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A061039
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Numerator of 1/9 - 1/n^2.
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32
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0, 7, 16, 1, 40, 55, 8, 91, 112, 5, 160, 187, 8, 247, 280, 35, 352, 391, 16, 475, 520, 7, 616, 667, 80, 775, 832, 11, 952, 1015, 40, 1147, 1216, 143, 1360, 1435, 56, 1591, 1672, 65, 1840, 1927, 224, 2107, 2200, 85, 2392, 2491, 32, 2695, 2800, 323, 3016, 3127
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OFFSET
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3,2
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COMMENTS
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The denominators are given in A061040.
From Paschen spectrum of hydrogen. Wavelengths in hydrogen spectrum are given by Rydberg's formula 1/wavelength = constant*(1/m^2 - 1/n^2).
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REFERENCES
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J. E. Brady and G. E. Humiston, General Chemistry, 3rd. ed., Wiley; p. 78.
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1).
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FORMULA
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a(n) <= n^2 - 9; if n is not divisible by 3 then a(n) = n^2 - 9. - Stefan Steinerberger, Apr 16 2006
a(n) = 3*a(n-27) - 3*a(n-54) + a(n-81) for n > 83. - Colin Barker, Oct 09 2016
a(n) = (n^2 - 9)/9^2 if n == 3 or 24 (mod 27), a(n) = (n^2 - 9)/(3*9) if n == 6 or 24 or 15 or 21 (mod 27), a(n) = (n^2 - 9)/9 if n == 0 (mod 9) and n^2 - 9 otherwise. From the period length 27 sequence gcd(n^2 - 9, 9*n^2). - Wolfdieter Lang, Mar 15 2018
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MAPLE
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MATHEMATICA
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PROG
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(Haskell)
import Data.Ratio ((%), numerator)
(Python)
from math import gcd
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CROSSREFS
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KEYWORD
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nonn,frac,nice,easy
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AUTHOR
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STATUS
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approved
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