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A006051 Square hex numbers.
(Formerly M5409)
6
1, 169, 32761, 6355441, 1232922769, 239180661721, 46399815451081, 9001325016847969, 1746210653453054881, 338755865444875798921, 65716891685652451935769, 12748738231151130799740241, 2473189499951633722697670961, 479786014252385791072548426169 (list; graph; refs; listen; history; text; internal format)
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
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.
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
KEYWORD
nonn,easy
AUTHOR
STATUS
approved

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Last modified April 16 22:07 EDT 2024. Contains 371755 sequences. (Running on oeis4.)