The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A157910

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A157910

a(n) = 18*n^2 - 1.
(history; published version)

#26 by Joerg Arndt at Tue Mar 07 02:26:59 EST 2023

STATUS

reviewed

approved

#25 by Michel Marcus at Tue Mar 07 02:21:15 EST 2023

STATUS

proposed

reviewed

#24 by Amiram Eldar at Tue Mar 07 02:09:24 EST 2023

STATUS

editing

proposed

#23 by Amiram Eldar at Tue Mar 07 01:48:02 EST 2023

DATA

17, 71, 161, 287, 449, 647, 881, 1151, 1457, 1799, 2177, 2591, 3041, 3527, 4049, 4607, 5201, 5831, 6497, 7199, 7937, 8711, 9521, 10367, 11249, 12167, 13121, 14111, 15137, 16199, 17297, 18431, 19601, 20807, 22049, 23327, 24641, 25991, 27377, 28799, 30257, 31751

LINKS

Vincenzo Librandi, <a href="https://web.archive.org/web/20090309225914/http://mathforum.org/kb/message.jspa?messageID=5785989&tstart=0">X^2-AY^2=1</a>, Math Forum, 2007. [Wayback Machine link]

FORMULA

From Vincenzo Librandi, Feb 08 2012: (Start)

G.f.: x*(-17 - 20*x + x^2)/(x - 1)^3. - _Vincenzo Librandi_, Feb 08 2012

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _Vincenzo Librandi_, Feb 08 2012(End)

#22 by Amiram Eldar at Tue Mar 07 01:46:46 EST 2023

FORMULA

From Amiram Eldar, Mar 07 2023: (Start)

Sum_{n>=1} 1/a(n) = (1 - cot(Pi/(3*sqrt(2)))*Pi/(3*sqrt(2)))/2.

Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/(3*sqrt(2)))*Pi/(3*sqrt(2)) - 1)/2. (End)

STATUS

approved

editing

#21 by Charles R Greathouse IV at Thu Sep 08 08:45:42 EDT 2022

PROG

(MAGMAMagma) I:=[17, 71, 161]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 08 2012

Discussion

Thu Sep 08

08:45

OEIS Server: https://oeis.org/edit/global/2944

#20 by Joerg Arndt at Wed Aug 22 07:49:56 EDT 2018

STATUS

proposed

approved

#19 by Jon E. Schoenfield at Wed Aug 22 07:12:15 EDT 2018

STATUS

editing

proposed

#18 by Jon E. Schoenfield at Wed Aug 22 07:12:13 EDT 2018

NAME

a(n) = 18*n^2 - 1.

COMMENTS

The identity (18*n^2 - 1)^2 - (81*n^2 - 9)*(2*n)^2 = 1 can be written as a(n)^2 - A157909(n)*A005843(n)^2 = 1. - _Vincenzo Librandi, _, Feb 08 2012

FORMULA

G.f.: x*(-17 - 20*x + x^2)/(x - 1)^3. - _Vincenzo Librandi, _, Feb 08 2012

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _Vincenzo Librandi, _, Feb 08 2012

PROG

(MAGMA) I:=[17, 71, 161]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // _Vincenzo Librandi, _, Feb 08 2012

(PARI) for(n=1, 40, print1(18*n^2 - 1", ")); \\ _Vincenzo Librandi, _, Feb 08 2012

STATUS

approved

editing

#17 by Charles R Greathouse IV at Sat Jun 17 02:52:39 EDT 2017

LINKS

<a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

Discussion

Sat Jun 17

02:52

OEIS Server: https://oeis.org/edit/global/2661