The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A341616

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A341616

Table read by ascending antidiagonals: T(n,j) = Fibonacci(n)*Lucas(n+j), product of the n-th term in the Fibonacci sequence (with F(1)=1 and F(2)=1) and the (n+j)-th term in the Lucas sequence (with L(1)=1 and L(2)=3 and j=0,1,2,...).
(history; published version)

#24 by N. J. A. Sloane at Sun Mar 28 00:22:29 EDT 2021

STATUS

editing

approved

#23 by N. J. A. Sloane at Sun Mar 28 00:22:25 EDT 2021

NAME

Table read by ascending antidiagonals: T(n,j) = Fibonacci(n)*Lucas(n+j), product of the n-th term in the Fibonacci sequence (with F(1)=1 and F(2)=1) and the (n+j)-th term in the Lucas sequence (with L(1)=1 and L(2)=3 and j=0,1,2,...) read by ascending antidiagonals.

STATUS

proposed

editing

#22 by Michel Marcus at Fri Mar 05 04:14:31 EST 2021

STATUS

editing

proposed

Discussion

Fri Mar 05

09:24

Jens Rasmussen: It is absolutely ok for me to leave out the comment. It seems only to be confusing.

09:28

Jens Rasmussen: Thank you, Marcus, for clarifying the limit formulas.

17:21

Kevin Ryde: I take it first limit n->oo and any given j, and the second vice versa j->oo any given n, if that could be expressed ...

#21 by Michel Marcus at Fri Mar 05 04:13:51 EST 2021

FORMULA

lim_Lim_{n, j -> oo} T(n+1,j)/T(n,j)-> = phi^2 = 2.618....for n and j large(A104457).

lim_Lim_{n, j -> oo} T(n,j+1)/T(n,j)-> = phi = 1.618... for n and j large(A001622).

CROSSREFS

Cf. A000045, A000032, A001622, A094214, A104457.

STATUS

proposed

editing

Discussion

Fri Mar 05

04:14

Michel Marcus: ok like this ?

04:14

Michel Marcus: what about the comment ?

#20 by Jon E. Schoenfield at Thu Mar 04 02:13:24 EST 2021

STATUS

editing

proposed

Discussion

Thu Mar 04

11:26

Michel Marcus: I meant this comment: j is the offset when combining terms from the two initial sequences.

#19 by Jon E. Schoenfield at Thu Mar 04 02:12:12 EST 2021

FORMULA

T(n,j) = (phi^n - tau^n)*(phi^(n+j) + tau^(n+j))/sqrt(5).

T(n,j) = Fibonacci(2n+j) -( (-1)^n)*Fibonacci(j).

lim_{T(n+1,j)/T(n,j)-> phi^2 = 2,.618....for n and j large.

STATUS

proposed

editing

Discussion

Thu Mar 04

02:13

Jon E. Schoenfield: I'm not sure what to do about the limit notation in the Formula section, but neither of the left curly braces has a matching right curly brace.

#18 by Michel Marcus at Tue Mar 02 13:06:37 EST 2021

STATUS

editing

proposed

Discussion

Tue Mar 02

15:54

Jens Rasmussen: Sorry. My comment was that I have added the array headings as Kevin suggested. I guess it is up to you, Michel, to decide the layout.

#17 by Michel Marcus at Tue Mar 02 13:05:52 EST 2021

EXAMPLE

T(4,3) = Fibonacci(2*4+3)-((-1)^4)*Fibonacci(3) = 89-2 = 87 or Fibonacci(4)*Lucas(4+3) = 3*29 = 87.

. .. .. .. .. .. ..

... .. .. .. .. .. ..

PROG

(PARI) T(n, j) = fibonacci(2*n+j) - (-1)^n*fibonacci(j);

matrix(7, 7, n, k, T(n, k-1)) \\ Michel Marcus, Mar 02 2021

STATUS

proposed

editing

Discussion

Tue Mar 02

13:06

Michel Marcus: I don't really understand the comment

#16 by Jens Rasmussen at Sat Feb 27 16:22:39 EST 2021

STATUS

editing

proposed

#15 by Jens Rasmussen at Sat Feb 27 16:18:52 EST 2021

EXAMPLE

n\ j =0 j=1 j=2 j=3 j=4 ..

n=1 1 3 4 7 11 ..

n=2 3 4 7 11 18 ..

n=3 8 14 22 36 58 ..

n=4 21 33 54 87 141 ..

STATUS

proposed

editing

Discussion

Sat Feb 27

16:22

Jens Rasmussen: Array headings more clear to understand