page2 transforms

Permutations with three variables:




The n values where these permutations occur give the eight sequences: (only first 20 values shown)

000: n: {1, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73}
100: n: {2, 4, 8, 14, 16, 22, 26, 28, 32, 34, 38, 44, 46, 52, 56, 58, 62, 64, 68, 74}
010: n: {3, 9, 21, 27, 33, 39, 51, 57, 63, 69, 81, 87, 93, 99, 111, 117, 123, 129, 141, 147}
110: n: {6, 12, 18, 24, 36, 42, 48, 54, 66, 72, 78, 84, 96, 102, 108, 114, 126, 132, 138, 144}
001: n: {5, 25, 35, 55, 65, 85, 95, 115, 125, 145, 155, 175, 185, 205, 215, 235, 245, 265, 275, 295}
101: n: {10, 20, 40, 50, 70, 80, 100, 110, 130, 140, 160, 170, 190, 200, 220, 230, 250, 260, 280, 290}
011: n: {15, 45, 75, 105, 135, 165, 195, 225, 255, 285, 315, 345, 375, 405, 435, 465, 495, 525, 555, 585}
111: n: {30, 60, 90, 120, 150, 180, 210, 240, 270, 300, 330, 360, 390, 420, 450, 480, 510, 540, 570, 600}

First three values in each sequence:
{{1, 7, 11}, {2, 4, 8}, {3, 9, 21}, {6, 12, 18}, {5, 25, 35}, {10, 20, 40}, {15, 45, 75}, {30, 60, 90}}

Difference Sequences:

The difference sequences of the above:

First differences sequences:

000: Differences1[n]: {6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2}
100: Differences1[n]: {2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6}
010: Differences1[n]: {6, 12, 6, 6, 6, 12, 6, 6, 6, 12, 6, 6, 6, 12, 6, 6, 6, 12, 6}
110: Differences1[n]: {6, 6, 6, 12, 6, 6, 6, 12, 6, 6, 6, 12, 6, 6, 6, 12, 6, 6, 6}
001: Differences1[n]: {20, 10, 20, 10, 20, 10, 20, 10, 20, 10, 20, 10, 20, 10, 20, 10, 20, 10, 20}
101: Differences1[n]: {10, 20, 10, 20, 10, 20, 10, 20, 10, 20, 10, 20, 10, 20, 10, 20, 10, 20, 10}
011: Differences1[n]: {30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30}
111: Differences1[n]: {30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30}




Transforms:

For sequence: 000: n: {1, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73}
with 8/30 proportion, or 4/15 reduced proportion.

The first 8 values of the sequence:
n={1, 7, 11, 13, 17, 19, 23, 29}

Taking the reduced proportion 4/15 and creating a list of (4/15)*k for 1<=k<=8 gives the list:
a4={4/15,8/15,4/5,16/15,4/3,8/5,28/15,32/15}

Taking the reduced proportion denominator 15 and creating a list of 15*k for 1<=k<=8 gives the list:
b={15,30,45,60,75,90,105,120}

Taking the reduced proportion numerator 4 and creating a list of 4*k for 1<=k<=8 gives the list:
c={4,8,12,16,20,24,28,32} (this is unused so far)

Taking the reduced proportion numerator 4 and n={1, 7, 11, 13, 17, 19, 23, 29} sequence and creating a list of 4*n[k] for 1<=k<=8 gives the list:
d={4*1,4*7,4*11,4*13,4*17,4*19,4*23,4*29}
d={4,28,44,52,68,76,92,116}

Taking the pairwise differences of the two lists:
b={15,30,45,60,75,90,105,120}
d={4,28,44,52,68,76,92,116}
Gives the sequence:
e={11,2,1,8,7,14,13,4} (oeis A309497)

Dividing the sequence e={11,2,1,8,7,14,13,4} by a1={4/15,2/15,4/45,1/15,4/75,2/45,4/105,1/30} gives f1={165/4,15,45/4,120,525/4,315,1365/4,120}
Dividing the sequence e={11,2,1,8,7,14,13,4} by a2={15/4,15/2,45/4,15,75/4,45/2,105/4,30} gives f2={44/15,4/15,4/45,8/15,28/75,28/45,52/105,2/15}
Dividing the sequence e={11,2,1,8,7,14,13,4} by a3={15/4,15/8,5/4,15/6,3/4,5/8,15/28,15/32} gives f3={44/15,16/15,4/5,128/15,28/3,112/5,364/15,128/15}
Dividing the sequence e={11,2,1,8,7,14,13,4} by a4={4/15,8/15,4/5,16/15,4/3,8/5,28/15,32/15} gives f4={165/4,15/4,5/4,15/2,21/4,35/4,195/28,15/8}
(interesting sequence similar or equivalent to {11,2,1,8,7,14,13,4} but in a fraction form)

