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

Author

Eric W. Weisstein

Keywords

• nonn
• easy

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).

Crossrefs

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

Cf. A001109, A001110 (partial sums).

Programs

GAP

List([0..20], n-> Lucas(2,-1, 4*n-2)[1]/2 ); # G. C. Greubel, Jan 13 2020

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

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 *)

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

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

Extensions

Chebyshev comments from Wolfdieter Lang, Nov 29 2002