A191113 -id:A191113 - cp's OEIS frontend

A191146 Integers in (2+A191113)/3; contains A191113 as a proper subsequence.

1, 2, 4, 6, 8, 10, 14, 16, 22, 28, 38, 40, 46, 54, 62, 64, 72, 82, 86, 110, 112, 118, 136, 150, 158, 160, 182, 184, 190, 200, 214, 244, 246, 254, 256, 326, 328, 334, 342, 352, 406, 438, 446, 448, 456, 470, 472, 478, 542, 544, 550, 568, 584, 598, 630, 638, 640, 726, 730, 734, 736, 758, 760, 766, 840, 854

Offset: 1

Clark Kimberling, May 27 2011

nonn

See A191113.

			A191113=(1,2,4,6,10,...), whence (2+A191113)/3=(1,4/2,2,8/3,4,...), so that, retaining only integers, A191146=(1,2,4,...).

Cf. A191113, A191149.

Mathematica
```
(See A191113.)
```

A191149 Integers in (2+A191113)/4; contains A191113 as a proper subsequence.

Original entry on oeis.org

1, 2, 3, 4, 6, 10, 12, 14, 16, 21, 22, 28, 30, 38, 40, 46, 48, 54, 62, 64, 82, 84, 86, 102, 110, 112, 118, 120, 136, 138, 150, 158, 160, 182, 183, 184, 190, 192, 214, 244, 246, 254, 256, 264, 326, 328, 334, 336, 342, 352, 354, 406, 408, 426, 438, 446, 448, 470, 472, 478, 480, 542, 544, 550, 552, 568, 570, 598, 630, 638, 640, 726, 730

Offset: 1

Clark Kimberling, May 27 2011

nonn

See A191113 and A191146.

			A191113=(1,2,4,6,10,...), whence (2+A191113)/4=(3/4,1,6/4,2,3,...), so that, retaining only integers, A191149=(1,2,3,...).

Cf. A191113, A191146.

Mathematica
```
(See A191113.)
```

A190803 Increasing sequence generated by these rules: a(1)=1, and if x is in a then 2x-1 and 3x-1 are in a.

Original entry on oeis.org

1, 2, 3, 5, 8, 9, 14, 15, 17, 23, 26, 27, 29, 33, 41, 44, 45, 50, 51, 53, 57, 65, 68, 77, 80, 81, 86, 87, 89, 98, 99, 101, 105, 113, 122, 129, 131, 134, 135, 149, 152, 153, 158, 159, 161, 170, 171, 173, 177, 194, 195, 197, 201, 203, 209, 225, 230, 239, 242

Offset: 1

Clark Kimberling, May 25 2011

nonn

This sequence represents a class of sequences generated by rules of the form "a(1)=1, and if x is in a then hx+i and jx+k are in a, where h,i,j,k are integers." If m>1, at least one of the numbers b(m)=(a(m)-i)/h and c(m)=(a(m)-k)/j is in the set N of natural numbers. Let d(n) be the n-th b(m) in N, and let e(n) be the n-th c(m) in N. Note that a is a subsequence of both d and e.
Examples, where [A......] indicates a conjecture:
A190803: (h,i,j,k)=(2,-1,3,-1); d=A190841, e=A190842
A190804: (h,i,j,k)=(2,-1,3,0); d=[A190803], e=A190844
A190805: (h,i,j,k)=(2,-1,3,1); d=A190845, e=[A190808]
A190806: (h,i,j,k)=(2,-1,3,2); d=[A190804], e=A190848
...
A190807: (h,i,j,k)=(2,0,3,-1); d=A190849, e=A190850
A003586: (h,i,j,k)=(2,0,3,0); d=e=A003586
A190808: (h,i,j,k)=(2,0,3,1); d=A190851, e=A190852
A190809: (h,i,j,k)=(2,0,3,2); d=A190853, e=A190854
...
A190810: (h,i,j,k)=(2,1,3,-1); d=A190855, e=A190856
A190811: (h,i,j,k)=(2,1,3,0); d=A002977, e=A190857
A002977: (h,i,j,k)=(2,1,3,1); d=A190858, e=A190859
A190812: (h,i,j,k)=(2,1,3,2); d=A069353, e=[A190812]
...
For h=j=3, see A191106; for h=3 and j=4, see A191113.

			1 -> 2 -> 3,5 -> 8,9,14 -> 15,17,23,26,27,41 -> ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
David Garth and Adam Gouge, Affinely Self-Generating Sets and Morphisms, Journal of Integer Sequences, 10 (2007) 1-13.

