A274160 -id:A274160 - cp's OEIS frontend

A274142 Number of integers in n-th generation of tree T(1/2) defined in Comments.

1, 1, 1, 2, 2, 4, 5, 8, 11, 17, 25, 37, 54, 81, 119, 177, 261, 388, 574, 851, 1260, 1868, 2767, 4101, 6077, 9006, 13347, 19781, 29315, 43448, 64392, 95436, 141444, 209636, 310705, 460501, 682519, 1011581, 1499295, 2222155, 3293534, 4881472, 7235018, 10723311, 15893460, 23556367, 34913897, 51747400

Offset: 0

Clark Kimberling, Jun 11 2016

nonn

Let T* be the infinite tree with root 0 generated by these rules:  if p is in T*, then p+1 is in T* and x*p is in T*.  Let g(n) be the set of nodes in the n-th generation, so that g(0) = {0}, g(1) = {1}, g(2) = {2,x}, g(3) = {3,2x,x+1,x^2}, etc.  Let T(r) be the tree obtained by substituting r for x.
Guide to related sequences:
r           sequence
1/2         A274142
1/3         A274143
1/4         A274144
2/3         A274145
3/4         A274146
-1/2        A274147
-1/3        A274148
-1/4        A274149
-2/3        A274150
-3/4        A274151
3/2         A274152
5/2         A274153
-3/2        A274154
-5/2        A274155
2^(1/2)     A000045 (Fibonacci numbers)
2^(1/3)     A000930
2^(1/4)     A003269
2^(-1/2)    A274156
3^(-1/2)    A274157
2^(-1/3)    A274158
3^(-1/3)    A274159
i           A274160
2i          A206743
3i          A274162
4i          A274163
i/2         A274149
i/3         A274165
i+1         A274166
i-1         A274167
(-1+3i)/2   A274168

			If r = 1/2, then g(3) = {3,2r,r+1, r^2}, in which the integers are 3 and 1, so that a(3) = 2.

Kenny Lau, Table of n, a(n) for n = 0..5847

Cf. A274143-A274160, A274162, A274163, A274165-A274168.

z = 18; t = Join[{{0}}, Expand[NestList[DeleteDuplicates[Flatten[Map[{# + 1, x*#} &, #], 1]] &, {1}, z]]];
u = Table[t[[k]] /. x -> 1/2, {k, 1, z}];
Table[Count[Map[IntegerQ, u[[k]]], True], {k, 1, z}]
(* second program: *)
T[0] = {0}; T[n_] := T[n] = Complement[Join[T[n-1]+1, x*T[n-1]], T[n-1]]; Reap[For[n = 0, n <= 25, n++, cnt = Count[T[n] /. x -> 1/2, Integer]; Print[n, " ", cnt]; Sow[cnt]]][[2, 1]] (* _Jean-François Alcover, Jun 14 2016 *)

More terms from Jean-François Alcover, Jun 14 2016

More terms from Kenny Lau, Jul 04 2016

A284222 T(n,k) = Number of n X k 0..1 arrays with the number of 1's horizontally or antidiagonally adjacent to some 0 equal to the number of 0's adjacent to some 1, with top left element zero.

Original entry on oeis.org

1, 2, 2, 3, 6, 4, 6, 13, 19, 8, 10, 44, 70, 62, 16, 19, 143, 499, 467, 212, 32, 33, 495, 3214, 6511, 3224, 751, 64, 62, 1746, 22729, 85887, 91260, 23218, 2719, 128, 112, 6399, 163708, 1214427, 2417194, 1317770, 170155, 10002, 256, 212, 23776, 1209005

Offset: 1

R. H. Hardin, Mar 23 2017

Table starts
...1.....2.......3..........6............10..............19.................33
...2.....6......13.........44...........143.............495...............1746
...4....19......70........499..........3214...........22729.............163708
...8....62.....467.......6511.........85887.........1214427...........17538010
..16...212....3224......91260.......2417194........68272108.........1970768869
..32...751...23218....1317770......69938626......3945536189.......227677212684
..64..2719..170155...19371893....2058141264....232036495984.....26772321109067
.128.10002.1264605..288303232...61304713234..13815034342734...3187484402643589
.256.37258.9488108.4330336770.1842647045686.830095968128023.383012404461341362

			Some solutions for n=4 k=4
..0..1..1..1. .0..1..0..1. .0..0..1..1. .0..1..1..1. .0..1..1..1
..0..0..0..1. .0..0..1..1. .1..0..0..0. .0..0..1..0. .0..0..0..1
..1..1..0..1. .0..1..0..1. .1..0..1..1. .1..1..0..1. .1..1..0..0
..0..0..1..0. .1..0..0..1. .0..1..1..1. .0..1..0..0. .1..0..1..0

R. H. Hardin, Table of n, a(n) for n = 1..198

Column 1 is A000079(n-1).

