[ Universal Sequence or K-Sequence , K for Kalkulieren ]
I found interesting family of sequences named " Universal Sequence " which computes any sequences like Universal Turing Machine
I wish Mathematicians study this interesting and strange sequence and reveal the reason why it behaves so
1. Definition of K-sequence
The following iteration represents any sequences
x(n) = [A*x(n-1) + B]/p^k
where [m] is Floor(m), p^k is the highest power of p dividing [A*x(n-1) + B]
2. Several classes of K-sequences
2. 0. Collatz's sequence
It is a generalization of Collatz's 3x+1 sequence.
Because, if p=2 and A=1.5*2^s and B=0.5*2^s+c, 0<=s, 0<=c<1
then f(x)=[1.5*2^s*x+0.5*2^s+c]/2^r
=(1.5*2^s*x+0.5*2^s)/2^r
=(3*x+1)/2^(r-s+1)
It becomes the definition of Collatz's sequence.
But for example 3,10,5,16,8,4,2,1,4,2,1,--- is represented more compactly as 3,5,1,1,1,1,1--- .
2. 1. ESS, for extremely strange sequence, is defined as follows :
K-sequence such that A=p^r + e , 0<e<k , k is about 0.01 , B=1.2
the case of p=2, r=1 is LLI.
ESS looks like a mosaic of periodic subsequences and linear subsequences and random behaving subsequence.
2. 2. LLI, an abbreviation of "a lot of linear parts which are isolated each other", is defined as follows :
ESS such that p=2, A=2+e, 0<e<k, k is about 0.01
For example, ESS, {x(0),p,A,B=1,2,4.001, 1.2} is similar to ESS, {x(0),p,A,B=1,3,3.00001,1.2} , but is absolutely different to LLI.
Only LLI is different from the other ESS.
2. 3. Another class : x0=3, p=2, A=1.6, 1.0<B<=1.2
These sequences seem to be unbounded.
The period of K-sequence such that {x(0),p,A,B=107,2,1.6,1.}1 is still unknown
Don Reble calculated x(2,000,000) = 852756...564079; it has 46892digits.
2. 4. Fibonacci like
LLI { x0, p, A, B } = { 1, 2, Phi, 1.0 }
Where Phi = (1 + 5^(1/2))/2
a(n) = a(n-1) + a(n-2) + 1, with a(0) = a(1) = 1. A001595 on OEIS
1, 1, 3, 5, 9, 15, 25, 41, 67, 109, 177, 287, 465, 753, 1219, 1973, 3193, 5167, 8361, 13529, 21891, 35421, 57313, 92735, 150049, 242785, 392835, 635621, 1028457, 1664079, 2692537, 4356617, 7049155, 11405773, 18454929, 29860703, 48315633, 78176337
2. 5. Many polynomial subsequences.
2. 5. 1. LLI { x(0),p,A,B } = { 1,2,2.00013,3.0 }, between 0-th term and 10000-th term, there are 44 linear subsequences such that 4<={number of terms}.
The first several are :
x(16) to x(3613) x(n)=x(n-1)+2
x(3650) to x(3655) x(n)=x(n-1)+1028
x(3820) to x(3848) x(n)=x(n-1)+66
..........
From a linear part to next linear part, LLI behaves as if it were a random sequence.
Besides them, LLI has two degree's parts, and three and four degree's parts.
