Simon> hello everybody, about that formula for 1/pi, it is nice , the coefficient a(k) is a binomial term The sequence a(k) is : 1, 1, 33, 97, 1729, 8001, 105441, 627873, 6989697, 48363649, 488206753, 3701949153, 35289342529, 283146701761, 2610495177057, 21695983405857, 196218339243777, 1667338615773441, 14917038493453089, 128562758660255073, 1143482133220664769, 9946278255903268929, 88205310329762729697, 771946983805271894433, 6837125121111415598721,... Which has this g.f. 1 -------------------- = 2 1/2 (1 - 2 x - 63 x ) Now, this has a closed binomial expression, of course anyhow, this series : can be programmed into : proc(k) local x; (x^2 + x + 16)^k; expand(%); coeff(%, x, k) end; and proc(k) (30*k + 7)*binomial(2*k, k)^2/(-256)^k;%*f(k); end; which gives a sequence : which converges to 24/Pi at a rate of approx. .25 digits per term. or in other words, 1000 terms = 250 digits of precision. best regards, Simon Plouffe ---------------- In matrixland, this identity is prod(matrix([-(k+1/2)^3/(8*(k+1)^3),-(k+1/2)^2/(16*(k+1)^2),30*(k+7/30)],[-63*k*(k+1/2)^2/(16*(k+1)^3),0,0],[0,0,1]),k,0,inf) = matrix([0,0,24/%pi],[0,0,0],[0,0,1]) [ 1 3 1 2 ] [ (k + -) (k + -) ] [ 2 2 7 ] [ - ---------- - ----------- 30 (k + --) ] inf [ 3 2 30 ] [ 24 ] /===\ [ 8 (k + 1) 16 (k + 1) ] [ 0 0 --- ] | | [ ] [ %pi ] | | [ 1 2 ] = [ ] | | [ 63 k (k + -) ] [ 0 0 0 ] k = 0 [ 2 ] [ ] [ - ------------- 0 0 ] [ 0 0 1 ] [ 3 ] [ 16 (k + 1) ] [ ] [ 0 0 1 ] which looks very much like a coordinate transformation of some very special case of the "3F2[1] Rosetta Stone". (http://gosper.org/stanfordn3.dvi) --rwg