Row 1 is A274160(n+1).

A284334 T(n,k) = Number of n X k 0..1 arrays with the number of 1's horizontally, diagonally or antidiagonally adjacent to some 0 equal to the number of 0's adjacent to some 1, with top left element zero.

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 6, 13, 13, 8, 10, 34, 49, 42, 16, 19, 118, 373, 415, 134, 32, 33, 419, 2259, 5130, 2243, 467, 64, 62, 1481, 18114, 68458, 70039, 19385, 1689, 128, 112, 5394, 120058, 966889, 1792007, 1000935, 122495, 6210, 256, 212, 19848, 959041, 13941789

Offset: 1

R. H. Hardin, Mar 25 2017

Table starts
...1.....2.......3..........6............10..............19.................33
...2.....4......13.........34...........118.............419...............1481
...4....13......49........373..........2259...........18114.............120058
...8....42.....415.......5130.........68458..........966889...........13941789
..16...134....2243......70039.......1792007........53435287.........1495300322
..32...467...19385....1000935......53977741......3063494246.......177476886878
..64..1689..122495...14581112....1544903012....179138039472.....20513380740103
.128..6210.1021999..215806808...46769835006..10629901713799...2467754689139967
.256.23178.7013557.3229703988.1390664014686.637390912071364.294625135884008616

			Some solutions for n=4, k=4
..0..0..0..0. .0..1..0..1. .0..0..1..1. .0..1..1..0. .0..0..1..0
..1..0..1..0. .1..0..1..1. .0..1..1..0. .1..0..0..1. .0..1..0..1
..1..0..0..0. .0..1..0..1. .1..1..1..0. .1..1..0..0. .1..0..1..1
..1..1..1..1. .0..0..1..0. .0..0..1..1. .0..1..1..0. .0..0..1..1

R. H. Hardin, Table of n, a(n) for n = 1..143

Column 1 is A000079(n-1).
Column 2 is A284159.
Row 1 is A274160(n+1).

A284113 T(n,k) = Number of n X k 0..1 arrays with the number of 1's king-move adjacent to some 0 equal to the number of 0's adjacent to some 1, with top left element zero.

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 6, 13, 13, 6, 10, 46, 25, 46, 10, 19, 160, 435, 435, 160, 19, 33, 571, 1230, 5896, 1230, 571, 33, 62, 2095, 20275, 80200, 80200, 20275, 2095, 62, 112, 7802, 67957, 1143533, 905327, 1143533, 67957, 7802, 112, 212, 29326, 1057769, 16442183

Offset: 1

R. H. Hardin, Mar 20 2017

Table starts
...1......2........3...........6.............10................19
...2......4.......13..........46............160...............571
...3.....13.......25.........435...........1230.............20275
...6.....46......435........5896..........80200...........1143533
..10....160.....1230.......80200.........905327..........61595318
..19....571....20275.....1143533.......61595318........3514396333
..33...2095....67957....16442183......835390074......202053817163
..62...7802..1057769...240146980....51786529350....11804599876991
.112..29326..4091201..3538905880...809052347723...696189477669561
.212.111040.57977326.52568393480.45439256969227.41397421665690592

			Some solutions for n=4, k=4
..0..0..0..1. .0..1..1..1. .0..0..0..0. .0..0..0..0. .0..0..1..0
..1..1..1..1. .1..1..0..1. .1..0..1..0. .0..1..1..1. .1..1..1..0
..1..0..0..1. .1..0..0..0. .1..1..1..1. .0..1..1..1. .0..1..1..0
..0..0..1..0. .0..0..0..1. .1..1..1..0. .1..0..1..0. .1..1..0..0

R. H. Hardin, Table of n, a(n) for n = 1..161

Column 1 is A274160(n+1).

A284165 T(n,k) = Number of n X k 0..1 arrays with the number of 1's horizontally or vertically adjacent to some 0 equal to the number of 0's adjacent to some 1, with top left element zero.