2. 5. 2. Two degree
LLI { x(0), p, A, B } = { 2443499297,2,2.00014,3.0 } :
the 68 terms from 0-th to 67-th are represented as follows
x(n) = 6*n^2+171040*n+2443499297
2. 5. 3. Three degree
LLI { x(0),p,A,B } = { 2417903893829791,2,2.00013,3.0 } :
the 5 terms from 0-th term to 4-th term are represented as follows
x(n) = 664/6*n^3+5107490*n^2+471475936498/3*n+2417903893829791
2. 5. 4. Four degree
LLI { x(0),p,A,B } = { 2015985557869547951,2,2.00013,3.2 }
the 8 terms from 0-th term to 7-th term re s follows
2015985557869547951, 2016116596930809473, 2016247644509609977, 2016378700606503103, 2016509765222042527, 2016640838356781961, 2016771920011275153, 2016903010186075887
2. 6. Oscillation like
ESS such that x(0) = 0 , p=2, A=4.001, B=1.2
0 , 5, 21, 85, 341, 1365, 2731, 10927, 5465, 10933, 1367, 2735, 10943, 5473, 10949, 1369, 2739, 10959, 5481, 10965, 1371, 2743, 10975, 5489, 10981, 1373, 2747, 10991, 5497, 10997, 1375, 2751, 11007, 5505, 11013, 1377, 2755, 11023
2. 7. Periodic
LLI { x(0), p, A, B } = { 1, 2, 2.00014, 3.0 } Period =73667
2. 8. Exponential
LLI { x(0), p, A, B } = { 1, 2, 2.00013, 3,0 }
A029580 on OEIS
Formula
x(n) = 2^(n+2) - 3 From 0th term To 11th term
1, 5, 13, 29, 61, 125, 253, 509, 1021, 2045, 4093, 8189, 8191, 8193, 8195, 8197, 8199, 8201, 8203, 8205, 8207, 8209, 8211, 8213, 8215, 8217, 8219, 8221, 8223, 8225, 8227, 8229, 8231, 8233, 8235, 8237, 8239, 8241, 8243, 8245, 8247, 8249, 8251, 8253, 8255
2. 9. Unbound
LLI { x(0), p, A, B } = { 7. 2, 8/5, 1.0 }
A152199 on OEIS
7, 12, 6, 3, 5, 9, 15, 25, 41, 66, 33, 53, 85, 137, 220, 110, 55, 89, 143, 229, 367, 588, 294, 147, 236, 118, 59, 95, 153, 245, 393, 629, 1007, 1612, 806, 403, 645, 1033, 1653, 2645, 4233, 6773, 10837, 17340, 8670, 4335, 6937, 11100, 5550, 2775, 4441, 7106
2. １０. Random
LLI { x(0), p, A, B } = { 15377, 2, 2.00013, 3.0 }
15379, 15381, 15383, 15385, 30775, 61557, 123125, 246269, 492573, 985213, 1970557, 3941373, 7883261, 15767549, 15768575, 15769601, 31541255, 63086613, 63090715, 63094817, 126197839, 126206043, 252428495, 31555613, 63115331, 126238869, 31561769, 15781911, 7891469, 7891983, 7892497, 15786023, 31574101, 63152309, 63156415, 126321043, 126329255, 252674935, 7896605, 15794239, 15795267, 15796295, 15797323, 15798351, 15799379, 15800407, 31602871, 63209853, 63213963, 63218073, 126444367, 126452587, 252921615, 31617257, 63238627, 63242739, 63246851, 126501927, 126510151, 253036751, 506106399, 506139297, 1012344395, 2024820397, 4049904023, 1012541817, 2025215267, 2025346907, 506369639, 1012805109, 2025741885, 4051747119, 1013002621, 2026136935, 2026268635, 4052800687, 8106128241, 16213310281, 32428728295, 64861672327, 64865888337, 64870104621, 64874321179, 64878538011, 64882755117, 129773944995, 259564760605, 519163264631, 1038394020489, 2076923032203, 4154116064403, 8308772163897, 8309312234089, 16619704678771, 33241569919153, 33243730621199, 66491782927381, 132992209786545, 266001708560365, 532037997342845, 1064145159625347, 266053582265181, 532141751496059, 1064352681419815, 2128843728688217, 2128982103530583, 4258240974734627, 2129258880198993, 21293972820262
2. 11. Chaotic
ESS { x(0), p, A, B } = { 13, 2, 4.001, 1,2 }
Program in PARI
pro(m) = { my( x=13 , A=4.001 , B=1.2, p=2, n=0 ); while(n<m, my( y=floor(A*x + B) ); while( Mod(y, p) == 0, y=y/p ); print1 (y, ",", " "); x=y; n++ ) };
pro(500)
