The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A130284

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A130284

Integers j > 0 such that (2j+1)^2(m^2-1) + 1 is a square for some integer m > 1.
(history; published version)

#13 by Susanna Cuyler at Sun Jun 13 10:19:20 EDT 2021

STATUS

reviewed

approved

#12 by Joerg Arndt at Sun Jun 13 08:50:49 EDT 2021

STATUS

proposed

reviewed

#11 by Jon E. Schoenfield at Sun Jun 13 08:44:46 EDT 2021

STATUS

editing

proposed

#10 by Jon E. Schoenfield at Sun Jun 13 08:44:42 EDT 2021

NAME

Integers nj > 0 such that (2n2j+1)^2(m^2-1) + 1 is a square for some integer m > 1.

FORMULA

A130284 = { P[k](m) ; k=1,2,3,..., m=2,3,4,... } where P[k] = (sqrt((X^2 Q[k]^2 - 1)/(X^2 - 1))-1)/2 and Q[0] = Q[ -1] = 1, Q[k+1] = (4X^2 -2)*Q[k] - Q[k-1]. Furthermore, (2P[k](m)+1)^2 (m^2 - 1)+1 = m^2 Q[k](m)^2, thus A130280(P[k](m)) <= m. So far, no case is known where we have strict inequality.

EXAMPLE

Up to k=17, a(k)=P[1](k+1) with P[1] = 2x^2 - 1, A130280(a(k)) = k+1.

a(18) = P[2](2) < P[1](19) with P[2] = 2x^2 *(4x^2 - 3), A130280(a(18)) = 2.

STATUS

approved

editing

#9 by Bruno Berselli at Fri May 12 08:33:21 EDT 2017

STATUS

reviewed

approved

#8 by Joerg Arndt at Fri May 12 08:32:53 EDT 2017

STATUS

proposed

reviewed

#7 by Jean-François Alcover at Fri May 12 08:28:57 EDT 2017

STATUS

editing

proposed

#6 by Jean-François Alcover at Fri May 12 08:28:00 EDT 2017

MATHEMATICA

r[n_] := Reduce[m>1 && k>1 && (2n+1)^2*(m^2-1)+1 == k^2, {m, k}, Integers];

Reap[For[n=1, n <= 5000, n++, If[r[n] =!= False, Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, May 12 2017 *)

STATUS

approved

editing

Discussion

Fri May 12

08:28

Jean-François Alcover: Next terms up to 5000 = 4231, 4417, 4607, 4801, 4850, 4999

#5 by Russ Cox at Sat Mar 31 13:48:24 EDT 2012

AUTHOR

_M. F. Hasler (Maximilian.Hasler(AT)gmail.com), _, May 24 2007, May 29 2007

Discussion

Sat Mar 31

13:48

OEIS Server: https://oeis.org/edit/global/893

#4 by N. J. A. Sloane at Sun Jul 11 03:00:00 EDT 2010

FORMULA

a(n)=2*n^2+8*n+7 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 22 2009]

EXAMPLE

n=0, a(0)=7; n=1, a(1)=17; n=2, a(2)=31 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 22 2009]

KEYWORD

nonn,new

nonn