Cf. A002977, A003586, A190804-A190812, A190841-A190860.

import Data.Set (singleton, deleteFindMin, insert)
a190803 n = a190803_list !! (n-1)
a190803_list = 1 : f (singleton 2)
   where f s = m : (f $ insert (2*m-1) $ insert (3*m-1) s')
             where (m, s') = deleteFindMin s
-- Reinhard Zumkeller, Jun 01 2011

h = 2; i = -1; j = 3; k = -1; f = 1; g = 10;
a = Union[Flatten[NestList[{h # + i, j # + k} &, f, g]]]  (* A190803 *)
b = (a + 1)/2; c = (a + 1)/3; r = Range[1, 300];
d = Intersection[b, r] (* A190841 *)
e = Intersection[c, r] (* A190842 *)
(* Regarding this program - useful for many choices of h,i,j,k,f,g - the depth g must be chosen with care - that is, large enough.  Assuming that h<=j, the least new terms in successive nests a are typically iterates of hx+i, starting from x=1.  If, for example, h=2 and i=0, the least terms are 2,4,8,...,2^g, so that g>=9 ensures inclusion of all the desired terms <=256. *)

a(34)=225 inserted by Reinhard Zumkeller, Jun 01 2011

A191203 Increasing sequence generated by these rules: a(1)=1, and if x is in a then 2x and 1+x^2 are in a.

Original entry on oeis.org

1, 2, 4, 5, 8, 10, 16, 17, 20, 26, 32, 34, 40, 52, 64, 65, 68, 80, 101, 104, 128, 130, 136, 160, 202, 208, 256, 257, 260, 272, 290, 320, 401, 404, 416, 512, 514, 520, 544, 580, 640, 677, 802, 808, 832, 1024, 1025, 1028, 1040, 1088, 1157, 1160, 1280, 1354, 1601, 1604, 1616, 1664, 2048, 2050, 2056, 2080, 2176, 2314, 2320, 2560

Offset: 1

Clark Kimberling, May 29 2011

nonn

The method generalizes: a finite set F={f} of functions f:N->N and finite set G of numbers generate a set S by these rules: (1) every element of G is in S, and (2) if x is in S then f(x) is in S for every f in F.  The sequence a results by taking the numbers in S in increasing order.
Examples include A190803, A191106, A191113, and these:
  A191203: 2x, 1+x^2
  A191211: 1+2x, 1+x^2
  A191281: 2x, x^2-x+1
  A191282: 2x, x^2+x+1
  A191283: 2x, x(x+1)/2
  A191284: floor(3x/2), 2x
  A191285: 3x, floor((x^2)/2)
  A191286: 3x, 1+x^2
  A191287: floor(3x/2), 3x
  A191288: 2x, floor((x^2)/3)
  A191289: 3x-1, x^2
  A191290: 2x+1, x(x+1)/2
For A191203 and other such sequences, the depth g for the NestList in the Mathematica program must be large enough to generate as many terms as required by the user.  For example, the rules 2x and 1+x^2, starting with x=1, successively generate set of numbers whose minima are powers of 2: 1->2->4-> ... 2^g -> ....

			1 -> 2 -> 4,5 -> 8,10,17,26 ->

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A190803, A191106, A191113.

import Data.Set (singleton, deleteFindMin, insert)
a191203 n = a191203_list !! (n-1)
a191203_list = f $ singleton 1 where
   f s = m : f (insert (2 * m) $ insert (m ^ 2 + 1) s')
         where (m, s') = deleteFindMin s
-- Reinhard Zumkeller, Apr 18 2014

g = 12; Union[Flatten[NestList[{2 #, 1 + #^2} &, 1, g]]]
(*  A191203; use g>11 to get all terms up to 4096 *)

A191131 Increasing sequence generated by these rules: a(1)=1, and if x is in a then 3x and 4x+3 are in a.

Original entry on oeis.org

1, 3, 7, 9, 15, 21, 27, 31, 39, 45, 63, 81, 87, 93, 111, 117, 127, 135, 159, 183, 189, 243, 255, 261, 279, 327, 333, 351, 375, 381, 405, 447, 471, 477, 511, 543, 549, 567, 639, 729, 735, 759, 765, 783, 837, 975, 981, 999, 1023, 1047, 1053, 1119, 1125, 1143, 1215, 1311, 1335, 1341, 1407, 1413, 1431, 1503, 1527, 1533, 1623

Offset: 1

Clark Kimberling, May 27 2011

nonn

See A191113.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A191113, A191186, A191187.

