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, 393146012008229658338304, 13355391270239113019379844

Offset: 0

Author

Amarnath Murthy, Jun 08 2003

Keywords

• nonn
• easy

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

• D. W. Wilson, Table of n, a(n-1) for n = 1..100 (offset=1)
• Emrah Kılıç, Yücel Türker Ulutaş, and Neşe Ömür, A Formula for the Generating Functions of Powers of Horadam's Sequence with Two Additional Parameters, J. Int. Seq. 14 (2011), Article 11.5.6, table 3, k=2.
• Thomas Koshy, Products Involving Reciprocals of Gibonacci Polynomials, The Fibonacci Quarterly, Vol. 60, No. 1 (2022), pp. 15-24.
• Index entries for linear recurrences with constant coefficients, signature (35,-35,1).

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).

Programs

Magma

[4*Evaluate(ChebyshevU(n), 3)^2: n in [0..30]]; // 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 *)

SageMath

[4*chebyshev_U(n-1, 3)^2 for n in (0..30)] # G. C. Greubel, Aug 18 2022

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.

Limit_{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

Product_{n>=2} (1 - 4/a(n)) = sqrt(2)/3 + 1/2 (Koshy, 2022, section 3, p. 19). - Amiram Eldar, Jan 23 2025

Extensions

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