Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Posting-Version: version B 2.10.3 4.3bsd-beta 6/6/85; site lll-crg.ARpA Path: utzoo!watmath!clyde!burl!ulysses!bellcore!decvax!decwrl!amdcad!lll-crg!cralle From: cralle@lll-crg.ARpA (Bob Cralle) Newsgroups: net.math Subject: Re: series sum Message-ID: <1226@lll-crg.ARpA> Date: Sun, 2-Feb-86 17:39:49 EST Article-I.D.: lll-crg.1226 Posted: Sun Feb 2 17:39:49 1986 Date-Received: Tue, 4-Feb-86 03:22:27 EST References: <2265@utcsstat.uucp> <854@spp2.UUCP> <361@mcgill-vision.UUCP> Reply-To: cralle@lll-crg.UUCP (Bob cralle) Organization: Lawrence Livermore Labs, CRG Group Lines: 81 > > I did something similar: I realized that the C of F D guaranteed >that a cubic would be sufficient, but not remembering the formula and >not feeling like deriving it, I wrote out > > a + b + c + d = 1 [ from an^3+bn^2+cn+d = S, n=1 ] > 8a + 4b + 2c + d = 4 [ ... and n=2 ... ] >27a + 9b + 3c + d = 10 [ n=3 ] >64a + 16b + 4c + d = 20 [ n=4 ] > What is the next term in the series: 1 2 4 8 16 ... ? If I told you I wanted 31 what would you say? Since, as has been pointed out, I can make it what ever I want, what do I get by your method? (Thirty years ago when I solved this, I justed differenced it.) At the end you may be interested to see what the series comes from. As math types need MACSYMA more than they sometimes admit I let it do all of the algebra. (c3) a*n^5+b*n^4+c*n^3+d*n^2+e*n+f=g; 5 4 3 2 (d3) a n + b n + c n + d n + e n + f = g (c4) d3,n:1,g:1; (d4) f + e + d + c + b + a = 1 (c5) d3,n:2,g:2; (d5) f + 2 e + 4 d + 8 c + 16 b + 32 a = 2 (c6) d3,n:3,g:4; (d6) f + 3 e + 9 d + 27 c + 81 b + 243 a = 4 (c7) d3,n:4,g:8; (d7) f + 4 e + 16 d + 64 c + 256 b + 1024 a = 8 (c8) d3,n:5,g:16; (d8) f + 5 e + 25 d + 125 c + 625 b + 3125 a = 16 (c9) d3,n:6,g:31; (d9) f + 6 e + 36 d + 216 c + 1296 b + 7776 a = 31 (c10) solve([d4,d5,d6,d7,d8,d9]); 3 23 1 1 (d10) [[f = 1, e = - -, d = --, c = - -, b = --, a = 0]] 4 24 4 24 (c13) d10[1]; 3 23 1 1 (d13) [f = 1, e = - -, d = --, c = - -, b = --, a = 0] 4 24 4 24 (c19) subst(":","=",d13); 3 23 1 1 (d19) [f : 1, e : - -, d : --, c : - -, b : --, a : 0] 4 24 4 24 (c22) d3,d19; 4 3 2 n n 23 n 3 n (d22) -- - -- + ----- - --- + 1 = g 24 4 24 4 (c28) factor(lhs(d22)); 4 3 2 n - 6 n + 23 n - 18 n + 24 (d29) ----------------------------- 24 and so the series goes: 1 2 4 8 16 31 57 99 163 256 386 562... [isn't the 256 curious--one late!] What does the series represent? If you have n points on a circle & and you connect every point to every other point the series gives the number of areas those lines cut the circle into. (more than two lines intersecting at a point not allowed.) Anyone desirous of a copy of a paper I wrote on this subject & a proof, send me your address. -- Regards, &c., Bob cralle@lll-crg R. K. Cralle LLNL MS L-73 POX 808 Livermore, CA 94550 fone 415/422-4041 FTS 415/532-4041 <<< Facts are no match for belief >>>