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A231187 Decimal expansion of the length ratio (largest diagonal)/side in the regular 7-gon (or heptagon). 6
2, 2, 4, 6, 9, 7, 9, 6, 0, 3, 7, 1, 7, 4, 6, 7, 0, 6, 1, 0, 5, 0, 0, 0, 9, 7, 6, 8, 0, 0, 8, 4, 7, 9, 6, 2, 1, 2, 6, 4, 5, 4, 9, 4, 6, 1, 7, 9, 2, 8, 0, 4, 2, 1, 0, 7, 3, 1, 0, 9, 8, 8, 7, 8, 1, 9, 3, 7, 0, 7, 3, 0, 4, 9, 1, 2, 9, 7, 4, 5, 6, 9, 1, 5, 1, 8, 8, 5, 0, 1, 4, 6, 5, 3, 1, 7, 0, 7, 4, 3, 3, 3, 4, 1 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The length ratio (largest diagonal)/side in the regular 7-gon (heptagon) is sigma(7) = S(2, rho(7)) = -1 + rho(7)^2, with rho(7) = 2*cos(Pi/7), which is approx. 1.8019377358 (see A160389 for its decimal expansion, and A049310 for the Chebyshev S-polynomials). sigma(7), approx. 2.2469796, is also the reciprocal of one of the solutions of the minimal polynomial C(7, x) = x^3 - x^2 - 2*x + 1 of rho(7) (see A187360), namely 1/(2*cos(3*Pi/7)).

sigma(7) is the limit of a(n+1)/a(n) for n->infinity for the sequences A006054 and A077998 which can be considered as analogs of the Fibonacci sequence in the pentagon. Thus sigma(7) plays in the heptagon the role of the golden section in the pentagon.

See the Steinbach link.

LINKS

Table of n, a(n) for n=1..104.

Peter Steinbach, Golden Fields: A Case for the Heptagon, Mathematics Magazine, Vol. 70, No. 1, Feb. 1997.

FORMULA

sigma(7) = -1 + (2*cos(Pi/7))^2 = 1/(2*cos(3*Pi/7)).

sigma(7) = exp(asinh(cos(Pi/7))). - Geoffrey Caveney, Apr 23 2014

cos(Pi/7) + sqrt(1+cos(Pi/7)^2). - Geoffrey Caveney, Apr 23 2014

EXAMPLE

2.24697960371746706105000976800847962126454946179280421073109887819...

CROSSREFS

Cf. A160389, A006054, A077998. Also -1 + A116425.

Sequence in context: A153964 A001010 A091966 * A055529 A222735 A299408

Adjacent sequences:  A231184 A231185 A231186 * A231188 A231189 A231190

KEYWORD

nonn,cons,easy

AUTHOR

Wolfdieter Lang, Nov 21 2013

STATUS

approved

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Last modified February 20 21:15 EST 2020. Contains 332084 sequences. (Running on oeis4.)