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Self-Summing Ratio

The term self-summing ratio is used in this work for the unique real number r_n>1 satisfying

r_n=sum_(k=n)^inftyr_n^(-k)

for a nonnegative integer n. Summing the geometric series shows that this is equivalent to

r_n^(n+1)-r_n^n-1=0.

The first five self-summing ratios are all special Pisot numbers and are given in the following table (Smyth 1999, p. 1).

nself-summing ratioOEISvaluedefining polynomial
022x-2
1golden ratio phiA0016221.6180339887...x^2-x-1
2supergolden ratio psiA0925261.4655712318...x^3-x^2-1
3superplastic ratio chiA0861061.3802775690...x^4-x^3-1
4plastic constant rhoA0600061.3247179572...x^5-x^4-1

For n=4, the defining polynomial factors as x^5-x^4-1=(x^3-x-1)(x^2-x+1), so r_4 is the plastic constant, whose minimal polynomial is x^3-x-1. The equation r_n^n(r_n-1)=1 shows that r_n strictly decreases with n. Since r_4 is the plastic constant, the smallest Pisot number (Siegel 1944), no r_n with n>=5 is a Pisot number.

The values with n>=1 appear as beta_n in Baker (2017), and the self-summing representation was considered by Pegg (2019).

See also

Golden Ratio, Pisot Number, Plastic Constant, Special Pisot Number, Supergolden Ratio, Superplastic Ratio

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References

Baker, S. "Exceptional Digit Frequencies and Expansions in Non-Integer Bases." 28 Nov 2017. https://arxiv.org/abs/1711.10397.Pegg, E. Jr. "Shattering the Plane with Twelve New Substitution Tilings Using 2, phi, psi, chi, rho." Mar. 7, 2019. https://blog.wolfram.com/2019/03/07/shattering-the-plane-with-twelve-new-substitution-tilings-using-2-phi-psi-chi-rho/.Siegel, C. L. "Algebraic Integers Whose Conjugates Lie in the Unit Circle." Duke Math. J. 11, 597-602, 1944. https://doi.org/10.1215/S0012-7094-44-01152-X.Sloane, N. J. A. Sequences A001622, A060006, A086106, and A092526 in "The On-Line Encyclopedia of Integer Sequences."Smyth, C. J. "There Are Only Eleven Special Pisot Numbers." Bull. London Math. Soc. 31, 1-5, 1999. https://doi.org/10.1112/S0024609398005141.

Cite this as:

Weisstein, Eric W. "Self-Summing Ratio." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Self-SummingRatio.html

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