A084703 Squares k such that 2*k+1 is also a square.

0, 4, 144, 4900, 166464, 5654884, 192099600, 6525731524, 221682772224, 7530688524100, 255821727047184, 8690408031080164, 295218051329678400, 10028723337177985444, 340681375412721826704, 11573138040695364122500

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*k+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, and 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

a(n) = A001653(k+n)*A001653(k-n) - A001653(k)^2, for k >= n >= 0; 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: a(n) = A089928(4*n-2), for n > 0. - Alex Ratushnyak, Apr 12 2013

a(n) = 4*A001109(n)^2. - G. C. Greubel, Aug 18 2022

Mathematica

b[n_]:= b[n]= If[n<2, n, 34*b[n-1] -b[n-2] +2]; (* b=A001110 *)
a[n_]:= 4*b[n]; Table[a[n], {n, 0, 30}]
4*ChebyshevU[Range[-1, 30], 3]^2 (* G. C. Greubel, Aug 18 2022 *)

Prog

(Magma) [4*Evaluate(ChebyshevU(n), 3)^2: n in [0..30]]; // G. C. Greubel, Aug 18 2022
(SageMath) [4*chebyshev_U(n-1, 3)^2 for n in (0..30)] # G. C. Greubel, Aug 18 2022

Crossrefs

Cf. A000129, A001109, A001110, A001333, A001541, A001542, A001652, A001653.

Cf. A014105, A029549, A046090, A046176, A055792, A075114, A079291, A084702.

Cf. A089928, 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

Author

Amarnath Murthy, Jun 08 2003

Extensions

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

Status

approved