A140253 -id:A140253 - cp's OEIS frontend

A266161 Hamming weights of A266089.

0, 1, 2, 1, 2, 1, 2, 3, 2, 1, 2, 3, 2, 3, 4, 3, 2, 1, 2, 3, 2, 3, 4, 3, 2, 3, 4, 3, 4, 3, 4, 5, 4, 3, 2, 1, 2, 3, 2, 3, 2, 3, 4, 3, 4, 3, 4, 5, 4, 3, 2, 3, 4, 3, 4, 5, 4, 3, 4, 5, 4, 5, 6, 5, 4, 3, 2, 1, 2, 3, 2, 3, 2, 3, 4, 3, 4, 3, 4, 5, 4, 3, 2, 3, 4, 3

Offset: 0

Reinhard Zumkeller, Dec 22 2015

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A000120, A140253, A266089.

Haskell
```
a266161 = a000120 . a266089
```

s[0] = 0; s[n_] := s[n] = Module[{bw = DigitCount[s[n - 1], 2, 1], k = 1}, While[!FreeQ[Array[s, n - 1], k] || Abs[DigitCount[k, 2, 1] - bw] != 1, k++]; k]; DigitCount[Array[s, 100, 0], 2, 1] (* Amiram Eldar, Jul 18 2023 *)

a(n) = A000120(A266089(n)).
abs(a(n+1) - a(n)) = 1 by definition of A266089.
For n > 0: a(A140253(n)) = n and a(m) < n for m < A140253(n).

A290259 Triangle read by rows: row n (>=1) contains in increasing order the integers for which the binary representation has length n, the first run of 1's has odd length, and all the other runs of 1's have even length.

Original entry on oeis.org

1, 2, 4, 7, 8, 11, 14, 16, 19, 22, 28, 31, 32, 35, 38, 44, 47, 56, 59, 62, 64, 67, 70, 76, 79, 88, 91, 94, 112, 115, 118, 124, 127, 128, 131, 134, 140, 143, 152, 155, 158, 176, 179, 182, 188, 191, 224, 227, 230, 236, 239, 248, 251, 254, 256, 259, 262, 268, 271, 280, 283, 286, 304, 307, 310, 316, 319, 352, 355, 358, 364, 367, 376, 379, 382, 448, 451, 454, 460, 463, 472, 475, 478, 496, 499, 502, 508, 511

Offset: 1

Emeric Deutsch, Sep 12 2017

The viabin numbers of integer partitions having only odd parts. The viabin number of an integer partition is defined in the following way. Consider the southeast border of the Ferrers board of the integer partition and consider the binary number obtained by replacing each east step with 1 and each north step, except the last one, with 0. The corresponding decimal form is, by definition, the viabin number of the given integer partition. "Viabin" is coined from "via binary". For example, consider the integer partition [7,5]. The southeast border of its Ferrers board yields 11111011 (length is 8), leading to the viabin number 251 (a term in row 8).
Number of entries in row n is the Fibonacci number F(n) = A000045(n).
T(n,k) = A290258(n+1,k) - 2^n.
Last entry in row n = A140253(n).

			115 is in the sequence; indeed, its binary representation, namely 1110011, has first run of 1's of odd length and the other runs of 1's have even length.
Triangle begins:
   1;
   2;
   4,  7;
   8, 11, 14;
  16, 19, 22, 28, 31;
  32, 35, 38, 44, 47, 56, 59, 62;
  ...

Cf. A000045, A140253, A290258.

A[1] := {1}; A[2] := {2}; for n from 3 to 10 do A[n] := `union`(map(proc (x) 2*x end proc, A[n-1]), map(proc (x) 4*x+3 end proc, A[n-2])) end do; # yields sequence in triangular form

nmax = 10; A[1] = {1}; A[2] = {2}; For[n = 3, n <= nmax, n++, A[n] = Union[2 A[n-1], 4 A[n-2] + 3]]; Table[A[n], {n, 1, nmax}] // Flatten (* Jean-François Alcover, Aug 26 2024, after Maple program *)

The entries in row n (n>=3) are (i) 2x, where x is in row n-1, and (ii) 4y + 3, where y is in row n-2. The Maple program is based on this.

A155997 Triangle read by rows: T(n, k) = f(n, k) + f(n, n-k), where f(n, k) = binomial(n, k)*(1 + (-1)^k)/2.

Original entry on oeis.org

2, 1, 1, 2, 0, 2, 1, 3, 3, 1, 2, 0, 12, 0, 2, 1, 5, 10, 10, 5, 1, 2, 0, 30, 0, 30, 0, 2, 1, 7, 21, 35, 35, 21, 7, 1, 2, 0, 56, 0, 140, 0, 56, 0, 2, 1, 9, 36, 84, 126, 126, 84, 36, 9, 1, 2, 0, 90, 0, 420, 0, 420, 0, 90, 0, 2, 1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1

Offset: 0

Roger L. Bagula, Feb 01 2009

Row sums are: {2, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024,...}

			Triangle begins as:
  2;
  1, 1;
  2, 0,  2;
  1, 3,  3,  1;
  2, 0, 12,  0,   2;
  1, 5, 10, 10,   5,   1;
  2, 0, 30,  0,  30,   0,   2;
  1, 7, 21, 35,  35,  21,   7,  1;
  2, 0, 56,  0, 140,   0,  56,  0,  2;
  1, 9, 36, 84, 126, 126,  84, 36,  9, 1;
  2, 0, 90,  0, 420,   0, 420,  0, 90, 0, 2;

