A127676: partial sums of a series for Pi*sqrt(2)/4 Rationals r(n):=sum((-1)^floor(k/2)/(2*k+1),k=0..n). r(n), n=0..25: [1, 4/3, 17/15, 104/105, 347/315, 4132/3465, 50251/45045, 47248/45045, 848261/765765, 16882724/14549535, 16189889/14549535, 357817912/334639305, 1856017421/1673196525, 5753962988/5019589575, 161845337077/145568097675, 4871637351712/4512611027925, 5008383140437/4512611027925, 5137314884092/4512611027925, 185568039683479/166966608033225, 181286844605704/166966608033225, 7599727236867089/6845630929362225, 333633902114647052/294362129962575675, 327092521448812037/294362129962575675, 15078986378131590064/13835020108241056725, 107529336090955567123/96845140757687397075, 109428260419537672948/96845140757687397075] The numerator sequence is this sequence a(n), n=0..25: [1, 4, 17, 104, 347, 4132, 50251, 47248, 848261, 16882724, 16189889, 357817912, 1856017421, 5753962988, 161845337077, 4871637351712, 5008383140437, 5137314884092, 185568039683479, 181286844605704, 7599727236867089, 333633902114647052, 327092521448812037, 15078986378131590064, 107529336090955567123, 109428260419537672948] The denominator sequence is d(n), n=0..25: [1, 3, 15, 105, 315, 3465, 45045, 45045, 765765, 14549535, 14549535, 334639305, 1673196525, 5019589575, 145568097675, 4512611027925, 4512611027925, 4512611027925, 166966608033225, 166966608033225, 6845630929362225, 294362129962575675, 294362129962575675, 13835020108241056725, 96845140757687397075, 96845140757687397075] The denominator sequence coincides with A025547(n+1) for n=0..41, however: d(42)=7422822568422519986207785205976075 but the corresponding entry is A025547(43)=126187983663182839765532348501593275 with 36 digits. The limit of this sequence of rationals, lim(r(n),n->infinity)= Pi*sqrt(2)/4. The numbers for r(10^k), k=0..4 are (Maple10 with 10 digits): [1.333333333, 1.112742710, 1.110696234, 1.110720485, 1.110720732] This should be compared with the approximate value for Pi*sqrt(2)/4: 1.110720734 ##################################### e.o.f. ################################## ############################################################################