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A084703 Squares n such that 2*n+1 is also a square. 12
0, 4, 144, 4900, 166464, 5654884, 192099600, 6525731524, 221682772224, 7530688524100, 255821727047184, 8690408031080164, 295218051329678400, 10028723337177985444, 340681375412721826704, 11573138040695364122500 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

With the exception of 0, a subsequence of A075114. - R. J. Mathar, Dec 15 2008

Consequently, A014105(k) is a square if and only if k = a(n). - Bruno Berselli, Oct 14 2011

From M. F. Hasler, Jan 17 2012: (Start)

Bisection of A079291. The squares 2*n+1 are given in A055792.

A204576 is this sequence written in binary. (End)

a(n+1),n >= 0, is the perimeter squared (x(n)+y(n)+z(n))^2 of the ordered primitive Pythagorean triple (x(n), y(n)=x(n)+1, z(n)). The first two terms are (x(0)=0, y(0)=1, z(0)=1), a(1) = 2^2, and (x(1)=3, y(1)=4, z(1)=5), a(2) = 12^2. - George F. Johnson, Nov 02 2012

LINKS

Table of n, a(n) for n=0..15.

D. W. Wilson, Table of n, a(n-1) for n = 1..100 (offset=1)

E. Kilic, Y. T. Ulutas, N. Omur, A Formula for the Generating Functions of Powers of Horadam's Sequence with Two Additional Parameters, J. Int. Seq. 14 (2011) #11.5.6, table 3, k=2.

Index entries for linear recurrences with constant coefficients, signature (35,-35,1).

FORMULA

a(n) = 4*A001110(n) = A001542(n)^2.

a(n+1) = A001652(n)*A001652(n+1) + A046090(n)*A046090(n+1) = A001542(n+1)^2. - Charlie Marion, Jul 01 2003

For k>=n>=0, a(n) = A001653(k+n)*A001653(k-n) - A001653(k)^2; e.g. 144 = 5741*5 - 169^2. - Charlie Marion, Jul 16 2003

G.f.: 4*x*(1+x)/((1-x)*(1-34*x+x^2)). - R. J. Mathar, Dec 15 2008

a(n) = A079291(2n). - M. F. Hasler, Jan 16 2012

From George F. Johnson, Nov 02 2012: (Start)

a(n) = ((17+12*sqrt(2))^n + (17-12*sqrt(2))^n - 2)/8.

a(n+1) = 17*a(n) + 4 + 12*sqrt(a(n)*(2*(a(n) + 1)).

a(n-1) = 17*a(n) + 4 - 12*sqrt(a(n)*(2*(a(n) + 1)).

a(n-1)*a(n+1) = (a(n) - 4)^2.

2*a(n) + 1 = (A001541(n))^2.

a(n+1) = 34*a(n) - a(n-1) + 8 for n>1, a(0)=0, a(1)=4.

a(n+1) = 35*a(n) - 35*a(n-1) + a(n-2) for n>0, a(0)=0, a(1)=4, a(2)=144.

a(n)*a(n+1) = (4*A029549(n))^2.

a(n+1) - a(n) = 4*A046176(n).

a(n) + a(n+1) = 4*(6*A029549(n) + 1).

a(n) = (2*A001333(n)*A000129(n))^2.

Lim_{n -> infinity} a(n)/a(n-r) = (17+12*sqrt(2))^r.

(End)

Empirical: for n>0, a(n) = A089928(4*n-2). - Alex Ratushnyak, Apr 12 2013

MATHEMATICA

a[0] = 0; a[1] = 1; a[n_] := 34a[n - 1] - a[n - 2] + 2; Table[ 4a[n], {n, 0, 15}]

CROSSREFS

Cf. A001110, A055792, A075114, A079291, A084702, A204576.

Cf. similar sequences with closed form ((1 + sqrt(2))^(4*r) + (1 - sqrt(2))^(4*r))/8 + k/4: this sequence (k=-1), A076218 (k=3), A278310 (k=-5).

Sequence in context: A186720 A060870 A268894 * A186418 A122747 A069135

Adjacent sequences:  A084700 A084701 A084702 * A084704 A084705 A084706

KEYWORD

nonn,easy,changed

AUTHOR

Amarnath Murthy, Jun 08 2003

EXTENSIONS

Edited and extended by Robert G. Wilson v, Jun 15 2003

STATUS

approved

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Last modified August 24 00:29 EDT 2017. Contains 291052 sequences.