%I #80 Aug 12 2022 09:22:06

%S 0,5,3,21,2,45,15,77,6,117,35,165,12,221,63,285,20,357,99,437,30,525,

%T 143,621,42,725,195,837,56,957,255,1085,72,1221,323,1365,90,1517,399,

%U 1677,110,1845,483,2021,132,2205,575,2397,156,2597,675

%N Numerator of 1/4 - 1/n^2.

%C From Balmer spectrum of hydrogen. Wavelengths in hydrogen spectrum are given by Rydberg's formula 1/wavelength = constant*(1/m^2 - 1/n^2).

%C a(-2) = 0, a(-1) = a(1) = -3. - _Paul Curtz_, Feb 19 2011

%C Can be thought of as 4 interlocking sequences, each of the form a(n) = 3a(n - 1) - 3a(n - 2) + a(n - 3). - _Charles R Greathouse IV_, May 27 2011

%D J. E. Brady and G. E. Humiston, General Chemistry, 3rd. ed., Wiley; p. 78.

%H Harry J. Smith, <a href="/A061037/b061037.txt">Table of n, a(n) for n=2..1000</a>

%H J. J. O'Connor and E. F. Robertson, <a href="http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Rydberg.html">Johannes Robert Rydberg</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Balmer_series">Balmer series</a>.

%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,3,0,0,0,-3,0,0,0,1).

%F G.f.: x^2(-3x^11-x^10-3x^9+14x^7+6x^6+30x^5+2x^4+21x^3+3x^2+5x)/(1-x^4)^3.

%F a(4n+2) = n(n+1), a(2n+3) = (2n+1)(2n+5), a(4n+4) = (2n+1)(2n+3). - _Ralf Stephan_, Jun 10 2005

%F a(n+2) = A060819(n) * A060819(n+4).

%F a(n) = (n^2-4)*(3*i^n+3*(-i)^n-27*(-1)^n+37)/64, where i is the imaginary unit. - _Bruno Berselli_, Feb 10 2011

%F a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12). - _Paul Curtz_, Feb 28 2011

%F a(n+2) = n*(n+4)/(period 4: 16, 1, 4, 1 = A146160(n)) = A028347(n+2) / A146160(n). - _Paul Curtz_, Mar 24 2011 [edited by _Franklin T. Adams-Watters_, Mar 25 2011]

%F a(n) = (n^2-4) / gcd(4*n^2, (n^2-4)). - _Colin Barker_, Jan 13 2014

%F Sum_{n>=3} 1/a(n) = 11/6. - _Amiram Eldar_, Aug 12 2022

%t f[n_] := n/GCD[n, 4]; Array[f[#] f[# + 4] &, 51, 0]

%t f[n_] := Numerator[(n - 2) (n + 2)/(4 n^2)]; Array[f, 51, 2] (* Or *)

%t a[n_] := 3 a[n - 4] - 3 a[n - 8] + a[n - 12]; a[1] = -3; a[2] = 0; a[3] = 5; a[4] = 3; a[5] = 21; a[6] = 2; a[7] = 45; a[8] = 15; a[9] = 77; a[10] = 6; a[11] = 117; a[12] = 35; Array[a, 51, 2] (* _Robert G. Wilson v_ *)

%t Numerator[1/4-1/Range[2,60]^2] (* _Harvey P. Dale_, Aug 18 2011 *)

%o (PARI) { for (n=2, 1000, write("b061037.txt", n, " ", numerator(1/4 - 1/n^2)) ) } \\ _Harry J. Smith_, Jul 17 2009

%o (Magma) [ Numerator(1/4-1/n^2): n in [2..52] ]; // _Bruno Berselli_, Feb 10 2011

%o (Haskell)

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

%o a061037 n = numerator (1%4 - 1%n^2) -- _Reinhard Zumkeller_, Dec 17 2011

%Y Cf. A061035-A061050, A126252, A028347.

%K nonn,frac,nice,easy

%O 2,2

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