A108035 -id:A108035 - cp's OEIS frontend

A108036 Triangle read by rows: the triangle in A108035 surrounded by a border of 0's.

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 3, 3, 0, 0, 5, 5, 5, 5, 0, 0, 8, 8, 8, 8, 8, 0, 0, 13, 13, 13, 13, 13, 13, 0, 0, 21, 21, 21, 21, 21, 21, 21, 0, 0, 34, 34, 34, 34, 34, 34, 34, 34, 0, 0, 55, 55, 55, 55, 55, 55, 55, 55, 55, 0, 0, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 0, 0, 144, 144, 144, 144

Offset: 0

N. J. A. Sloane, Jun 01 2005

			0; 0,0; 0,1,0; 0,2,2,0; 0,3,3,3,0; 0,5,5,5,5,0; ...

Cf. A039913, A108035.

from math import isqrt
from sympy import fibonacci
def A108036(n): return 0 if 0<=(k:=n+1<<1)-(r:=(m:=isqrt(k))*(m+1))<=2 or n<=1 else int(fibonacci(m-(k<=r))) # Chai Wah Wu, Nov 07 2024

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

A023607 a(n) = n * Fibonacci(n+1).

Original entry on oeis.org

0, 1, 4, 9, 20, 40, 78, 147, 272, 495, 890, 1584, 2796, 4901, 8540, 14805, 25552, 43928, 75258, 128535, 218920, 371931, 630454, 1066464, 1800600, 3034825, 5106868, 8580897, 14398412, 24129160, 40388070, 67527579, 112786496, 188195271

Offset: 0

Clark Kimberling

Convolution of Fibonacci numbers and Lucas numbers.
Central terms of the triangle in A119457 for n>0. - Reinhard Zumkeller, May 20 2006
d/dx(1 + x + 2x^2 + 3x^3 + 5x^4 + 8x^5 + ...) = (1 + 4x + 9x^2 + ...). - Gary W. Adamson, Jun 27 2009
For n > 0: sums of rows of the triangle in A108035. - Reinhard Zumkeller, Oct 08 2012

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
M. Griffiths, A Restricted Random Walk defined via a Fibonacci Process, Journal of Integer Sequences, Vol. 14 (2011), #11.5.4.
Milan Janjić, Hessenberg Matrices and Integer Sequences, J. Int. Seq. 13 (2010) # 10.7.8, section 3.
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

First differences of A094584.
Second column of triangle A016095.
Cf. A000045, A104796.

a023607 n = a023607_list !! n
a023607_list = zipWith (*) [0..] $ tail a000045_list
-- Reinhard Zumkeller, Oct 08 2012

A023607 := proc(n)
    n*combinat[fibonacci](n+1) ;
end proc:
seq(A023607(n),n=0..10) ; # R. J. Mathar, Jul 15 2017

Times@@@Thread[{Range[0, 50], Fibonacci[Range[51]]}]  (* Harvey P. Dale, Mar 08 2011 *)
Table[n*Fibonacci[n + 1], {n, 0, 50}]

a(n)=n*fibonacci(n+1) \\ Charles R Greathouse IV, Sep 24 2015

O.g.f.: x(2x+1)/(1-x-x^2)^2. - Len Smiley, Dec 11 2001
a(n) = n*Sum_{k=0..n} binomial(k,n-k). - Paul Barry, Sep 25 2004
a(n) = A215082(2n-2) + A215082(2n-1). - Philippe Deléham, Aug 03 2012
a(n) = Sum_{i=1..n} A000045(i)*A000032(n-i+1). - Vladimir Kruchinin, Nov 08 2013

Simpler description from Samuel Lachterman (slachterman(AT)fuse.net), Sep 19 2003

Name improved by T. D. Noe, Mar 08 2011

A039913 Triangular "Fibonacci array".

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 3, 3, 2, 3, 5, 4, 5, 3, 5, 8, 7, 7, 8, 5, 8, 13, 11, 12, 11, 13, 8, 13, 21, 18, 19, 19, 18, 21, 13, 21, 34, 29, 31, 30, 31, 29, 34, 21, 34, 55, 47, 50, 49, 49, 50, 47, 55, 34, 55, 89, 76, 81, 79, 80, 79, 81, 76, 89, 55, 89, 144, 123, 131, 128, 129, 129, 128

Offset: 0

N. J. A. Sloane

Sum of n-th row = 2*A001629(n+1). - Reinhard Zumkeller, Oct 07 2012

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
L. Carlitz, A Fibonacci array, Fib. Quart. 1(#2) (1963), 217-28.

Cf. A108035.

a039913 n k = a039913_tabl !! n !! k
a039913_row n = a039913_tabl !! n
a039913_tabl = [[0], [1, 1]] ++ f [0] [1, 1] where
   f us@(u:us') vs@(v:vs') = ws : f vs ws where
     ws = [u + v, u + v + v] ++ zipWith (+) us vs'
-- Reinhard Zumkeller, Oct 07 2012

a(0, n)=Fib(n), a(1, n)=Fib(n+2), a(r, n)=a(r-1, n)+a(r-2, n), r >= 2.

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

More terms from Larry Reeves (larryr(AT)acm.org), Sep 28 2000

A108037 Triangle read by rows: n-th row is n-th nonzero Fibonacci number repeated n+1 times.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 5, 5, 5, 5, 5, 5, 8, 8, 8, 8, 8, 8, 8, 13, 13, 13, 13, 13, 13, 13, 13, 21, 21, 21, 21, 21, 21, 21, 21, 21, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 144, 144

Offset: 0

N. J. A. Sloane, Jun 01 2005

			0; 1,1; 1,1,1; 2,2,2,2; 3,3,3,3,3; 5,5,5,5,5,5; ...

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened

Cf. A039913, A108036, A108035.

Cf. A099920 (row sums).

a108037 n k = a108037_tabl !! n !! k
a108037_row n = a108037_tabl !! n
a108037_tabl = zipWith replicate [1..] a000045_list
-- Reinhard Zumkeller, Oct 07 2012

Table[Table[Fibonacci[n],{n+1}],{n,0,12}]//Flatten (* Harvey P. Dale, May 07 2017 *)

from math import isqrt
from sympy import fibonacci
def A108037(n): return int(fibonacci((m:=isqrt(k:=n+1<<1))-(k<=m*(m+1)))) # Chai Wah Wu, Nov 07 2024

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

cp's OEIS Frontend

A108036 Triangle read by rows: the triangle in A108035 surrounded by a border of 0's.

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A023607 a(n) = n * Fibonacci(n+1).

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A039913 Triangular "Fibonacci array".

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A108037 Triangle read by rows: n-th row is n-th nonzero Fibonacci number repeated n+1 times.

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