Note that A191131, A261524, A261871, and A282572 are very similar and easily confused with each other.

import Data.Set (singleton, deleteFindMin, insert)
a191131 n = a191131_list !! (n-1)
a191131_list = f $ singleton 1
   where f s = m : (f $ insert (3*m) $ insert (4*m+3) s')
             where (m, s') = deleteFindMin s
-- Reinhard Zumkeller, Jun 01 2011

h = 3; i = 0; j = 4; k = 3; f = 1; g = 9;
a = Union[Flatten[NestList[{h # + i, j # + k} &, f, g]]]   (* A191131 *)
b = a/3; c = (a - 3)/4; r = Range[1, 1500];
d = Intersection[b, r] (* A191186 *)
e = Intersection[c, r] (* A191187 *)

A191136 Increasing sequence generated by these rules: a(1)=1, and if x is in a then 3x+1 and 4x+1 are in a.

Original entry on oeis.org

1, 4, 5, 13, 16, 17, 21, 40, 49, 52, 53, 64, 65, 69, 85, 121, 148, 157, 160, 161, 193, 196, 197, 208, 209, 213, 256, 257, 261, 277, 341, 364, 445, 472, 481, 484, 485, 580, 589, 592, 593, 625, 628, 629, 640, 641, 645, 769, 772, 773, 784, 785, 789, 832, 833, 837, 853, 1024, 1025, 1029, 1045, 1093, 1109, 1336, 1365, 1417, 1444

Offset: 1

Clark Kimberling, May 28 2011

nonn

See A191113.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A191113.

import Data.Set (singleton, deleteFindMin, insert)
a191136 n = a191136_list !! (n-1)
a191136_list = f $ singleton 1
   where f s = m : (f $ insert (3*m+1) $ insert (4*m+1) s')
             where (m, s') = deleteFindMin s
-- Reinhard Zumkeller, Jun 01 2011

h = 3; i = 1; j = 4; k = 1; f = 1; g = 9;
a = Union[Flatten[NestList[{h # + i, j # + k} &, f, g]]]   (* A191136 *)
b = (a - 1)/3; c = (a - 1)/4; r = Range[1, 1500];
d = Intersection[b, r] (* A191196 *)
e = Intersection[c, r] (* A191197 *)

A191323 Increasing sequence generated by these rules: a(1)=1, and if x is in a then [3x/2]+1 and 3x+1 are in a, where [ ]=floor.

Original entry on oeis.org

1, 2, 4, 7, 11, 13, 17, 20, 22, 26, 31, 34, 40, 47, 52, 61, 67, 71, 79, 92, 94, 101, 103, 107, 119, 121, 139, 142, 152, 155, 157, 161, 179, 182, 184, 202, 209, 214, 229, 233, 236, 238, 242, 269, 274, 277, 283, 304, 310, 314, 322, 344, 350, 355, 358, 364, 404, 412, 416, 418, 425, 427, 457, 466, 472, 484, 517, 526, 533, 538, 547, 553

Offset: 1

Clark Kimberling, May 30 2011

nonn

This sequence represents a class of sequences generated by rules of the form "a(1)=1, and if x is in a then floor(hx+i) and floor(jx+k) are in a, where h and j are rational numbers and i and k are positive integers."  In the following examples, the floor function is denoted by [ ].
A191323:  [3x/2]+1, 3x+1
A191324:  [3x/2]+1, 3x+2
A191325:  [3x/2], [5x/2]
A191326:  [3x/2], [7x/2]
A191327:  [5x/2], [7x/2]
A191328:  [5x/3], [7x/3]
Other families of sequences generated by "rules" are listed at A191803, A191106, A101113 and A191203.

			1 -> 2,4 -> 6,7,13 -> 10,11,19,20,22,40 -> ...

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A190503, A191100, A191113, A191203.

h = 3; i = 1; j = 3; k = 1; f = 1; g = 12;
a=Union[Flatten[NestList[{Floor[h#/2]+i,j#+k}&,f,g]]]
(* A191323 *)

A191119 Increasing sequence generated by these rules: a(1)=1, and if x is in a then 3x-1 and 4x-3 are in a.

Original entry on oeis.org

1, 2, 5, 14, 17, 41, 50, 53, 65, 122, 149, 158, 161, 194, 197, 209, 257, 365, 446, 473, 482, 485, 581, 590, 593, 626, 629, 641, 770, 773, 785, 833, 1025, 1094, 1337, 1418, 1445, 1454, 1457, 1742, 1769, 1778, 1781, 1877, 1886, 1889, 1922, 1925, 1937, 2309, 2318, 2321, 2354, 2357, 2369, 2498, 2501, 2513, 2561, 3074, 3077

Offset: 1

Clark Kimberling, May 27 2011

nonn

See A191113.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A191113.

import Data.Set (singleton, deleteFindMin, insert)
a191119 n = a191119_list !! (n-1)
a191119_list = 1 : f (singleton 2)
   where f s = m : (f $ insert (3*m-1) $ insert (4*m-3) s')
             where (m, s') = deleteFindMin s
-- Reinhard Zumkeller, Jun 01 2011

h = 3; i = -1; j = 4; k = -3; f = 1; g = 9;
a = Union[Flatten[NestList[{h # + i, j # + k} &, f, g]]]  (* A191119 *)
b = (a + 1)/3; c = (a + 3)/4; r = Range[1, 1500];
d = Intersection[b, r] (* A191120 *)
e = Intersection[c, r] (* A191163 *)

A191120 Increasing sequence generated by these rules: a(1)=1, and if x is in a then 3x-1 and 4x-2 are in a.

