A061045 - OEIS

%I #28 Mar 03 2023 16:51:39

%S -35,-2,-1,-5,-11,0,13,7,5,4,85,1,133,10,7,55,253,2,325,91,5,28,493,5,

%T 589,40,77,187,805,2,925,247,13,70,1189,35,1333,88,55,391,1645,4,1813,

%U 475,221,130,2173,7,2365,154,95,667,2773,20,2989,775,119,208,3445,11,3685,238,437,1015,4189,10

%N Numerator of 1/36 - 1/n^2.

%C Sixth case of Rydberg's formula. From Humphrey's spectrum of hydrogen. See A045944, A000567, A061043, A061046, A061047. - _Paul Curtz_, Dec 08 2008

%H Reinhard Zumkeller, <a href="/A061045/b061045.txt">Table of n, a(n) for n = 1..10000</a>

%H Curtis J. Humphreys, <a href="https://doi.org/10.1364/JOSA.42.000432">The Sixth Series in the Hydrogen Spectrum</a>, Journal of the Optical Society of America, 1952, 42, p. 432.

%e The fractions are -35/36, -2/9, -1/12, -5/144, -11/900, 0, 13/1764, 7/576, 5/324, 4/225, 85/4356, 1/48, 133/6084, 10/441, 7/300, 55/2304, 253/10404, 2/81, 325/12996, ...

%t Numerator[(1/36-1/Range[100]^2)] (* _Harvey P. Dale_, Mar 17 2013 *)

%o (Haskell)

%o import Data.Ratio ((%), numerator)

%o a061045 = numerator . (1 % 36 -) . recip . (^ 2) . fromIntegral

%o -- _Reinhard Zumkeller_, Jan 06 2014

%o (Magma) [Numerator(1/6^2 -1/n^2): n in [1..80]]; // _G. C. Greubel_, Feb 24 2023

%o (SageMath)

%o def A061045(n): return ((n^2-36)/(6*n)^2).numerator()

%o [A061045(n) for n in range(1,81)] # _G. C. Greubel_, Feb 24 2023

%Y A061046 gives denominators.

%Y Cf. A000567, A045944, A061035 - A061050.

%K sign,frac

%O 1,1

%A _N. J. A. Sloane_, May 26 2001