[math-fun] Can someone please explain how Somos4,
initialized to 1,1,-1,x, Nest[Append[Rest[#], Factor[(#[[4]] #[[2]] + #[[3]]^2)/prnt@#[[1]]]] &, {1, 1, -1, x}, 17]; 1 1 -1 x 1+x -1-x+x^2 -1-x-x^3 -x (2+3 x) 1+3 x+3 x^2-x^4+x^5 -(1+x) (-1-2 x+2 x^2+3 x^3-x^4+x^5) -1-3 x-3 x^2-5 x^3-9 x^4-3 x^5+2 x^6-x^7 -x (-1-x+x^2) (3+9 x+9 x^2+5 x^3+2 x^4+x^6) 1+6 x+15 x^2+16 x^3-6 x^5+12 x^6+13 x^7+x^9+x^10 -(1+x+x^3) (1+5 x+4 x^2-15 x^3-30 x^4-13 x^5+6 x^6+x^7-3 x^8+x^9) -(1+x) (1+5 x+10 x^2+20 x^3+50 x^4+63 x^5+12 x^6-19 x^7+8 x^8+5 x^9-2 x^10+5 x^11-2 x^12+x^13) x (2+3 x) (-2-12 x-27 x^2-27 x^3-6 x^4+18 x^5+23 x^6+3 x^7-2 x^8+14 x^9+6 x^10-3 x^11+2 x^12) 1+10 x+45 x^2+110 x^3+135 x^4+45 x^5-2 x^6+211 x^7+429 x^8+311 x^9+105 x^10+69 x^11+56 x^12+5 x^13-9 x^14+6 x^15+4 x^16-3 x^17+x^18 . . . never manages to produce a denominator? (Very) similarly for 1,i,1,x: Nest[Append[Rest[#], Factor[(#[[4]] #[[2]] + #[[3]]^2)/prnt@#[[1]], Extension -> I]] &, {1, I, 1, x}, 17] 1 i 1 x 1+i x -i+x-i x^2 1+i x-i x^3 -i x (-2 i+3 x) 1+3 i x-3 x^2-x^4+i x^5 (1+i x) (i-2 x+2 i x^2-3 x^3+i x^4+x^5) 1+3 i x-3 x^2-5 i x^3+9 x^4+3 i x^5+2 x^6-i x^7 -x (1+i x+x^2) (-3-9 i x+9 x^2+5 i x^3-2 x^4+x^6) 1+6 i x-15 x^2-16 i x^3-6 i x^5-12 x^6-13 i x^7+i x^9-x^10 (1+i x-i x^3) (-i+5 x+4 i x^2+15 x^3+30 i x^4-13 x^5+6 i x^6-x^7+3 i x^8+x^9) -(-i+x) (-i+5 x+10 i x^2-20 x^3-50 i x^4+63 x^5+12 i x^6+19 x^7-8 i x^8+5 x^9-2 i x^10-5 x^11+2 i x^12+x^13) x (-2 i+3 x) (-2 i+12 x+27 i x^2-27 x^3-6 i x^4-18 x^5-23 i x^6+3 x^7-2 i x^8-14 x^9-6 i x^10-3 x^11+2 i x^12) 1+10 i x-45 x^2-110 i x^3+135 x^4+45 i x^5+2 x^6-211 i x^7+429 x^8+311 i x^9-105 x^10-69 i x^11+56 x^12+5 i x^13+9 x^14-6 i x^15+4 x^16-3 i x^17-x^18 --rwg
This is part of the "Laurent Phenomenon" of Somos sequences, which led Sergey Fomin and Andrei Zelevinsky to develop the theory of cluster algebras. (https://arxiv.org/abs/math/0104241) Jim Propp will surely have more to say on the subject. --Michael On Wed, Jul 5, 2017 at 12:05 AM, Bill Gosper <billgosper@gmail.com> wrote:
initialized to 1,1,-1,x, Nest[Append[Rest[#], Factor[(#[[4]] #[[2]] + #[[3]]^2)/prnt@#[[1]]]] &, {1, 1, -1, x}, 17]; 1 1 -1 x 1+x -1-x+x^2 -1-x-x^3 -x (2+3 x) 1+3 x+3 x^2-x^4+x^5 -(1+x) (-1-2 x+2 x^2+3 x^3-x^4+x^5) -1-3 x-3 x^2-5 x^3-9 x^4-3 x^5+2 x^6-x^7 -x (-1-x+x^2) (3+9 x+9 x^2+5 x^3+2 x^4+x^6) 1+6 x+15 x^2+16 x^3-6 x^5+12 x^6+13 x^7+x^9+x^10 -(1+x+x^3) (1+5 x+4 x^2-15 x^3-30 x^4-13 x^5+6 x^6+x^7-3 x^8+x^9) -(1+x) (1+5 x+10 x^2+20 x^3+50 x^4+63 x^5+12 x^6-19 x^7+8 x^8+5 x^9-2 x^10+5 x^11-2 x^12+x^13) x (2+3 x) (-2-12 x-27 x^2-27 x^3-6 x^4+18 x^5+23 x^6+3 x^7-2 x^8+14 x^9+6 x^10-3 x^11+2 x^12) 1+10 x+45 x^2+110 x^3+135 x^4+45 x^5-2 x^6+211 x^7+429 x^8+311 x^9+105 x^10+69 x^11+56 x^12+5 x^13-9 x^14+6 x^15+4 x^16-3 x^17+x^18 . . . never manages to produce a denominator? (Very) similarly for 1,i,1,x: Nest[Append[Rest[#], Factor[(#[[4]] #[[2]] + #[[3]]^2)/prnt@#[[1]], Extension -> I]] &, {1, I, 1, x}, 17]