Original entry on oeis.org

1, 2, 5, 6, 14, 17, 18, 22, 41, 50, 53, 54, 65, 66, 70, 86, 122, 149, 158, 161, 162, 194, 197, 198, 209, 210, 214, 257, 258, 262, 278, 342, 365, 446, 473, 482, 485, 486, 581, 590, 593, 594, 626, 629, 630, 641, 642, 646, 770, 773, 774, 785, 786, 790, 833, 834, 838, 854, 1025, 1026, 1030, 1046, 1094, 1110, 1337, 1366, 1418, 1445

Offset: 1

Clark Kimberling, May 27 2011

nonn

See A191113.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A191113, A191229, A191165.

import Data.Set (singleton, deleteFindMin, insert)
a191120 n = a191120_list !! (n-1)
a191120_list = f $ singleton 1
   where f s = m : (f $ insert (3*m-1) $ insert (4*m-2) s')
             where (m, s') = deleteFindMin s
-- Reinhard Zumkeller, Jun 01 2011

h = 3; i = -1; j = 4; k = -2; f = 1; g = 9;
a = Union[Flatten[NestList[{h # + i, j # + k} &, f, g]]]  (* A191120 *)
b = (a + 1)/3; c = (a + 2)/4; r = Range[1, 1500];
d = Intersection[b, r] (* A191229 *)
e = Intersection[c, r] (* A191165 *)

A191121 Increasing sequence generated by these rules: a(1)=1, and if x is in a then 3x-1 and 4x-1 are in a.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 11, 14, 19, 20, 23, 27, 31, 32, 41, 43, 55, 56, 59, 68, 75, 79, 80, 91, 92, 95, 107, 122, 123, 127, 128, 163, 164, 167, 171, 176, 203, 219, 223, 224, 235, 236, 239, 271, 272, 275, 284, 299, 315, 319, 320, 363, 365, 367, 368, 379, 380, 383, 427, 487, 488, 491, 500, 507, 511, 512, 527, 608, 651, 655, 656, 667, 668

Offset: 1

Clark Kimberling, May 27 2011

nonn

See A191113.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A191113, A191166, A191167.

import Data.Set (singleton, deleteFindMin, insert)
a191121 n = a191121_list !! (n-1)
a191121_list = f $ singleton 1
   where f s = m : (f $ insert (3*m-1) $ insert (4*m-1) s')
             where (m, s') = deleteFindMin s
-- Reinhard Zumkeller, Jun 01 2011

h = 3; i = -1; j = 4; k = -1; f = 1; g = 9;
a = Union[Flatten[NestList[{h # + i, j # + k} &, f, g]]]  (* A191121 *)
b = (a + 1)/3; c = (a + 1)/4; r = Range[1, 1500];
d = Intersection[b, r] (* A191166 *)
e = Intersection[c, r] (* A191167 *)

cp's OEIS Frontend

A191146 Integers in (2+A191113)/3; contains A191113 as a proper subsequence.

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Mathematica

A191149 Integers in (2+A191113)/4; contains A191113 as a proper subsequence.

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Mathematica

A190803 Increasing sequence generated by these rules: a(1)=1, and if x is in a then 2x-1 and 3x-1 are in a.

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Haskell

Mathematica

Extensions

A191203 Increasing sequence generated by these rules: a(1)=1, and if x is in a then 2x and 1+x^2 are in a.

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Haskell

Mathematica

A191131 Increasing sequence generated by these rules: a(1)=1, and if x is in a then 3x and 4x+3 are in a.

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Haskell

Mathematica

A191136 Increasing sequence generated by these rules: a(1)=1, and if x is in a then 3x+1 and 4x+1 are in a.

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Haskell

Mathematica

A191323 Increasing sequence generated by these rules: a(1)=1, and if x is in a then [3x/2]+1 and 3x+1 are in a, where [ ]=floor.

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Mathematica

A191119 Increasing sequence generated by these rules: a(1)=1, and if x is in a then 3x-1 and 4x-3 are in a.

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