%I #12 Oct 03 2023 01:47:54
%S 1,2,3,4,8,77929,17700337092966627,
%T 97049570583433629023486339792437254323980677153951,
%U 5843929411147236787850533935034401259024361461871518737494060444501523486808548249023180720047062910088805381472125716317734317997138497838459864230713
%N Write Pi = Sum_{n>=1} 1/sqrt(a(n)), where each a(n) is minimal and unique and the sum approaches Pi from below.
%C The first eight terms give Pi accurately to the first 75 decimal digits.
%F a(1) = 1, a(2) = 2, a(3) = 3, a(4) = 4; a(n) = ceiling((Pi - Sum_{i=1..n-1}(1/sqrt(a(i))))^-2).
%e Pi > 1/sqrt(1) + 1/sqrt(2) + 1/sqrt(3) + 1/sqrt(4) + 1/sqrt(8), but Pi < 1/sqrt(1) + 1/sqrt(2) + 1/sqrt(3) + 1/sqrt(4) + 1/sqrt(7).
%t a[1] = 1; a[2] = 2; a[3] = 3; a[4] = 4; a[n_] := Ceiling[(Pi - Sum[1/Sqrt[a[i]], {i, 1, n - 1}])^-2]; Table[ a[n], {n, 1, 9}]
%Y Cf. A000796.
%K nonn
%O 1,2
%A _Robert G. Wilson v_, Feb 13 2004