[seqfan] Re: Time-Indepenent Schrodinger Equation discrete version

Ron Hardin rhhardin at att.net
Mon Feb 11 17:37:47 CET 2013


I think the question amounts to whether all the values j=1..n of 
sin(j*2pi/n+offset) are rational for any other n than 1,2,3,4,6

 rhhardin at mindspring.com
rhhardin at att.net (either)



----- Original Message ----
> From: Ron Hardin <rhhardin at att.net>
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> Sent: Sun, February 10, 2013 5:19:57 PM
> Subject: [seqfan] Re: Time-Indepenent Schrodinger Equation discrete version
> 
> Apparently what is going on is that there can exist only cycles of lengths 
> 1,2,3,4 and 6 (anyway only those have turned  up).
> 
> czd
> T(n,k)=Number of -k..k arrays of length n whose end-around  centered second 
> difference is a constant times that array
> 
> (Time  independent Schrodinger equation)
> 
> Table  starts
> 
__3__5___7___9__11__13__15__17__19___21___23___25___27___29___31___33___35___37
> 
__5__9__13__17__21__25__29__33__37___41___45___49___53___57___61___65___69___73
> 
__9_23__43__69_101_139_183_233_289__351__419__493__573__659__751__849__953_1063
> 
_13_33__61__97_141_193_253_321_397__481__573__673__781__897_1021_1153_1293_1441
> 
__3__5___7___9__11__13__15__17__19___21___23___25___27___29___31___33___35___37
> 
_17_45__85_137_201_277_365_465_577__701__837__985_1145_1317_1501_1697_1905_2125
> 
__3__5___7___9__11__13__15__17__19___21___23___25___27___29___31___33___35___37
> 
_13_33__61__97_141_193_253_321_397__481__573__673__781__897_1021_1153_1293_1441
> 
__9_23__43__69_101_139_183_233_289__351__419__493__573__659__751__849__953_1063
> 
__5__9__13__17__21__25__29__33__37___41___45___49___53___57___61___65___69___73
> 
__3__5___7___9__11__13__15__17__19___21___23___25___27___29___31___33___35_____
> 
_25_69_133_217_321_445_589_753_937_1141_1365_1609_1873_2157_2461_______________
> 
> *Empirical*
> Columns  have period 12 following pattern 1 2 3 4 5 6 5 4 3 2 1 0 1 2 3 4 ...
> define  t(n,k) {
>     if(n%12==0)return 10*k^2+14*k+1;
>      if(n%12==1||n%12==5||n%12==7||n%12==11)return 2*k+1;
>      if(n%12==2||n%12==10)return 4*k+1;
>     if(n%12==3||n%12==9)return  3*k^2+5*k+1;
>     if(n%12==4||n%12==8)return 4*k^2+8*k+1;
>      if(n%12==6)return 6*k^2+10*k+1;
> }
> 
> 
> Restricting away  subcycles, apparently there can be only cycles of length 
> 1,2,3,4 and  6:
> 
> All solutions for cycles of length n (without shorter cycles) for  k=2.
> 
> Solutions are displayed alternately with their second  difference.
> n=1  k=2
> _-2____0____1____0____0____0____2____0___-1____0
> n=2  k=2
> _-1____4___-2____8____1___-4____2___-8
> __1___-4____2___-8___-1____4___-2____8
> n=3 k=2
> _-2____6___-1____3___-2____6____0____0____1___-3___-1____3____0____0____2___-6
> __0____0____0____0____1___-3___-2____6___-1____3____2___-6____1___-3___-2____6
> __2___-6____1___-3____1___-3____2___-6____0____0___-1____3___-1____3____0____0
> 
> __0____0____2___-6___-1____3____1___-3____1___-3___-2____6____2___-6___-1____3
> _-1____3___-1____3____1___-3___-2____6____0____0____2___-6____0____0___-1____3
> __1___-3___-1____3____0____0____1___-3___-1____3____0____0___-2____6____2___-6
> 
> __0____0____1___-3
> __2___-6____1___-3
> _-2____6___-2____6
> n=4 4  k=2
> __0____0____1___-2___-1____2____1___-2____2___-4___-2____4____1___-2___-2____4
> __2___-4____1___-2____2___-4___-2____4____0____0____2___-4____0____0___-2____4
> __0____0___-1____2____1___-2___-1____2___-2____4____2___-4___-1____2____2___-4
> _-2____4___-1____2___-2____4____2___-4____0____0___-2____4____0____0____2___-4
> 
> __0____0___-2____4___-1____2___-1____2___-2____4____2___-4____2___-4____2___-4
> _-1____2____0____0____0____0____1___-2___-1____2___-2____4____2___-4____1___-2
> __0____0____2___-4____1___-2____1___-2____2___-4___-2____4___-2____4___-2____4
> __1___-2____0____0____0____0___-1____2____1___-2____2___-4___-2____4___-1____2
> 
> _-1____2____1___-2____0____0____0____0____1___-2___-2____4___-1____2____2___-4
> _-2____4___-1____2___-2____4____1___-2____2___-4____1___-2___-1____2___-1____2
> __1___-2___-1____2____0____0____0____0___-1____2____2___-4____1___-2___-2____4
> __2___-4____1___-2____2___-4___-1____2___-2____4___-1____2____1___-2____1___-2
> n=5  k=2
> [no solution]
> n=6  k=2
> __2___-2___-1____1____0____0____0____0____2___-2____1___-1___-2____2____0____0
> __1___-1____1___-1___-1____1____1___-1____0____0____1___-1___-2____2___-2____2
> _-1____1____2___-2___-1____1____1___-1___-2____2____0____0____0____0___-2____2
> _-2____2____1___-1____0____0____0____0___-2____2___-1____1____2___-2____0____0
> _-1____1___-1____1____1___-1___-1____1____0____0___-1____1____2___-2____2___-2
> __1___-1___-2____2____1___-1___-1____1____2___-2____0____0____0____0____2___-2
> 
> _-1____1____1___-1____2___-2____1___-1___-2____2____0____0___-2____2____1___-1
> _-2____2____0____0____2___-2___-1____1___-1____1____2___-2____0____0____2___-2
> _-1____1___-1____1____0____0___-2____2____1___-1____2___-2____2___-2____1___-1
> __1___-1___-1____1___-2____2___-1____1____2___-2____0____0____2___-2___-1____1
> __2___-2____0____0___-2____2____1___-1____1___-1___-2____2____0____0___-2____2
> __1___-1____1___-1____0____0____2___-2___-1____1___-2____2___-2____2___-1____1
> 
> _-1____1___-1____1
> _-1____1____0____0
> __0____0____1___-1
> __1___-1____1___-1
> __1___-1____0____0
> __0____0___-1____1
> n=7  k=2
> [no solution]
> n=8 k=2
> [no solution]
> n=9 k=2
> [no  solution]
> n=10 k=2
> [no solution]
> n=11 k=2
> [no solution]
> n=12  k=2
> [no solution]
> 
> Can this cycle restriction be proved to extend to  any k? (Empirically seems to 
>
> be true)
> 
> The empirical formula suggests  that more values can fill the same cycles but 
>no 
>
> new cycles come up.
> Is  this true?
> 
> 
> rhhardin at mindspring.com
> rhhardin at att.net (either)
> 
> _______________________________________________
> 
> Seqfan  Mailing list - http://list.seqfan.eu/
> 


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