A046176 Indices of square numbers that are also hexagonal.

1, 35, 1189, 40391, 1372105, 46611179, 1583407981, 53789260175, 1827251437969, 62072759630771, 2108646576008245, 71631910824649559, 2433376321462076761, 82663163018885960315, 2808114166320660573949

Offset

1,2

Comments

Bisection (even part) of Chebyshev sequence with Diophantine property.

(3*b(n))^2 - 2*(2*a(n+1))^2 = 1 with companion sequence b(n) = A077420(n), n >= 0.

Sequence also refers to inradius of primitive Pythagorean triangles with consecutive legs, odd followed by even. - Lekraj Beedassy, Apr 23 2003

As n increases, this sequence is approximately geometric with common ratio r = lim_{n -> oo} a(n)/a(n-1) = (1 + sqrt(2))^4 = 17 + 12*sqrt(2). - Ant King, Nov 08 2011

Integers of the form sqrt((m+1)*(2*m+1)). The corresponding values of m form A078522. Subsequence of A284876. - Jonathan Sondow, Apr 07 2017

References

M. Rignaux, Query 2175, L'Intermédiaire des Mathématiciens, 24 (1917), 80.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Tanya Khovanova, Recursive Sequences.

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 4, k=1, t=3.

Serge Perrine, About the diophantine equation z^2 = 32y^2 - 16, SCIREA Journal of Mathematics (2019) Vol. 4, Issue 5, 126-139.

Maciej Ulas, On certain diophantine equations related to triangular and tetrahedral numbers, arXiv:0811.2477 [math.NT] (2008) , v_n in Theorem 5.6.

P. H. van der Kamp, Global classification of two-component..., Found. Comput. Math. 9 (5) (2009) 559-597 near Eq. (4.7).

Eric Weisstein's World of Mathematics, Hexagonal Square Number.

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (34,-1).

Formula

a(n) = 34*a(n-1) - a(n-2); a(0)=-1, a(1)=1.

a(n+1) = S(2*n, 6) = S(n, 34) + S(n-1, 34), n >= 1, with S(n, x) := U(n, x/2) Chebyshev's polynomials of the second kind. See A049310. S(n, 34) = A029547(n).

G.f.: x*(1+x)/(1-34*x+x^2).

a(n+1) = Sum_{k=0..n} (-1)^k*binomial(2*n-k, k)*6^(2*(n-k)), n >= 0.

a(n) = A001109(2n+1). - Lekraj Beedassy, Apr 23 2003

Define f[x,s] = s*x + sqrt((s^2-1)x^2+1); f[0,s]=0. a(n) = f[f[a(n-1),3],3]. - Marcos Carreira, Dec 27 2006

From Antonio Alberto Olivares, Mar 22 2008: (Start)

a(n) = (sqrt(2)/8)*(3 + 2*sqrt(2))*(17 + 12*sqrt(2))^(n-1) - (sqrt(2)/8)*(3 - 2*sqrt(2))*(17 - 12*sqrt(2))^(n-1).

a(n) = (sqrt(2)/8)*( (17+12*sqrt(2))^(n-1/2) - (17-12*sqrt(2))^(n-1/2) ).

a(n) = (sqrt(2)/8)*( (3+2*sqrt(2))^(2n-1) - (3-2*sqrt(2))^(2n-1) ).

a(n) = (sqrt(2)/8)*( (1+sqrt(2))^(4n-2) - (1-sqrt(2))^(4n-2) ).

a(n) = 35*a(n-1) - 35*a(n-2) + a(n-3). (End)

a(n+1) = 17*a(n) + 6*sqrt(8*a(n)^2+1) for n >= 0. - Richard Choulet, May 01 2009

a(n) = b such that (-1)^(n+1) * Integral_{x=-Pi/2..Pi/2} cos((2*n-1)*x)/(3-sin(x)) dx = c + b*log(2). - Francesco Daddi, Aug 01 2011

a(n) are the nonzero integer square roots of A227970. - Richard R. Forberg, Aug 01 2013

a(n) = y/5, where y are solutions to: y^2 = 2x^2 - x - 3. - Richard R. Forberg, Nov 24 2013

a(n) = sqrt((A078522(n)+1)*(2*A078522(n)+1)). - Jonathan Sondow, Apr 07 2017

a(n) = Pell(4*n-2)/2. - G. C. Greubel, Jan 13 2020

a(n) = A001653(n)*A002315(n). - Gerry Martens, Mar 23 2024

Maple

seq( simplify(ChebyshevU(2*(n-1), 3)), n = 1..20); # G. C. Greubel, Jan 13 2020

Mathematica

LinearRecurrence[{34, -1}, {1, 35}, 15] (* Ant King, Nov 08 2011 *)
Fibonacci[4*Range[20] -2, 2]/2 (* G. C. Greubel, Jan 13 2020 *)

Prog

(Magma) I:=[1, 35]; [n le 2 select I[n] else 34*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 22 2011
(PARI) vector(21, n, polchebyshev(2*(n-1), 2, 3) ) \\ G. C. Greubel, Jan 13 2020
(Sage) [lucas_number1(4*n-2, 2, -1)/2 for n in (1..20)] # G. C. Greubel, Jan 13 2020
(GAP) List([0..20], n-> Lucas(2, -1, 4*n-2)[1]/2 ); # G. C. Greubel, Jan 13 2020

Crossrefs

Cf. A000129, A008844, A046177, A078522, A284876.

Cf. A001109, A001110 (partial sums).

Sequence in context: A115492 A187538 A306685 * A352183 A162847 A029546

Adjacent sequences: A046173 A046174 A046175 * A046177 A046178 A046179

Keyword

nonn,easy

Author

Eric W. Weisstein

Extensions

Chebyshev comments from Wolfdieter Lang, Nov 29 2002

Status

approved