A175485 Numerators of averages of squares of the first n positive integers.

1, 5, 14, 15, 11, 91, 20, 51, 95, 77, 46, 325, 63, 145, 248, 187, 105, 703, 130, 287, 473, 345, 188, 1225, 221, 477, 770, 551, 295, 1891, 336, 715, 1139, 805, 426, 2701, 475, 1001, 1580, 1107, 581, 3655, 638, 1335, 2093, 1457, 760, 4753, 825, 1717, 2678

Offset

1,2

Comments

See A089128(n) for n >= 1 - denominators of averages of squares of the first n positive integers.

Sqrt (a(n) / A089128(n)) for n >= 1 is harmonic mean of the first n positive integers.

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.

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,3,0,0,0,0,0,-3,0,0,0,0,0,1).

Formula

a(n) = A000330(n) * A089128 / n = (n+1) * (2n+1) * GCD(6, n) / 6 for n >= 1.

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

Sum_{k=1..n} a(k) ~ (5/18)*n^3. - Amiram Eldar, Oct 07 2023

Example

a(10) = 11*21*2 / 6 = 77.

Mathematica

Module[{nn=60, sqs}, sqs=Range[nn]^2; Table[Numerator[Mean[Take[sqs, n]]], {n, nn}]] (* Harvey P. Dale, Nov 06 2021 *)

Prog

(PARI) a(n)=(n+1)*(2*n+1)*gcd(n, 6)/6 // Ralf Stephan, Sep 20 2013

Crossrefs

Cf. A089128 (denominators), A000330, A089128.

Sequence in context: A169811 A272970 A168213 * A174657 A231665 A160709

Adjacent sequences: A175482 A175483 A175484 * A175486 A175487 A175488

Keyword

nonn,frac

Author

Jaroslav Krizek, May 27 2010

Extensions

More terms from Ralf Stephan, Sep 20 2013

Status

approved