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A061036
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Triangle T(m,n) = denominator of 1/m^2 - 1/n^2, n >= 1, m=n,n-1,n-2,...,1.
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3
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1, 1, 4, 1, 36, 9, 1, 144, 16, 16, 1, 400, 225, 100, 25, 1, 900, 144, 12, 9, 36, 1, 1764, 1225, 784, 441, 196, 49, 1, 3136, 576, 1600, 64, 576, 64, 64, 1, 5184, 3969, 324, 2025, 1296, 81, 324, 81, 1, 8100, 1600, 4900, 225, 100, 400, 900, 25, 100, 1, 12100, 9801
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OFFSET
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1,3
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COMMENTS
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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. 77.
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LINKS
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_Reinhard Zumkeller_, Rows n=..100 of triangle, flattened
J. J. O'Connor and E. F. Robertson, Johannes Robert Rydberg
Eric Weisstein's World of Physics, Balmer Formula
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EXAMPLE
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Triangle 1/m^2-1/n^2, m >= 1, 1<=n<=m, (i.e. with rows reversed) begins
0
3/4, 0
8/9, 5/36, 0
15/16, 3/16, 7/144, 0
24/25, 21/100, 16/225, 9/400, 0
35/36, 2/9, 1/12, 5/144, 11/900, 0
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MATHEMATICA
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t[m_, n_] := Denominator[1/m^2 - 1/n^2]; Table[t[m, n], {n, 1, 12}, {m, n, 1, -1}] // Flatten (* Jean-François Alcover, Oct 17 2012 *)
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PROG
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(Haskell)
import Data.Ratio ((%), denominator)
a061036 n k = a061036_tabl !! (n-1) !! (k-1)
a061036_row = map denominator . balmer where
balmer n = map (subtract (1 % n ^ 2) . (1 %) . (^ 2)) [n, n-1 .. 1]
a061036_tabl = map a061036_row [1..]
-- Reinhard Zumkeller, Apr 12 2012
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CROSSREFS
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Cf. A061035. Rows give A061037-A061050.
Cf. A126252.
Sequence in context: A144284 A144285 A091741 * A217020 A144267 A011801
Adjacent sequences: A061033 A061034 A061035 * A061037 A061038 A061039
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KEYWORD
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nonn,tabl,easy,nice,frac
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AUTHOR
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N. J. A. Sloane, May 26 2001
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EXTENSIONS
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More terms from Naohiro Nomoto, Jul 15 2001
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STATUS
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approved
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