The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A158638 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A158638

a(n) = 48*n^2 + 1.
(history; published version)

#29 by Michel Marcus at Sun Mar 19 02:47:30 EDT 2023

STATUS

reviewed

approved

#28 by Joerg Arndt at Sun Mar 19 02:33:58 EDT 2023

STATUS

proposed

reviewed

#27 by Amiram Eldar at Sun Mar 19 02:01:00 EDT 2023

STATUS

editing

proposed

#26 by Amiram Eldar at Sun Mar 19 01:50:56 EDT 2023

DATA

1, 49, 193, 433, 769, 1201, 1729, 2353, 3073, 3889, 4801, 5809, 6913, 8113, 9409, 10801, 12289, 13873, 15553, 17329, 19201, 21169, 23233, 25393, 27649, 30001, 32449, 34993, 37633, 40369, 43201, 46129, 49153, 52273, 55489, 58801, 62209, 65713, 69313, 73009, 76801

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 Amiram Eldar, Mar 19 2023: (Start)

Sum_{n>=0} 1/a(n) = (coth(Pi/(4*sqrt(3)))*Pi/(4*sqrt(3)) + 1)/2.

Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/(4*sqrt(3)))*Pi/(4*sqrt(3)) + 1)/2. (End)

CROSSREFS

Cf. A000290, A005843, A158638.

STATUS

approved

editing

#25 by Charles R Greathouse IV at Thu Sep 08 08:45:43 EDT 2022

PROG

(MAGMAMagma) I:=[1, 49, 193]; [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 17 2012

Discussion

Thu Sep 08

08:45

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

#24 by Joerg Arndt at Tue Aug 28 02:11:47 EDT 2018

STATUS

proposed

approved

#23 by Jon E. Schoenfield at Tue Aug 28 00:46:36 EDT 2018

STATUS

editing

proposed

#22 by Jon E. Schoenfield at Tue Aug 28 00:46:33 EDT 2018

NAME

48na(n) = 48*n^2 + 1.

COMMENTS

The identity (48*n^2 + 1)^2 - (576*n^2 + 24)*(2*n)^2 = 1 can be written as a(n)^2 - A158637(n)*A005843(n)^2 = 1.

FORMULA

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

G.f.: -(1 + 46*x + 49*x^2)/(x-1)^3.

a(n) = 48*A000290(n) + 1. - Wesley Ivan Hurt, Dec 06 2013

PROG

(MAGMA) I:=[1, 49, 193]; [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 17 2012

(PARI) for(n=0, 40, print1(48*n^2 + 1", ")); \\ _Vincenzo Librandi, _, Feb 17 2012

STATUS

approved

editing

#21 by Charles R Greathouse IV at Sat Jun 17 02:52:49 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

#20 by Charles R Greathouse IV at Sat Jun 13 00:53:05 EDT 2015

LINKS

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

Discussion

Sat Jun 13

00:53

OEIS Server: https://oeis.org/edit/global/2439