Taking Mod[b={15,30,45,60,75,90,105,120}, n={1,7,11,13,17,19,23,29}] gives the sequence l={0,2,1,8,7,14,13,4} which is similar to the sequence e={11,2,1,8,7,14,13,4}
Taking Quotient[b={15,30,45,60,75,90,105,120}, n={1,7,11,13,17,19,23,29}] gives the sequence o={15,4,4,4,4,4,4,4} which only has p1 and p2 as values

Transforms in table form for sequence: 000: n: {1, 7, 11, 13, 17, 19, 23, 29} with proportion 8/30 and reduced proportion 4/15.
proportion 8/30=4/15
 k
 n
 diff[n]
 a1=p5/k
 a2=p6*k
 a3=p6/k
 a4=p5*k
 a5=N[a1]
 a6=D[a1]
 a7=N[a3]
 a8=D[a3]
 b=p2*k=a2*p1
 c=p1*k
 d=p1*n[k] e=b-d=p1*g diff[e]
 Abs[diff[e]]
 f1=e/a1
 f2=e/a2
 f3=e/a3
 f4=e/a4 g=a2-n[k]
 h=N[g]
 i=D[g]
 j=(empty)
 diff[g]
 Abs[diff[g]]
 l=Mod[b,n]
 m=e-l
 o=Quotient[b,n]
p1=4
 1
 1
 6
 4/15
 15/4
 15/4
 4/15 4
 15
 15
 4
 15 4 4 11
 -9
 9
 165/4
 44/15
 44/15
 165/4 11/4
 11
 4
 null
 -9/4
 9/4
 0
 11
 15
p2=15
 2
 7
 4
 2/15
 15/2
 15/8
 8/15 2
 15
 15
 8
 30
 8
 28
 2
 -1
 1
 15
 4/15
 16/15
 15/4
 1/2
 1
 2
 null -1/4
 1/4
 2
 0
 4
p3=8
 3
 11
 2
 4/45
 45/4
 5/4
 4/5
 4
 45
 5
 4
 45
 12
 44
 1
 7
 7
 45/4
 4/45
 4/5
 5/4
 1/4
 1
 4
 null 7/4
 7/4
 1
 0
 4
p4=30
 4
 13
 4
 1/15
 15
 15/6
 16/15
 1
 15
 15
 6
 60
 16
 52
 8
 -1
 1
 120
 8/15
 128/15
 15/2
 2
 2
 1
 null -1/4
 1/4
 8
 0
 4
p5=4/15
 5
 17
 2
 4/75
 75/4
 3/4
 4/3
 4
 75
 3
 4
 75
 20
 68
 7
 7
 7
 525/4
 28/75
 28/3
 21/4
 7/4
 7
 4
 null 7/4
 7/4
 7
 0
 4
p6=15/4
 6
 19
 4
 2/45
 45/2
 5/8
 8/5
 2
 45
 5
 8
 90
 24
 76
 14
 -1
 1
 315
 28/45
 112/5
 35/4
 7/2
 7
 2
 null -1/4
 1/4
 14
 0
 4
p7=8/30
 7
 23
 6
 4/105
 105/4
 15/28
 28/15
 4
 105
 15
 28
 105
 28
 92
 13
 -9
 9
 1365/4
 52/105
 364/15
 195/28
 13/4
 13
 4
 null -9/4
 9/4
 13
 0
 4
p8=30/8
 8
 29
 null
 1/30
 30
 15/32
 32/15
 1
 30
 15
 32
 120
 32
 116
 4
 null
 null
 120
 2/15
 128/15
 15/8
 1
 1
 1
 null null
 null
 4
 0
 4
calculations





























