%I #39 May 01 2024 17:02:16
%S 1,5,14,15,11,91,20,51,95,77,46,325,63,145,248,187,105,703,130,287,
%T 473,345,188,1225,221,477,770,551,295,1891,336,715,1139,805,426,2701,
%U 475,1001,1580,1107,581,3655,638,1335,2093,1457,760,4753,825,1717,2678
%N Numerators of averages of squares of the first n positive integers.
%C See A089128(n) for n >= 1 - denominators of averages of squares of the first n positive integers.
%C Sqrt (a(n) / A089128(n)) for n >= 1 is harmonic mean of the first n positive integers.
%C For n = 337 holds: a(n) is square (= 38025 = 195^2) and simultaneously A089128(n) = 1, i.e. number k = 195 is quadratic mean (root mean square) of first 337 positive integers. There are other such numbers - see A084231 and A084232.
%H Harvey P. Dale, <a href="/A175485/b175485.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_18">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,3,0,0,0,0,0,-3,0,0,0,0,0,1).
%F a(n) = A000330(n) * A089128 / n = (n+1) * (2n+1) * GCD(6, n) / 6 for n >= 1.
%F G.f.: (x^17 + x^15 + 5*x^14 + 7*x^13 + 6*x^12 + 52*x^11 + 13*x^10 + 32*x^9 + 53*x^8 + 36*x^7 + 17*x^6 + 91*x^5 + 11*x^4 + 15*x^3 + 14*x^2 + 5*x + 1)/(1-x^6)^3. - _Ralf Stephan_, Sep 20 2013
%F Sum_{k=1..n} a(k) ~ (5/18)*n^3. - _Amiram Eldar_, Oct 07 2023
%e a(10) = 11*21*2 / 6 = 77.
%t Module[{nn=60,sqs},sqs=Range[nn]^2;Table[Numerator[Mean[Take[sqs,n]]],{n,nn}]] (* _Harvey P. Dale_, Nov 06 2021 *)
%o (PARI) a(n)=(n+1)*(2*n+1)*gcd(n,6)/6 \\ _Ralf Stephan_, Sep 20 2013
%Y Cf. A089128 (denominators), A000330, A089128.
%K nonn,frac
%O 1,2
%A _Jaroslav Krizek_, May 27 2010
%E More terms from _Ralf Stephan_, Sep 20 2013