13, 53, 213, 853, 1707, 3415, 427, 1709, 3419, 855, 1711, 3423, 107, 429, 1717, 3435, 859, 1719, 3439, 215, 861, 1723, 3447, 431, 1725, 3451, 863, 1727, 3455, 27, 109, 437, 1749, 3499, 875, 1751, 3503, 219, 877, 1755, 3511, 439, 1757, 3515, 879, 1759, 3519, 55, 221, 885, 1771, 3543, 443, 1773, 3547, 887, 1775, 3551, 111, 445, 1781, 3563, 891, 1783, 3567, 223, 893, 1787, 3575, 447, 1789, 3579, 895, 1791, 3583, 7, 29, 117, 469, 1877, 7511, 7513, 7515, 7517, 7519, 7521, 7523, 7525, 7527, 7529, 7531, 7533, 7535, 7537, 7539, 7541, 7543, 7545, 7547, 7549, 7551, 7553, 7555, 7557, 7559, 7561, 7563, 7565, 7567, 7569, 7571, 7573, 7575, 7577, 7579, 7581, 7583, 7585, 7587, 7589, 7591, 7593, 7595, 7597, 7599,
3. A program of K-sequence in Mathematica
In[1]:=
g[x_] := If[Mod[Floor[A*x + B], p] == 0,
Floor[a*x b]/p^IntegerExponent[Floor[a*x + b], p], Floor[a*x + b]];
In[2]:=
a = 5.0001; b = 1.0; p = 5;
NestList[g, 1, 200]
Out[3]=
{1, 6, 31, 156, 781, 3906, 19531, 97657, 97659, 97661, 97663, 19533, 97667, \
97669, 97671, 97673, 3907, 19536, 97682, 97684, 97686, 97688, 19538, 97692,\
97694, 97696, 97698, 3908, 19541, 97707, 97709, 97711, 97713, 19543, 97717,\
97719, 97721, 97723, 3909, 19546, 97732, 97734, 97736, 97738, 19548, 97742,\
97744, 97746, 97748, 782, 3911, 19556, 97782, 97784, 97786, 97788, 19558, \
97792, 97794, 97796, 97798, 3912, 19561, 97807, 97809, 97811, 97813, 19563, \
97817, 97819, 97821, 97823, 3913, 19566, 97832, 97834, 97836, 97838, 19568, \
97842, 97844, 97846, 97848, 3914, 19571, 97857, 97859, 97861, 97863, 19573, \
97867, 97869, 97871, 97873, 783, 3916, 19581, 97907, 97909, 97911, 97913, \
19583, 97917, 97919, 97921, 97923, 3917, 19586, 97932, 97934, 97936, 97938, \
19588, 97942, 97944, 97946, 97948, 3918, 19591, 97957, 97959, 97961, 97963, \
19593, 97967, 97969, 97971, 97973, 3919, 19596, 97982, 97984, 97986, 97988, \
19598, 97992, 97994, 97996,
4.
I am sure polynomial sequence of five degree
Could any one find it ? It might exist on LLI {x0, p, A, B } = {1, 2, 2.000014, 3.0
Graph of LLI
graph of the first 4000 terms of a iteration such that :
x(n)=[A*x(n-1)+B]/p^r , x(0)=1 , A=2.00013 , B=3.0 , p=2
where p^r is the highest power of p dividing [A*x(n-1)+B] , [x]=integer part of x
It is a cyclic sequence whose period is 1790641.
Graph of ESS
I found interesting family of sequences named " Universal Sequence " which computes any sequences like Universal Turing Machine
I wish Mathematicians study this interesting and strange sequence and reveal the reason why it behaves so
1. Definition of K-sequence
The following iteration represents any sequences
x(n) = [A*x(n-1) + B]/p^k
where [m] is Floor(m), p^k is the highest power of p dividing [A*x(n-1) + B]
2. Several classes of K-sequences
2. 0. Collatz's sequence
It is a generalization of Collatz's 3x+1 sequence.
Because, if p=2 and A=1.5*2^s and B=0.5*2^s+c, 0<=s, 0<=c<1
then f(x)=[1.5*2^s*x+0.5*2^s+c]/2^r
=(1.5*2^s*x+0.5*2^s)/2^r
=(3*x+1)/2^(r-s+1)
It becomes the definition of Collatz's sequence.
But for example 3,10,5,16,8,4,2,1,4,2,1,--- is represented more compactly as 3,5,1,1,1,1,1--- .
2. 1. ESS, for extremely strange sequence, is defined as follows :
K-sequence such that A=p^r + e , 0<e<k , k is about 0.01 , B=1.2
the case of p=2, r=1 is LLI.
ESS looks like a mosaic of periodic subsequences and linear subsequences and random behaving subsequence.
2. 2. LLI, an abbreviation of "a lot of linear parts which are isolated each other", is defined as follows :
ESS such that p=2, A=2+e, 0<e<k, k is about 0.01
For example, ESS, {x(0),p,A,B=1,2,4.001, 1.2} is similar to ESS, {x(0),p,A,B=1,3,3.00001,1.2} , but is absolutely different to LLI.
Only LLI is different from the other ESS.
2. 3. Another class : x0=3, p=2, A=1.6, 1.0<B<=1.2
These sequences seem to be unbounded.
The period of K-sequence such that {x(0),p,A,B=107,2,1.6,1.}1 is still unknown
Don Reble calculated x(2,000,000) = 852756...564079; it has 46892digits.