total
 36
 120
 28
 761/1050
 135
 2283/224
 48/5
 22
 345
 88
 104
 540
 144
 480
 60
 -7
 35
 1095
 8578/1575
 1168/15
 4289/56 15
 43
 22
 null -7/4
 35/4
 49
 11
 43
mean
 9/2
 15
 4
 761/8400
 135/8
 2283/1792
 6/5
 11/4
 345/8
 11
 13
 135/2
 18
 60
 15/2
 -1
 5
 1095/8
 4289/6300
 146/15
 4289/448 15/8
 43/8
 11/4
 null -1/4
 5/4
 49/8
 11/8
 43/8
min
 1
 1
 2
 1/30
 15/4
 15/32
 4/15
 1
 15
 3
 4
 15
 4
 4
 1
 -9
 1
 45/4
 4/45
 4/5
 5/4
 1/4
 1
 1
 null -9/4
 1/4
 0
 0
 4
max
 8
 29
 6
 4/15
 30
 15/4
 32/15
 4
 105
 15
 32
 120
 32
 116
 14
 7
 9
 1365/4
 44/15
 364/15
 165/4
 7/2
 13
 4
 null 7/4
 9/4
 14
 11
 15
first
 1
 1
 6




















null




second
 2
 7
 4




















null




penultimate
 7
 23
 4




















null




last
 8
 29
 6




















null




distinct count
 8
 8
 3
 8
 8
 8
 8
 3
 5
 3
 5
 8
 8
 8
 8
 3
 3
 7
 8
 7
 8
 8
 5
 3
 null 3
 3
 8
 2
 2
first differences distinct count
 1
 3
 2
 7
 1
 7
 1
 4
 5
 5
 5
 1
 1
 3
 3
 2
 4
 7
 7
 7
 7
 3
 6
 4
 null 2
 4
 4
 2
 2
link to Mathematica code for above table
(make this code for the table into a function or module and create a table of tables, then calculate values across tables ie for the 1, 1, 2, 4, 8, 16, 26, 50, 88, 176, 352, 576, 824, 1248, 2040, 4080, 8160, 16320, 21728, 34480, 53520, 69648, 111808, 223616, 326464 sequence etc)
(also store the tables to disk instead of ram)
(also make tables for the below table variant that uses the A324591 recycled sequence)

notes:
--write g and h formula at top of table in terms of multipled k factor etc, also add another rational sequence, in addition to a,g,h, that starts with 15/4 and then gets smaller. (then have two rational sequences starting at 4/15 and two starting at 15/4)
--(add a section using the deleted sequence that is similar to A309497 with similar transforms and related sequences)
https://github.com/jmorken/math/tree/master/20200814%20A324591%20sequence%20recycled%20related%20to%20A309497)

Sequences from the above table: (add any comments from above and new comments for each sequence too)
k={1, 2, 3, 4, 5, 6, 7, 8}
n={1, 7, 11, 13, 17, 19, 23, 29}
diff[n]={6, 4, 2, 4, 2, 4, 6, null}
a1={4/15, 2/15, 4/45, 1/15, 4/75, 2/45, 4/105, 1/30} (a1=1/a2=A335261/A335260)
a2={15/4, 15/2, 45/4, 15, 75/4, 45/2, 105/4, 30} (a2=1/a1=A335260/A335261)
a3={15/4, 15/8, 5/4, 15/6, 3/4, 5/8, 15/28, 15/32}
a4={4/15,8/15,4/5, 16/15, 4/3, 8/5, 28/15, 32/15}
a5={4, 2, 4, 1, 4, 2, 4, 1} (a5=Numerator[a1]=Denominator[a2])
a6={15, 15, 45, 15, 75, 45, 105, 30} (a6=Denominator[a1]=Numerator[a2])
a7={15, 15, 5, 15, 3, 5, 15, 15} (a7=Numerator[a3]=Denominator[a4])
a8={4, 8, 4, 6, 4, 8, 28, 32} (a8=Denominator[a3]=Numerator[a4])
b={15, 30, 45, 60, 75, 90, 105, 120}
c={4, 8, 12, 16, 20, 24, 28, 32}
d={4, 28, 44, 52, 68, 76, 92, 116}
e={11, 2, 1, 8, 7, 14, 13, 4} (e=b-d=p1*g=p1*(a2-n[k])=p1*((p6*k)-n[k])=p2*k-p1*n[k]=A309497)
diff[e]={-9, -1, 7, -1, 7, -1, -9, null}
Abs[diff[e]]={9, 1, 7, 1, 7, 1, 9, null}
f1={165/4, 15, 45/4, 120, 525/4, 315, 1365/4, 120} (f1=e/a1)
f2={44/15, 4/15, 4/45, 8/15, 28/75, 28/45, 52/105, 2/15} (f2=e/a2)
f3={44/15, 16/15, 4/5, 128/15, 28/3, 112/5, 364/15, 128/15} (f3=e/a3)
f4={165/4, 15/4, 5/4, 15/2, 21/4, 35/4, 195/28, 15/8} (f4=e/a4)
g={}
h={}
i={}
j={}(empty)
l={} (for l, check positions where l != e, also check count and proportion of values in l that match values in e etc, also see m and o)
m={}
o={}