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Flat(List([0..12], n-> List([0..n], k-> Binomial(n, k)*(2 + (-1)^k*(1 + (-1)^n))/2 ))); # G. C. Greubel, Dec 01 2019

[Binomial(n, k)*(2+(-1)^k*(1+(-1)^n))/2: k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 01 2019

seq(seq( binomial(n, k)*(2+(-1)^k*(1+(-1)^n))/2, k=0..n), n=0..12); # G. C. Greubel, Dec 01 2019

f[n_, k_]:= (Binomial[n, k] + (-1)^k*Binomial[n, k])/2; Table[f[n,k]+f[n,n-k], {n,0,10}, {k,0,n}]//Flatten
Table[Binomial[n, k]*(2+(-1)^k*(1+(-1)^n))/2, {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 01 2019 *)

T(n,k) = binomial(n, k)*(2 + (-1)^k*(1 + (-1)^n))/2; \\ G. C. Greubel, Dec 01 2019

[[binomial(n, k)*(2+(-1)^k*(1+(-1)^n))/2 for k in (0..n)] for n in (0..12)] # G. C. Greubel, Dec 01 2019

T(n, k) = f(n, k) + f(n, n-k), where f(n, k) = binomial(n, k)*(1 + (-1)^k)/2.
From G. C. Greubel, Dec 01 2019: (Start)
T(n, k) = binomial(n, k)*(2 + (-1)^k*(1 + (-1)^n))/2.
Sum_{k=0..n} T(n,k) = 2^n for n >= 1.
Sum_{k=0..n-1} T(n,k) = (2^(n+1) - 3 - (-1)^n)/2 = A140253(n), n >= 2. (End)

A290191 Binary representation of the diagonal from the corner to the origin of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 673", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 10, 111, 1110, 11111, 111110, 1111111, 11111110, 111111111, 1111111110, 11111111111, 111111111110, 1111111111111, 11111111111110, 111111111111111, 1111111111111110, 11111111111111111, 111111111111111110, 1111111111111111111, 11111111111111111110

Offset: 0

Robert Price, Jul 23 2017

Initialized with a single black (ON) cell at stage zero.

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Robert Price, Table of n, a(n) for n = 0..126
Robert Price, Diagrams of first 20 stages
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Wolfram Research, Wolfram Atlas of Simple Programs
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata

Cf. A288335, A140253, A140252.

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 673; stages = 128;
rule = IntegerDigits[code, 2, 10];
g = 2 * stages + 1; (* Maximum size of grid *)
a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
ca = a;
ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
PrependTo[ca, a];
(* Trim full grid to reflect growth by one cell at each stage *)
k = (Length[ca[[1]]] + 1)/2;
ca = Table[Table[Part[ca[[n]] [[j]],Range[k + 1 - n, k - 1 + n]], {j, k + 1 - n, k - 1 + n}], {n, 1, k}];
Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 10], {i, 1, stages - 1}]

Conjecture: a(n) is the binary representation of 2^(n - 1) - 1 - (n mod 2).

A208324 Triangle T(n,k), read by rows, given by (2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (4, -2, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 2, 4, 3, 10, 8, 4, 18, 28, 16, 5, 28, 64, 72, 32, 6, 40, 120, 200, 176, 64, 7, 54, 200, 440, 576, 416, 128, 8, 70, 308, 840, 1456, 1568, 960, 256, 9, 88, 448, 1456, 3136, 4480, 4096, 2176, 512, 10, 108, 624, 2352, 6048, 10752, 13056, 10368, 4864

Offset: 0

Philippe Deléham, Feb 25 2012

Row sums are A134931(n).
Diagonal sums are A140253(n).
Compare this sequence with A207627.
Column k is divisible by 2^k.

			Triangle begins :
1
2, 4
3, 10, 8
4, 18, 28, 16
5, 28, 64, 72, 32
6, 40, 120, 200, 176, 64
7, 54, 200, 440, 576, 416, 128
8, 70, 308, 840, 1456, 1568, 960, 256
9, 88, 448, 1456, 3136, 4480, 4096, 2176, 512
10, 108, 624, 2352, 6048, 10752, 13056, 10368, 4864, 1024

Cf. A207627

T(n,0) = n+1.
T(n,1) = 2*T(n,0) + T(n-1,1).
T(n,k) = 2*T(n-1,k-1) + T(n-1,k) for k>1.
T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k) - 2*T(n-2,k-1) with T(0,0) = 1, T(1,0) = 2, T(1,1) = 4.
G.f.: (1+2*y*x)/(1-2*(1+y)*x+(1+2*y)*x^2).

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A266161 Hamming weights of A266089.

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A290259 Triangle read by rows: row n (>=1) contains in increasing order the integers for which the binary representation has length n, the first run of 1's has odd length, and all the other runs of 1's have even length.

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A155997 Triangle read by rows: T(n, k) = f(n, k) + f(n, n-k), where f(n, k) = binomial(n, k)*(1 + (-1)^k)/2.

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A290191 Binary representation of the diagonal from the corner to the origin of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 673", based on the 5-celled von Neumann neighborhood.

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A208324 Triangle T(n,k), read by rows, given by (2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (4, -2, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

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