A329591 - OEIS

%I #18 Apr 16 2021 00:09:06

%S 1,6,2,4,9,2,7,1,3,7,8,1,3,3,2,5,9,4,5,1,7,0,1,1,1,6,9,1,8,7,8,8,6,6,

%T 1,0,3,8,9,2,4,5,0,0,1,4,6,6,9,2,4,9,1,6,6,8,4,5,4,7,5,9,0,8,1,5,4,1,

%U 9,2,5,9,7,3,6,7,2,4,1,2,3,8,7,4,0,2,9,6,4,2,2,9,2,3,1,6,5,3,9

%N Decimal expansion of sqrt(34 + 2*sqrt(17))/4 = sqrt(8 + A222132)/2.

%C The present cp := sqrt(34 + 2*sqrt(17))/4 is used, together with cm := sqrt(34 - 2*sqrt(17))/4 = sqrt(9 - A222132)/2 = A329592, for the roots of the integer polynomial P(4, x) := x^4 + x^3 - 6*x^2 - x + 1 which are x1 = 4 + cp - 2*cp^2, x2 = 4 - cp - 2*cp^2, x3 = 4 + cm - 2*cm^2, and x4 = 4 - cm - 2*cm^2. The approximate values of these zeros are 0.344150732, -2.905703544, 2.049481177, and -0.4879283650, respectively.

%C In the power basis of cp (denoted by (...) and cm (denoted by [...]) the roots of P(4, x) are therefore: (4, +1, -2), (4, -1, -2), [4, +1, -2] and [4, -1, -2], respectively.

%F cp := sqrt(34 + 2*sqrt(17))/4 = sqrt(8 + w(17))/2, where w(17) = (1 - sqrt(17))/2 = A222132.

%e 1.62492713781332594517011169187886610389245001466924916684547590815419...

%t RealDigits[Sqrt[34 + 2*Sqrt[17]]/4, 10, 100][[1]] (* _Amiram Eldar_, Feb 17 2020 *)

%Y Cf, A222132, A329592.

%K nonn,cons,easy

%O 1,2

%A _Wolfdieter Lang_, Feb 17 2020