1 i 1 x 1+i x -i+x-i x^2 1+i x-i x^3 -i x (-2 i+3 x) 1+3 i x-3 x^2-x^4+i x^5 (1+i x) (i-2 x+2 i x^2-3 x^3+i x^4+x^5) 1+3 i x-3 x^2-5 i x^3+9 x^4+3 i x^5+2 x^6-i x^7 -x (1+i x+x^2) (-3-9 i x+9 x^2+5 i x^3-2 x^4+x^6) 1+6 i x-15 x^2-16 i x^3-6 i x^5-12 x^6-13 i x^7+i x^9-x^10 (1+i x-i x^3) (-i+5 x+4 i x^2+15 x^3+30 i x^4-13 x^5+6 i x^6-x^7+3 i x^8+x^9) -(-i+x) (-i+5 x+10 i x^2-20 x^3-50 i x^4+63 x^5+12 i x^6+19 x^7-8 i x^8+5 x^9-2 i x^10-5 x^11+2 i x^12+x^13) x (-2 i+3 x) (-2 i+12 x+27 i x^2-27 x^3-6 i x^4-18 x^5-23 i x^6+3 x^7-2 i x^8-14 x^9-6 i x^10-3 x^11+2 i x^12) 1+10 i x-45 x^2-110 i x^3+135 x^4+45 i x^5+2 x^6-211 i x^7+429 x^8+311 i x^9-105 x^10-69 i x^11+56 x^12+5 i x^13+9 x^14-6 i x^15+4 x^16-3 i x^17-x^18
--rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Forewarned is worth an octopus in the bush.
Sorry not to have bitten the bait sooner (I was on a math-free vacation with my wife). There are various ways to prove that the solution to a rational recurrence relation, a priori a sequence of rational functions, is in fact a sequence of Laurent polynomials. My favorite approach is to find a way to interpret the coefficients of those polynomials as having enumerative significance; then we get positivity of those coefficients as a bonus. That is not the case for Bill's recurrence, but maybe the coefficients provide signed enumerations of some sorts of combinatorial objects. My favorite way to reverse-engineer those combinatorial interpretations is to add more variables to the setup without sacrificing Laurentness. As you do this, you get Laurent polynomials with more terms and smaller coefficients. Add enough variables and all the coefficients are equal to 1. Then the monomials that occur can be seen as algebraic encodings of the combinatorial objects that are being enumerated. I wish I could say other relevant things or provide relevants links, but this is stuff I haven't thought about for over a decade. Jim Propp On Wednesday, July 5, 2017, Michael Kleber <michael.kleber@gmail.com> wrote:
This is part of the "Laurent Phenomenon" of Somos sequences, which led Sergey Fomin and Andrei Zelevinsky to develop the theory of cluster algebras. (https://arxiv.org/abs/math/0104241) Jim Propp will surely have more to say on the subject.
