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

Ron Hardin rhhardin at att.net
Sun Feb 10 23:19:49 CET 2013


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 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)


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