2. 4. Fibonacci like
LLI { x0, p, A, B } = { 1, 2, Phi, 1.0 }
Where Phi = (1 + 5^(1/2))/2
a(n) = a(n-1) + a(n-2) + 1, with a(0) = a(1) = 1. A001595 on OEIS
1, 1, 3, 5, 9, 15, 25, 41, 67, 109, 177, 287, 465, 753, 1219, 1973, 3193, 5167, 8361, 13529, 21891, 35421, 57313, 92735, 150049, 242785, 392835, 635621, 1028457, 1664079, 2692537, 4356617, 7049155, 11405773, 18454929, 29860703, 48315633, 78176337
2. 5. Many polynomial subsequences.
2. 5. 1. LLI { x(0),p,A,B } = { 1,2,2.00013,3.0 }, between 0-th term and 10000-th term, there are 44 linear subsequences such that 4<={number of terms}.
The first several are :
x(16) to x(3613) x(n)=x(n-1)+2
x(3650) to x(3655) x(n)=x(n-1)+1028
x(3820) to x(3848) x(n)=x(n-1)+66
..........
From a linear part to next linear part, LLI behaves as if it were a random sequence.
Besides them, LLI has two degree's parts, and three and four degree's parts.
2. 5. 2. Two degree
LLI { x(0), p, A, B } = { 2443499297,2,2.00014,3.0 } :
the 68 terms from 0-th to 67-th are represented as follows
x(n) = 6*n^2+171040*n+2443499297
2. 5. 3. Three degree
LLI { x(0),p,A,B } = { 2417903893829791,2,2.00013,3.0 } :
the 5 terms from 0-th term to 4-th term are represented as follows
x(n) = 664/6*n^3+5107490*n^2+471475936498/3*n+2417903893829791
2. 5. 4. Four degree
LLI { x(0),p,A,B } = { 2015985557869547951,2,2.00013,3.2 }
the 8 terms from 0-th term to 7-th term re s follows
2015985557869547951, 2016116596930809473, 2016247644509609977, 2016378700606503103, 2016509765222042527, 2016640838356781961, 2016771920011275153, 2016903010186075887
2. 6. Oscillation like
ESS such that x(0) = 0 , p=2, A=4.001, B=1.2
0 , 5, 21, 85, 341, 1365, 2731, 10927, 5465, 10933, 1367, 2735, 10943, 5473, 10949, 1369, 2739, 10959, 5481, 10965, 1371, 2743, 10975, 5489, 10981, 1373, 2747, 10991, 5497, 10997, 1375, 2751, 11007, 5505, 11013, 1377, 2755, 11023
2. 7. Periodic
LLI { x(0), p, A, B } = { 1, 2, 2.00014, 3.0 } Period =73667
2. 8. Exponential
LLI { x(0), p, A, B } = { 1, 2, 2.00013, 3,0 }
A029580 on OEIS
Formula
x(n) = 2^(n+2) - 3 From 0th term To 11th term
1, 5, 13, 29, 61, 125, 253, 509, 1021, 2045, 4093, 8189, 8191, 8193, 8195, 8197, 8199, 8201, 8203, 8205, 8207, 8209, 8211, 8213, 8215, 8217, 8219, 8221, 8223, 8225, 8227, 8229, 8231, 8233, 8235, 8237, 8239, 8241, 8243, 8245, 8247, 8249, 8251, 8253, 8255
2. 9. Unbound
LLI { x(0), p, A, B } = { 7. 2, 8/5, 1.0 }
A152199 on OEIS
7, 12, 6, 3, 5, 9, 15, 25, 41, 66, 33, 53, 85, 137, 220, 110, 55, 89, 143, 229, 367, 588, 294, 147, 236, 118, 59, 95, 153, 245, 393, 629, 1007, 1612, 806, 403, 645, 1033, 1653, 2645, 4233, 6773, 10837, 17340, 8670, 4335, 6937, 11100, 5550, 2775, 4441, 7106
2. １０. Random
LLI { x(0), p, A, B } = { 15377, 2, 2.00013, 3.0 }
15379, 15381, 15383, 15385, 30775, 61557, 123125, 246269, 492573, 985213, 1970557, 3941373, 7883261, 15767549, 15768575, 15769601, 31541255, 63086613, 63090715, 63094817, 126197839, 126206043, 252428495, 31555613, 63115331, 126238869, 31561769, 15781911, 7891469, 7891983, 7892497, 15786023, 31574101, 63152309, 63156415, 126321043, 126329255, 252674935, 7896605, 15794239, 15795267, 15796295, 15797323, 15798351, 15799379, 15800407, 31602871, 63209853, 63213963, 63218073, 126444367, 126452587, 252921615, 31617257, 63238627, 63242739, 63246851, 126501927, 126510151, 