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) = (1/8)*(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
KEYWORD
nonn,easy
AUTHOR
STATUS
approved