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 A084703 Squares k such that 2*k+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*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) 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

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Last modified February 4 03:47 EST 2023. Contains 360045 sequences. (Running on oeis4.)