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 6, 13, 13, 6, 10, 42, 62, 42, 10, 19, 134, 461, 461, 134, 19, 33, 467, 2872, 5888, 2872, 467, 33, 62, 1689, 21790, 80072, 80072, 21790, 1689, 62, 112, 6210, 154108, 1142173, 2179255, 1142173, 154108, 6210, 112, 212, 23178, 1174235

Offset: 1

R. H. Hardin, Mar 21 2017

Table starts
...1.....2.......3..........6............10..............19.................33
...2.....4......13.........42...........134.............467...............1689
...3....13......62........461..........2872...........21790.............154108
...6....42.....461.......5888.........80072.........1142173...........16699997
..10...134....2872......80072.......2179255........63520018.........1852406216
..19...467...21790....1142173......63520018......3648835932.......213897742464
..33..1689..154108...16699997....1852406216....213897742464.....25064817057781
..62..6210.1174235..247899694...55270158210..12712253739523...2981736353245523
.112.23178.8682209.3717494360.1656400271264.762954434250117.357844048109256955

			Some solutions for n=4, k=4
..0..1..0..1. .0..0..0..1. .0..0..0..1. .0..0..0..1. .0..1..1..0
..0..0..0..1. .1..0..0..1. .0..1..1..1. .0..1..1..1. .1..0..1..0
..1..1..0..1. .1..0..1..1. .0..1..1..1. .0..0..1..1. .0..1..1..0
..0..1..0..1. .0..1..1..1. .0..0..0..1. .1..0..1..1. .0..1..0..1

R. H. Hardin, Table of n, a(n) for n = 1..143

Column 1 is A274160(n+1).

A284199 T(n,k) = Number of n X k 0..1 arrays with the number of 1's horizontally, antidiagonally or vertically adjacent to some 0 equal to the number of 0's adjacent to some 1, with top left element zero.

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 6, 13, 13, 6, 10, 40, 43, 40, 10, 19, 131, 412, 412, 131, 19, 33, 455, 1919, 5268, 1919, 455, 33, 62, 1611, 18842, 71181, 71181, 18842, 1611, 62, 112, 5818, 105367, 1001047, 1488503, 1001047, 105367, 5818, 112, 212, 21328, 983238

Offset: 1

R. H. Hardin, Mar 22 2017

Table starts
...1.....2.......3..........6............10..............19.................33
...2.....4......13.........40...........131.............455...............1611
...3....13......43........412..........1919...........18842.............105367
...6....40.....412.......5268.........71181.........1001047...........14392557
..10...131....1919......71181.......1488503........54333131.........1321614728
..19...455...18842....1001047......54333131......3083244672.......178138187113
..33..1611..105367...14392557....1321614728....178138187113.....18564422929880
..62..5818..983238..210449237...45952259333..10468760423253...2429325477564655
.112.21328.6123441.3113715736.1219757876905.622301968698475.271315506003954612

			Some solutions for n=4, k=4
..0..1..1..0. .0..1..0..0. .0..1..1..0. .0..1..1..0. .0..0..0..0
..1..0..1..0. .1..0..0..1. .1..0..1..1. .1..0..0..1. .0..0..0..0
..1..0..0..1. .1..0..0..1. .0..1..1..0. .1..0..0..0. .0..0..1..1
..1..0..0..1. .1..1..1..0. .0..0..1..0. .1..1..0..1. .1..1..1..0

R. H. Hardin, Table of n, a(n) for n = 1..180

Column 1 is A274160(n+1).

cp's OEIS Frontend

A274142 Number of integers in n-th generation of tree T(1/2) defined in Comments.

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A284222 T(n,k) = Number of n X k 0..1 arrays with the number of 1's horizontally or antidiagonally adjacent to some 0 equal to the number of 0's adjacent to some 1, with top left element zero.

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A284334 T(n,k) = Number of n X k 0..1 arrays with the number of 1's horizontally, diagonally or antidiagonally adjacent to some 0 equal to the number of 0's adjacent to some 1, with top left element zero.

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A284113 T(n,k) = Number of n X k 0..1 arrays with the number of 1's king-move adjacent to some 0 equal to the number of 0's adjacent to some 1, with top left element zero.

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A284165 T(n,k) = Number of n X k 0..1 arrays with the number of 1's horizontally or vertically adjacent to some 0 equal to the number of 0's adjacent to some 1, with top left element zero.

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Keywords

Comments

Examples

Links

Crossrefs

A284199 T(n,k) = Number of n X k 0..1 arrays with the number of 1's horizontally, antidiagonally or vertically adjacent to some 0 equal to the number of 0's adjacent to some 1, with top left element zero.

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