The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A000032 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A000032

Lucas numbers beginning at 2: L(n) = L(n-1) + L(n-2), L(0) = 2, L(1) = 1.
(history; published version)

#710 by Alois P. Heinz at Wed Sep 25 16:10:42 EDT 2024

COMMENTS

The deviation being the reciprocal of the corresponding power of phi:

a(n) - phi^n = 1/(-phi)^n, e.g.

n = 1; a(1) = 2; phi = 1.618..; 1/phi = 0.618.. -> 1 - phi^1 = -1/phi^1;

n = 2; a(2) = 3; phi^2 = 2.618..; 1/phi^2 = 0.382.. -> 3 - phi^2 = 1/phi^2;

n = 3; a(3) = 4; phi^3 = 4.236..; 1/phi^3 = 0.236.. -> 4 - phi^3 = -1/phi^3;

and - trivially - for n = 0; a(0) = 2 -> 2 - phi^0 = 1/phi^0).

STATUS

editing

approved

#709 by Alois P. Heinz at Wed Sep 25 16:07:51 EDT 2024

STATUS

proposed

editing

Discussion

Wed Sep 25

16:09

Alois P. Heinz: again: the comment is not signed ... again: this is redundant ...

16:09

Alois P. Heinz: we stand by our decision ...

16:10

Alois P. Heinz: do not propose this again ...

#708 by Jörg Zurkirchen at Wed Sep 25 08:59:34 EDT 2024

STATUS

editing

proposed

Discussion

Wed Sep 25

10:36

Stefano Spezia: Your comment should be signed and moved to last of all the other comments

10:36

Stefano Spezia: To sign, please append - ~~~~

16:07

Alois P. Heinz: his has been rejected before, see history ...

#707 by Jörg Zurkirchen at Wed Sep 25 08:57:43 EDT 2024

COMMENTS

The deviation being the reciprocal of the corresponding power of phi:

a(n) - phi^n = 1/(-phi)^n, e.g.

n = 1; a(1) = 2; phi = 1.618..; 1/phi = 0.618.. -> 1 - phi^1 = -1/phi^1;

n = 2; a(2) = 3; phi^2 = 2.618..; 1/phi^2 = 0.382.. -> 3 - phi^2 = 1/phi^2;

n = 3; a(3) = 4; phi^3 = 4.236..; 1/phi^3 = 0.236.. -> 4 - phi^3 = -1/phi^3;

and - trivially - for n = 0; a(0) = 2 -> 2 - phi^0 = 1/phi^0).

STATUS

approved

editing

#706 by Alois P. Heinz at Tue Sep 24 09:52:50 EDT 2024

COMMENTS

The deviation being the reciprocal of the corresponding power of phi:

a(n) - phi^n = 1 / (-phi)^n,

e.g. 1 - phi^1 = -1/phi^1; 3 - phi^2 = 1/phi^2; 4 - phi^3 = -1/phi^3

(and - trivially - starting with the Lucas number sequence, a(0) = 2: 2 - phi^0 = 1/phi^0).

KEYWORD

nonn,nice,easy,core,changed

STATUS

editing

approved

#705 by Alois P. Heinz at Tue Sep 24 09:39:47 EDT 2024

STATUS

proposed

editing

Discussion

Tue Sep 24

09:48

Alois P. Heinz: see formula below: 
a(n) = (phi)^n + ( - phi)^( - n). - Paul Barry, Mar 12 2005

09:48

Alois P. Heinz: and also: 
https://oeis.org/wiki/Style_Sheet#Signing_your_name_when_you_contribute_to_an_existing_sequence

09:52

Alois P. Heinz: the new comment is not signed ... but also: it is redundant ...

09:52

Alois P. Heinz: rejected ...

#704 by Jörg Zurkirchen at Tue Sep 24 09:31:43 EDT 2024

STATUS

editing

proposed

#703 by Jörg Zurkirchen at Tue Sep 24 09:28:08 EDT 2024

COMMENTS

The deviation being the reciprocal of the corresponding power of phi:

a(n) - phi^n = 1 / (-phi)^n,

e.g. 1 - phi^1 = -1/phi^1; 3 - phi^2 = 1/phi^2; 4 - phi^3 = -1/phi^3

(and - trivially - starting with the Lucas number sequence, a(0) = 2: 2 - phi^0 = 1/phi^0).

STATUS

approved

editing

#702 by N. J. A. Sloane at Tue Sep 03 00:58:03 EDT 2024

STATUS

proposed

approved

#701 by Stefano Spezia at Mon Sep 02 14:54:32 EDT 2024

STATUS

editing

proposed