(mathematica test code for Mod A309497 matches)

(mathematica code for a1 to a8 and f1 to f4)



Table adjusted for A309497 alternate sequence (ie ruler shifted by half interval)
"try to set bb and dd so that bb-dd=ee, ie ll=Mod[bb,n] might help if assume ll can be similar sequence to ee (ie can find values of bb that can work then)"
"ee=bb-dd(check formula) also check other formulas for ee etc"
"checked correct formula: ee=(p1*2)*gg" (make sure it is p1*2 and not p3, ie both are 8 for this table, pretty sure it is p1 though)
"ee=recycled A324591 (add more info from that recycled sequence too)"
"recycled A324591: https://github.com/jmorken/math/blob/master/20200814%20A324591%20sequence%20recycled%20related%20to%20A309497)/sequence%20info%20that%20was%20recycled.txt"
"also show formula of ee=e-Reverse[e] etc"
"find new b, c, d (bb, cc, dd) that can generate correct ee and new ll, mm, oo"
"find correct formulas for aa1,aa3,aa4,bb,cc,dd, etc"
"for formulas for constraints for rows of A309497, in terms of other oeis sequences, try to relate those back to formulas for A309497 etc, ie circular formulas"
"bb=aa2*p1 I think (but doesn't give a very good Mod[bb,n] result still)"
"try Mod[Numerator[bb], n] for larger rows maybe etc"
"can indicate formulas are not for sure correct, and then edit later in Mathematica, good to get the multi-table Mathematica code/function done"
"found dd by first finding ee and bb, ie: dd=bb-ee (not sure if useful or correct)"
proportion 8/30=4/15
 k
 n
 diff[n]
 aa1=(p5/2)/k?=p3/(p2 + (p2*(k - 1)*2))
 aa2=p6/2+p6*(k-1)=(p2 + (p2*(k - 1)*2))/p3
 aa3=(p6/2)/k
 aa4=(p5*2)*k
 aa5=N[a1]
 aa6=D[a1]
 aa7=N[a3]
 aa8=D[a3]
 bb=p2*k?=aa2*p1
 cc=p1*k?
 dd=p1*n[k]? (dd=bb-ee)
 ee=bb-dd=(p1*2)*gg diff[ee]
 Abs[diff[ee]]
 ff1=ee/aa1
 ff2=ee/aa2
 ff3=ee/aa3
 ff4=ee/aa4 gg=aa2-n[k] hh=N[gg]
 ii=D[gg]
 jj=(empty)
 diff[gg]
 Abs[diff[gg]]
 ll=Mod[bb,n]
 mm=ee-ll
 oo=Quotient[bb,n]
p1=4
 1
 1
 6
 8/15
 15/8
 15/8
 8/15



15/2
1/2
 7





7/8

null

1/2

p2=15
 2
 7
 4
 8/45
 45/8
 15/16
 16/15



45/2
67/2
 -11





-11/8

null

3/2

p3=8
 3
 11
 2
 8/75
 75/8
 5/8
 8/5



75/2
101/2
 -13





-13/8

null

9/2

p4=30
 4
 13
 4
 8/105
 105/8
 15/32
 32/15



105/2
103/2
 1





1/8

null

1/2

p5=4/15
 5
 17
 2
 8/135
 135/8
 3/8
 8/3



135/2
137/2
 -1





-1/8

null

33/2

p6=15/4
 6
 19
 4
 8/165
 165/8
 5/16
 16/5



165/2
139/2
 13





13/8

null

13/2

p7=8/30
 7
 23
 6
 8/195
 195/8
 15/56
 56/15



195/2
173/2
 11





11/8

null

11/2

p8=30/8
 8
 29
 null
 8/225
 225/8
 15/64
 64/15



225/2
239/2
 -7
 null
 null



-7/8

null null
 null
 51/2

calculations





























total
 36
 120
 28




















null




mean
 9/2
 15
 4




















null




min
 1
 1
 2




















null




max
 8
 29
 6




















null




first
 1
 1
 6




















null




second
 2
 7
 4




















null




penultimate
 7
 23
 4




















null




last
 8
 29
 6




















null




distinct count
 8
 8
 3




















null




first differences distinct count
 1
 3
 2




















null