253036751, 506106399, 506139297, 1012344395, 2024820397, 4049904023, 1012541817, 2025215267, 2025346907, 506369639, 1012805109, 2025741885, 4051747119, 1013002621, 2026136935, 2026268635, 4052800687, 8106128241, 16213310281, 32428728295, 64861672327, 64865888337, 64870104621, 64874321179, 64878538011, 64882755117, 129773944995, 259564760605, 519163264631, 1038394020489, 2076923032203, 4154116064403, 8308772163897, 8309312234089, 16619704678771, 33241569919153, 33243730621199, 66491782927381, 132992209786545, 266001708560365, 532037997342845, 1064145159625347, 266053582265181, 532141751496059, 1064352681419815, 2128843728688217, 2128982103530583, 4258240974734627, 2129258880198993, 21293972820262
2. 11. Chaotic
ESS { x(0), p, A, B } = { 13, 2, 4.001, 1,2 }
Program in PARI
pro(m) = { my( x=13 , A=4.001 , B=1.2, p=2, n=0 ); while(n<m, my( y=floor(A*x + B) ); while( Mod(y, p) == 0, y=y/p ); print1 (y, ",", " "); x=y; n++ ) };
pro(500)
13, 53, 213, 853, 1707, 3415, 427, 1709, 3419, 855, 1711, 3423, 107, 429, 1717, 3435, 859, 1719, 3439, 215, 861, 1723, 3447, 431, 1725, 3451, 863, 1727, 3455, 27, 109, 437, 1749, 3499, 875, 1751, 3503, 219, 877, 1755, 3511, 439, 1757, 3515, 879, 1759, 3519, 55, 221, 885, 1771, 3543, 443, 1773, 3547, 887, 1775, 3551, 111, 445, 1781, 3563, 891, 1783, 3567, 223, 893, 1787, 3575, 447, 1789, 3579, 895, 1791, 3583, 7, 29, 117, 469, 1877, 7511, 7513, 7515, 7517, 7519, 7521, 7523, 7525, 7527, 7529, 7531, 7533, 7535, 7537, 7539, 7541, 7543, 7545, 7547, 7549, 7551, 7553, 7555, 7557, 7559, 7561, 7563, 7565, 7567, 7569, 7571, 7573, 7575, 7577, 7579, 7581, 7583, 7585, 7587, 7589, 7591, 7593, 7595, 7597, 7599,
3. A program of K-sequence in Mathematica
In[1]:=
g[x_] := If[Mod[Floor[A*x + B], p] == 0,
Floor[a*x b]/p^IntegerExponent[Floor[a*x + b], p], Floor[a*x + b]];
In[2]:=
a = 5.0001; b = 1.0; p = 5;
NestList[g, 1, 200]
Out[3]=
{1, 6, 31, 156, 781, 3906, 19531, 97657, 97659, 97661, 97663, 19533, 97667, \
97669, 97671, 97673, 3907, 19536, 97682, 97684, 97686, 97688, 19538, 97692,\
97694, 97696, 97698, 3908, 19541, 97707, 97709, 97711, 97713, 19543, 97717,\
97719, 97721, 97723, 3909, 19546, 97732, 97734, 97736, 97738, 19548, 97742,\
97744, 97746, 97748, 782, 3911, 19556, 97782, 97784, 97786, 97788, 19558, \
97792, 97794, 97796, 97798, 3912, 19561, 97807, 97809, 97811, 97813, 19563, \
97817, 97819, 97821, 97823, 3913, 19566, 97832, 97834, 97836, 97838, 19568, \
97842, 97844, 97846, 97848, 3914, 19571, 97857, 97859, 97861, 97863, 19573, \
97867, 97869, 97871, 97873, 783, 3916, 19581, 97907, 97909, 97911, 97913, \
19583, 97917, 97919, 97921, 97923, 3917, 19586, 97932, 97934, 97936, 97938, \
19588, 97942, 97944, 97946, 97948, 3918, 19591, 97957, 97959, 97961, 97963, \
19593, 97967, 97969, 97971, 97973, 3919, 19596, 97982, 97984, 97986, 97988, \
19598, 97992, 97994, 97996,
4.
I am sure polynomial sequence of five degree
Could any one find it ? It might exist on LLI {x0, p, A, B } = {1, 2, 2.000014, 3.0
Graph of LLI
graph of the first 4000 terms of a iteration such that :
x(n)=[A*x(n-1)+B]/p^r , x(0)=1 , A=2.00013 , B=3.0 , p=2
where p^r is the highest power of p dividing [A*x(n-1)+B] , [x]=integer part of x
It is a cyclic sequence whose period is 1790641.
Graph of ESS
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