%I #41 Oct 09 2023 11:05:47
%S 1,144,225,12,441,576,81,900,1089,48,1521,1764,75,2304,2601,324,3249,
%T 3600,147,4356,4761,64,5625,6084,729,7056,7569,100,8649,9216,363,
%U 10404,11025,1296,12321,12996,507,14400,15129,588,16641,17424
%N Denominator of 1/9 - 1/n^2.
%C See A061039 (numerators) for comments, references and links.
%H Reinhard Zumkeller, <a href="/A061040/b061040.txt">Table of n, a(n) for n = 3..1000</a>
%H Friedrich Paschen, <a href="https://doi.org/10.1002/andp.19083321303">Zur Kenntnis ultraroter Linienspektra</a>, Annalen der Physik 27, pp. 537-570 (1908).
%H <a href="/index/Rec#order_81">Index entries for linear recurrences with constant coefficients</a>, 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).
%F a(n) = denominator(n^2 - 9)/(9*n^2), n >= 3.
%F a(n) = (n^2)/9 if n == 3 or 24 (mod 27), a(n) = (n^2)/3 if n == 6 or 12 or 15 or 21 (mod 27), a(n) = n^2 if n == 0 (mod 9) and 9*n^2 otherwise. From the period length 27 sequence gcd(n^2 - 9, 9*n^2). - _Wolfdieter Lang_, Mar 15 2018
%t Denominator[1/9-1/Range[3,50]^2] (* _Harvey P. Dale_, Jan 16 2012 *)
%o (Haskell)
%o import Data.Ratio ((%), denominator)
%o a061040 n = denominator $ 1%9 - 1%n^2 -- _Reinhard Zumkeller_, Jan 03 2012
%o (PARI) a(n)=denominator(1/9 - 1/n^2) \\ _Charles R Greathouse IV_, Feb 07 2017
%o (Python)
%o from math import gcd
%o def A061040(n): return 9*n**2//gcd(n**2-9,9*n**2) # _Chai Wah Wu_, Apr 02 2021
%o (Sage) [denominator(1/9 -1/n^2) for n in (3..50)] # _G. C. Greubel_, Mar 10 2022
%Y Cf. A061039.
%K nonn,frac,nice,easy
%O 3,2
%A _N. J. A. Sloane_, May 26 2001