A342817 Power series expansion of AQM(1,1-8x) where AQM denotes the arithmetic-quadratic mean.
1, -4, 4, 16, 52, 112, -48, -1984, -11212, -33360, 6224, 713536, 4441872, 13004480, -17374656, -432012032, -2525831628, -6454496208, 21147389392, 326358047552, 1794285832464, 4124461926592, -19727734694848, -263598020446976, -1416694290412784, -3151402998261312
Offset: 0
Keywords
Examples
First steps of iteration of ((u+v)/2, sqrt((u^2+v^2)/2)) are (1, 1-8x), (1 - 4*x, 1 - 4*x + 8*x^2 + 32*x^3 + 96*x^4 + O(x^5)), then (1 - 4*x + 4*x^2 + 16*x^3 + 48*x^4 + O(x^5), 1 - 4*x + 4*x^2 + 16*x^3 + 56*x^4 + O(x^5)) and (1 - 4*x + 4*x^2 + 16*x^3 + 52*x^4 + O(x^5), 1 - 4*x + 4*x^2 + 16*x^3 + 52*x^4 + O(x^5)), so the first terms of this sequence are 1, -4, 4, 16, 52.
Crossrefs
Compare A060691 for the arithmetic-geometric mean.
Programs
-
PARI
seq(n)={my(p=1, q=1-8*x+O(x*x^n)); while(p!=q, my(t=p+q); q = sqrt((p^2 + q^2)/2); p=t/2); Vec(p)} \\ Andrew Howroyd, Mar 22 2021 -
Sage
R.
= PowerSeriesRing(QQ, default_prec=50) (a,b) = (1,1-8*x) for i in range(50): (a,b) = ((a+b)/2, sqrt((a^2+b^2)/2)) a.coefficients()
Comments