--Michael
On Wed, Jul 5, 2017 at 12:05 AM, Bill Gosper <billgosper@gmail.com <javascript:;>> wrote:
initialized to 1,1,-1,x, Nest[Append[Rest[#], Factor[(#[[4]] #[[2]] + #[[3]]^2)/prnt@#[[1]]]] &, {1, 1, -1, x}, 17]; 1 1 -1 x 1+x -1-x+x^2 -1-x-x^3 -x (2+3 x) 1+3 x+3 x^2-x^4+x^5 -(1+x) (-1-2 x+2 x^2+3 x^3-x^4+x^5) -1-3 x-3 x^2-5 x^3-9 x^4-3 x^5+2 x^6-x^7 -x (-1-x+x^2) (3+9 x+9 x^2+5 x^3+2 x^4+x^6) 1+6 x+15 x^2+16 x^3-6 x^5+12 x^6+13 x^7+x^9+x^10 -(1+x+x^3) (1+5 x+4 x^2-15 x^3-30 x^4-13 x^5+6 x^6+x^7-3 x^8+x^9) -(1+x) (1+5 x+10 x^2+20 x^3+50 x^4+63 x^5+12 x^6-19 x^7+8 x^8+5 x^9-2 x^10+5 x^11-2 x^12+x^13) x (2+3 x) (-2-12 x-27 x^2-27 x^3-6 x^4+18 x^5+23 x^6+3 x^7-2 x^8+14 x^9+6 x^10-3 x^11+2 x^12) 1+10 x+45 x^2+110 x^3+135 x^4+45 x^5-2 x^6+211 x^7+429 x^8+311 x^9+105 x^10+69 x^11+56 x^12+5 x^13-9 x^14+6 x^15+4 x^16-3 x^17+x^18 . . . never manages to produce a denominator? (Very) similarly for 1,i,1,x: Nest[Append[Rest[#], Factor[(#[[4]] #[[2]] + #[[3]]^2)/prnt@#[[1]], Extension -> I]] &, {1, I, 1, x}, 17]
1 i 1 x 1+i x -i+x-i x^2 1+i x-i x^3 -i x (-2 i+3 x) 1+3 i x-3 x^2-x^4+i x^5 (1+i x) (i-2 x+2 i x^2-3 x^3+i x^4+x^5) 1+3 i x-3 x^2-5 i x^3+9 x^4+3 i x^5+2 x^6-i x^7 -x (1+i x+x^2) (-3-9 i x+9 x^2+5 i x^3-2 x^4+x^6) 1+6 i x-15 x^2-16 i x^3-6 i x^5-12 x^6-13 i x^7+i x^9-x^10 (1+i x-i x^3) (-i+5 x+4 i x^2+15 x^3+30 i x^4-13 x^5+6 i x^6-x^7+3 i x^8+x^9) -(-i+x) (-i+5 x+10 i x^2-20 x^3-50 i x^4+63 x^5+12 i x^6+19 x^7-8 i x^8+5 x^9-2 i x^10-5 x^11+2 i x^12+x^13) x (-2 i+3 x) (-2 i+12 x+27 i x^2-27 x^3-6 i x^4-18 x^5-23 i x^6+3 x^7-2 i x^8-14 x^9-6 i x^10-3 x^11+2 i x^12) 1+10 i x-45 x^2-110 i x^3+135 x^4+45 i x^5+2 x^6-211 i x^7+429 x^8+311 i x^9-105 x^10-69 i x^11+56 x^12+5 i x^13+9 x^14-6 i x^15+4 x^16-3 i x^17-x^18
--rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Forewarned is worth an octopus in the bush. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (3)
-
Bill Gosper -
James Propp -
Michael Kleber