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A006062

Star-hex numbers.

1, 37, 1261, 42841, 1455337, 49438621, 1679457781, 57052125937, 1938092824081, 65838103892821, 2236557439531837, 75977114840189641, 2580985347126915961, 87677524687474953037, 2978454854027021487301, 101179787512231255615201, 3437134320561835669429537, 116761387111590181504989061

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

REFERENCES

Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing (2012), pages 60-61.

Martin Gardner, Time Travel and Other Mathematical Bewilderments. Freeman, NY, 1988, p. 22.

Harvey J. Hindin, Stars, hexes, triangular numbers and Pythagorean triples, J. Rec. Math., 16 (1983/1984), 191-193.

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

LINKS

Paolo Xausa, Table of n, a(n) for n = 1..650

Harvey J. Hindin, Stars, hexes, triangular numbers and Pythagorean triples, J. Rec. Math., 16 (1983/1984), 191-193. (Annotated scanned copy)

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.

Index entries for linear recurrences with constant coefficients, signature (35,-35,1).

FORMULA

a(n) = 34*a(n-1) - a(n-2) + 4.

G.f.: -x*(x + 1)^2/(x - 1)/(x^2 - 34*x + 1). - Simon Plouffe, see Maple code.

From Charlie Marion, Aug 03 2005: (Start)

a(n) = A001109(n-1)^2 + A001109(n)^2; e.g., 1261 = 6^2 + 35^2.

a(n) = A001652(n-1)*A046090(n-1) + A001653(n-1)^2; e.g., 1261 = 20*21 + 29^2. (End)

E.g.f.: exp(x)*(3*exp(16*x)*(3*cosh(12*sqrt(2)*x) - 2*sqrt(2)*sinh(12*sqrt(2)*x)) - 1)/8 - 1. - Stefano Spezia, Oct 18 2025

MAPLE

A006062:=-(z+1)**2/(z-1)/(z**2-34*z+1); # Simon Plouffe in his 1992 dissertation for offset zero.

MATHEMATICA