[ 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, K-sequence is the iteration of f(x)=[a*x+b]/p^r, and 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.
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
Only LLI is different from the other ESS.
For example, ESS, x0,p,a,b=1,2,4.001, 1.2 is similar to ESS, x0,p,a,b=1,3,3.00001,1.2 , but is absolutely different to LLI.
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 x0,p,a,b=107,2,1.6,1.1 is still unknown
Don Reble calculated x(2,000,000) = 852756...564079; it has 46892digits.
Several families
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 { x0,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 second power's parts, and third and fourth power's parts.
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.
2. 5. 2. 2nd power
LLI { x0, 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. 3rd power
LLI { x0,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. 4th power
LLI { x0,p,a,b=2015985557869547951,2,2.00013,3.2 }
a subsequence of 4th power exists, from 0-th term to 7-th term.
2015985557869547951, 2016116596930809473, 2016247644509609977, 2016378700606503103, 2016509765222042527, 2016640838356781961, 2016771920011275153, 2016903010186075887
2. 6. Oscillation like
ESS such that a(n+1) = [ A*a(n)+B ]/p^r, where p^r is the highest power of p dividing [ A*a(n)+B ] and p=2, A=4.001, B=1.2
0, 1, 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
Graph of ESS
It represents the first 1000 terms of ESS {x0, p, a, b = 1, 2, 4.001, 1.2} It behaves Periodic -> Chaotic -> Linear -> Random * Exponential
2. 7. Periodic
LLI {x0, p, a, b = 1, 2, 2.00014, 3.0} Period =73667
2. 8. Exponential
LLI {x0, p, a, b = 1, 2, 2.0001-, 3,0}
A029580 on OEIS
Formula
a(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 {x0, 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 {x0, p, A, B } = {15377, 2, 2.0001-, 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 {x0, p, a, b = -3, 2, 4.001, 1,2}
Program in PARI
pro(m) = { my( A=4.001 , B=1.2, p=2, x=-3, 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 5 power exists
Could any one find it ? It might exist on LLI {x0, p, A, B } = {1, 2, 2.000014, 3.0
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, K-sequence is the iteration of f(x)=[a*x+b]/p^r, and 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.
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
Only LLI is different from the other ESS.
For example, ESS, x0,p,a,b=1,2,4.001, 1.2 is similar to ESS, x0,p,a,b=1,3,3.00001,1.2 , but is absolutely different to LLI.
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 x0,p,a,b=107,2,1.6,1.1 is still unknown
Don Reble calculated x(2,000,000) = 852756...564079; it has 46892digits.
Several families
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 { x0,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 second power's parts, and third and fourth power's parts.
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.
2. 5. 2. 2nd power
LLI { x0, 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. 3rd power
LLI { x0,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. 4th power
LLI { x0,p,a,b=2015985557869547951,2,2.00013,3.2 }
a subsequence of 4th power exists, from 0-th term to 7-th term.
2015985557869547951, 2016116596930809473, 2016247644509609977, 2016378700606503103, 2016509765222042527, 2016640838356781961, 2016771920011275153, 2016903010186075887
2. 6. Oscillation like
ESS such that a(n+1) = [ A*a(n)+B ]/p^r, where p^r is the highest power of p dividing [ A*a(n)+B ] and p=2, A=4.001, B=1.2
0, 1, 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
Graph of ESS
It represents the first 1000 terms of ESS {x0, p, a, b = 1, 2, 4.001, 1.2} It behaves Periodic -> Chaotic -> Linear -> Random * Exponential
2. 7. Periodic
LLI {x0, p, a, b = 1, 2, 2.00014, 3.0} Period =73667
2. 8. Exponential
LLI {x0, p, a, b = 1, 2, 2.0001-, 3,0}
A029580 on OEIS
Formula
a(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 {x0, 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 {x0, p, A, B } = {15377, 2, 2.0001-, 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 {x0, p, a, b = -3, 2, 4.001, 1,2}
Program in PARI
pro(m) = { my( A=4.001 , B=1.2, p=2, x=-3, 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 5 power exists
Could any one find it ? It might exist on LLI {x0, p, A, B } = {1, 2, 2.000014, 3.0
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