-1,k-1) + T(n-1,k);
  end if;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 20 2023

A108617 := proc(n,k) option remember;
    if k = 0 or k=n then
        combinat[fibonacci](n) ;
    elif k <0 or k > n then
        0 ;
    else
        procname(n-1,k-1)+procname(n-1,k) ;
    end if;
end proc: # R. J. Mathar, Oct 05 2012

a[1]:={0}; a[n_]:= a[n]= Join[{Fibonacci[#]}, Map[Total, Partition[a[#],2,1]], {Fibonacci[#]}]&[n-1]; Flatten[Map[a, Range[15]]] (* Peter J. C. Moses, Apr 11 2013 *)

def T(n,k): # T = A108617
    if (k==0 or k==n): return fibonacci(n)
    else: return T(n-1,k-1) + T(n-1,k)
flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Oct 20 2023

T(n,0) = T(n,n) = A000045(n);
T(n,1) = T(n,n-1) = A000045(n+1) for n>0;
T(n,2) = T(n,n-2) = A000045(n+2) - 2 = A001911(n-1) for n>1;
Sum_{k=0..n} T(n,k) = 2*A027934(n-1) for n>0.
Sum_{k=0..n} (-1)^k*T(n, k) = 2*((n+1 mod 2)*Fibonacci(n-2) + [n=0]). - G. C. Greubel, Oct 20 2023

A108035 Triangle read by rows: n-th row consists of n copies of the n-th nonzero Fibonacci number.

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 5, 5, 5, 5, 8, 8, 8, 8, 8, 13, 13, 13, 13, 13, 13, 21, 21, 21, 21, 21, 21, 21, 34, 34, 34, 34, 34, 34, 34, 34, 55, 55, 55, 55, 55, 55, 55, 55, 55, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 144, 144, 144, 144, 144, 144, 144, 144, 144, 144, 144, 233, 233, 233, 233, 233, 233, 233, 233, 233, 233, 233, 233

Offset: 1

N. J. A. Sloane, Jun 01 2005

			1; 2,2; 3,3,3; 5,5,5,5; 8,8,8,8,8; ...

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened

Cf. A000045, A039913, A108036, A108037.

Cf. A023607 (row sums).

a108035 n k = a108035_tabl !! (n-1) !! (n-1)
a108035_row n = a108035_tabl !! (n-1)
a108035_tabl = zipWith replicate [1..] $ drop 2 a000045_list
-- Reinhard Zumkeller, Oct 07 2012

Flatten[Table[Table[Fibonacci[n],{n-1}],{n,13}]] (* Harvey P. Dale, Jul 18 2015 *)

from math import isqrt
from sympy import fibonacci
def A108035(n): return int(fibonacci(1+(m:=isqrt(k:=n<<1))+(k>m*(m+1)))) # Chai Wah Wu, Nov 07 2024

G.f.: (1+x+y)/((1-x-x^2)*(1-y-y^2)). [U coordinates]

Definition clarified by N. J. A. Sloane, Nov 09 2024

cp's OEIS Frontend

A099920 a(n) = (n+1)*F(n), F(n) = Fibonacci numbers A000045.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

PARI

Formula

Extensions

A108617 Triangle read by rows: T(n,k) = T(n-1,k-1) + T(n-1,k) for 0 < k < n, T(n,0) = T(n,n) = n-th Fibonacci number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

SageMath

Formula

A108035 Triangle read by rows: n-th row consists of n copies of the n-th nonzero Fibonacci number.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

Python

Formula

Extensions