A006051 Square hex numbers. (Formerly M5409)

1, 169, 32761, 6355441, 1232922769, 239180661721, 46399815451081, 9001325016847969, 1746210653453054881, 338755865444875798921, 65716891685652451935769, 12748738231151130799740241, 2473189499951633722697670961, 479786014252385791072548426169

Offset

1,2

Comments

Numbers n of the form n = y^2 = 3*x^2 - 3*x + 1.

References

M. Gardner, Time Travel and Other Mathematical Bewilderments. Freeman, NY, 1988, p. 19.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

G. C. Greubel, Table of n, a(n) for n = 1..435

M. Gardner & N. J. A. Sloane, Correspondence, 1973-74

Giovanni Lucca, Integer Sequences and Circle Chains Inside a Circular Segment, Forum Geometricorum, Vol. 18 (2018), 47-55.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

Sociedad Magic Penny Patagonia, Leonardo en Patagonia

Eric Weisstein's World of Mathematics, Hex Number.

Index entries for linear recurrences with constant coefficients, signature (195,-195,1).

Formula

a(n) = A001570(n)^2.

a(1 - n) = a(n).

G.f.: x * (1 - 26*x + x^2) / ((1 - x) * (1 - 194*x + x^2)). - Simon Plouffe in his 1992 dissertation

a(n) = 194*a(n-1) - a(n-2) - 24, a(1)=1, a(2)=169. - James A. Sellers, Jul 04 2000

a(n+1) = A003215(A001921(n)). - Joerg Arndt, Jan 02 2017

a(n) = (1/8)*(1 + 7*(ChebyshevU(n-1, 97) - ChebyshevU(n-2, 97))). - G. C. Greubel, Oct 07 2022

Example

G.f. = x + 169*x^2 + 32761*x^3 + 6355441*x^4 + 1232922769*x^5 + ...

Mathematica

Rest@ CoefficientList[Series[x(1-26x+x^2)/((1-x)(1-194x+x^2)), {x, 0, 20}], x] (* Michael De Vlieger, Jan 02 2017 *)
LinearRecurrence[{195, -195, 1}, {1, 169, 32761}, 20] (* Harvey P. Dale, Nov 03 2017 *)

Prog

(PARI) {a(n) = sqr( real( (2 + quadgen( 12)) ^ (2*n - 1)) / 2)} /* Michael Somos, Feb 15 2011 */
(Magma) [(7*Evaluate(ChebyshevSecond(n), 97) - 7*Evaluate(ChebyshevU(n-1), 97) + 1)/8: n in [1..30]]; // G. C. Greubel, Nov 04 2017; Oct 07 2022
(SageMath)
def A006051(n): return (7*chebyshev_U(n-1, 97) - 7*chebyshev_U(n-2, 97) + 1)/8
[A006051(n) for n in range(1, 31)] # G. C. Greubel, Oct 07 2022

Crossrefs

Cf. A003500.

Intersection of A000290 and A003215.

Values of x are given by A001922, values of y by A001570.

Sequence in context: A051477 A227692 A260862 * A069742 A069743 A210087

Adjacent sequences: A006048 A006049 A006050 * A006052 A006053 A006054

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved