A083420 -id:A083420 - cp's OEIS frontend

A156760 5*4^n-1.

4, 19, 79, 319, 1279, 5119, 20479, 81919, 327679, 1310719, 5242879, 20971519, 83886079, 335544319, 1342177279, 5368709119, 21474836479, 85899345919, 343597383679, 1374389534719, 5497558138879, 21990232555519, 87960930222079, 351843720888319

Offset: 0

Paul Curtz, Feb 15 2009

Second column of the array A132207, or, if this array is flattened, a(n)=A132207(A007583(n)).

			Binary.......................................Decimal
100................................................4
10011.............................................19
1001111...........................................79
100111111........................................319
10011111111.....................................1279
1001111111111...................................5119
100111111111111................................20479
10011111111111111..............................81919
1001111111111111111...........................327679
100111111111111111111........................1310719
10011111111111111111111......................5242879
1001111111111111111111111...................20971519
100111111111111111111111111.................83886079
10011111111111111111111111111..............335544319
1001111111111111111111111111111...........1342177279
... - _Philippe Deléham_, Feb 23 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..500

[5*4^n-1: n in [0..30]]; // Vincenzo Librandi, Jul 02 2011

Table[5*4^n - 1, {n, 0, 18}] (* Michael De Vlieger, Apr 20 2015 *)

a(n) mod 9 = A070403(n+2).
a(n+1) = 10*A083420(n)+9 .
a(n) = 5*A000302(n)-1.
a(n) = ( A024036(n+1)+A140529(n) )/2.
a(n) = 4a(n-1)+3, a(0)=4.
a(n) = A003947(n+1)-1 = 5*a(n-1)-4*a(n-2). G.f.: (4-x)/((1-x)(1-4x)). - R. J. Mathar, Feb 23 2009
a(n) = A198693(n) + 2^(2n+1). - Bob Selcoe, Apr 20 2015

Edited and extended by R. J. Mathar, Feb 23 2009

A132680 Number of ones in binary representation of odious numbers.

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 3, 3, 1, 3, 3, 3, 3, 3, 3, 5, 1, 3, 3, 3, 3, 3, 3, 5, 3, 3, 3, 5, 3, 5, 5, 5, 1, 3, 3, 3, 3, 3, 3, 5, 3, 3, 3, 5, 3, 5, 5, 5, 3, 3, 3, 5, 3, 5, 5, 5, 3, 5, 5, 5, 5, 5, 5, 7, 1, 3, 3, 3, 3, 3, 3, 5, 3, 3, 3, 5, 3, 5, 5, 5, 3, 3, 3, 5, 3, 5, 5, 5, 3, 5, 5, 5, 5, 5, 5, 7, 3, 3, 3, 5, 3, 5, 5, 5, 3

Offset: 1

Reinhard Zumkeller, Aug 26 2007

Reinhard Zumkeller, Table of n, a(n) for n = 1..10001
Eric Weisstein's World of Mathematics, Odious Number.
Eric Weisstein's World of Mathematics, Thue-Morse Sequence.
Index entries for sequences related to binary expansion of n.

Cf. A000069, A000120, A010060, A005408, A083420, A129771.

Select[DigitCount[Range[200], 2, 1], OddQ] (* Amiram Eldar, Jul 22 2023 *)

a(n)=hammingweight(2*n-1-hammingweight(n-1)%2) \\ Charles R Greathouse IV, Mar 26 2013

a(n) = a(floor(n/2)) + 2*(n mod 2)*(1-A010060(n)).
a(n) = A000120(A000069(n)).
a(A129771(n)) = a((A129771(n)-1)/2) + 2.
a(A083420(n)) = A005408(n).

A259663 Square array T(n,k) read by antidiagonals upwards: "Dropping Times" in reduced Collatz sequences. (See "Comments" for definitions and explanation.)

Original entry on oeis.org

1, 13, 11, 5, 19, 7, 53, 3, 55, 47, 21, 35, 87, 79, 31, 213, 99, 23, 143, 223, 191, 85, 483, 407, 15, 95, 319, 127, 853, 739, 663, 271, 351, 63, 895, 767, 341, 1251, 1175, 1807, 863, 1599, 1407, 1279, 511

Offset: 2

Bob Selcoe, Jul 02 2015

Array starts T(2,1).
Coefficients are least residues in congruence classes T(n,k) mod 2^(n+k). Let T"(n,k) be all members of that class.
Let reduced Collatz sequences (R) be Collatz sequences (C) showing only odd terms; and let S be the initial term in R and C, denoted as R(S) and C(S), respectively.
Define "Dropping Time" (D(S)) as the first term in R(S) < its preimage (P(S)). For example, S=159: R(159) starts [159, 239, 359, 539, 809, 607, ...]; therefore, D(159) = 607 and P(159) = 809.
When S is in T"(n,k): k is the number of terms in R(S) from S to D(S), and n is the number of halving steps in C(S) from P(S) to D(S). So for S=159: since 159 == 31 mod 128 and T(2,5) = 31, there are 5 steps from S=159 to D(159) = 607 (239, 359, 539, 809, 607) and 2 halving steps from P(159) = 809 to D(159), i.e., 809*3+1 = 2428; 2428/2 = 1214, 1214/2 = 607.
Generally, when any term in R is in T"(n,k) k>=2, trajectories follow T"(n,k-i) {i=1..k-1}. At T"(n,1), the next term's congruence class is not clearly predictable. So for example S=159: 159 == 31 mod 128 (in T"(2,5)), 239 == 47 mod 64 (in T"(2,4)), 359 == 7 mod 32 (in T"(2,3)), 539 == 11 mod 16 (in T"(2,2)) and 809 == 1 mod 8 (in T"(2,1)). Since 607 == 95 mod 512 (in T"(4,5)), we know the subsequent term will be in T"(4,4), followed by terms in T"(4,3), T"(4,2) and T"(4,1).
Let T"(k) be members of all T"(n,k) holding k constant. Then T"(k) == (2^k-1) mod 2^(k+1). However, since S is the only possible term in R(S) that could be congruent to 0 mod 3, it makes sense to consider only terms congruent to {1,2} mod 3 when evaluating T"(k). Therefore, after the initial term in R(S), all subsequent terms in T"(k) are congruent to either:
i. {(2^k - 1), (6*4^((k-1)/2) - 1)} mod 3*2^(k+1) when k is odd; or
ii. {(3*4^(k/2) - 1), (5*4^(k/2) - 1)} mod 3*2^(k+1) when k is even.
The array yields a wide variety of interesting patterns and sub-patterns associated with the residues and quotients of the congruence classes. Perhaps analysis of these patterns could shed light on the nature of Collatz sequences, including the Collatz conjecture (i.e., all Collatz sequences terminate at 1).
From Bob Selcoe, Sep 30 2019: (Start)
From equations i and ii above, terms in T"(k) can be described as follows:
ia. for odd k: {T(2,k), T(m+3,k)} mod 3*2^(k+1) when k == 2^m - 1 mod 2^(m+1), m >= 1; or
iia. for even k: {T(2,k), T(3,k)} mod 3*2^(k+1).
(End)

			Array starts T(2,1):
n\k   1    2     3     4      5      6      7      8      9 ...
2:    1    11    7     47     31     191    127    767    511
3:    13   19    55    79     223    319    895    1279   3583
4:    5    3     87    143    95     63     1407   2303   1535
5:    53   35    23    15     351    1599   2431   4351   13823
6:    21   99    407   271    863    575    383    255    22015
7:    213  483   663   1807   3935   2623   12671  8447   5631
8:    85   739   1175  783    5983   14911  20863  57599  38399
9:    853  1251  2199  6927   10079  6719   4479   90367  235007
10:   341  227   151   11023  18271  55871  37247  24831  366079
For n >= 4: e.g., n=4, so j == 2 (mod 4). Select j=6, i=2 to find T(4,4). T(4,6) = 2^6 - 1 = 63. 2^(6-2)*3^2 - 1 mod 2^(4+6-2) = 143 mod 256 = T(4,4) = 143. Now instead select j=10, i=6 to find T(4,4). T(4,10) = 2^10 - 1 = 1023. 2^(10-6)*3^6 - 1 mod 2^(4+10-6) = 11663 mod 256 = 143. - _Bob Selcoe_, Jul 15 2017

Cf. A083420, A156760, A198693, A206372.

T(n, k) = if (n==2, if (k%2, 2^k-1, 3*2^k-1), if (n==3, if (k%2, 7*2^k-1, 5*2^k-1), mj = 2^(n-3) % 2^(n-2); mk = k % 2^(n-2); (2^k*3^(mj-mk) - 1) % 2^(n+k)));
tabl(nn) = matrix(nn, nn, n, k, T(n+1,k)); \\ Michel Marcus, Jul 10 2018

From Bob Selcoe, Jul 15 2017: (Start)
The array is constructed by the following:
T(2,k) = 2^k-1 when k is odd, T(2,k) = 3*2^k-1 when k is even; i.e., A083420((k-1)/2) and A198693(k/2) interleaved.
T(3,k) = 7*2^k-1 when k is odd; T(3,k) = 5*2^k-1 when k is even; i.e., A206372((k-1)/2) and A156760(k/2) interleaved.
For n >= 4: T(n,j) = 2^j-1, j == 2^(n-3) (mod 2^(n-2)); T(n,j-i) = least residue of 2^(j-i)*3^i - 1 mod 2^(n+j-i), 1 <= i < j. (See Example.)
(End)

A279872 Decimal representation of the x-axis, from the left edge to the origin, (and also from the origin to the right edge) of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 209", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 0, 7, 0, 31, 0, 127, 0, 511, 0, 2047, 0, 8191, 0, 32767, 0, 131071, 0, 524287, 0, 2097151, 0, 8388607, 0, 33554431, 0, 134217727, 0, 536870911, 0, 2147483647, 0, 8589934591, 0, 34359738367, 0, 137438953471, 0, 549755813887, 0, 2199023255551, 0

Offset: 0

Robert Price, Dec 21 2016

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

The nonzero bisection appears to be A083420. - Tom Copeland, Dec 27 2016

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. A279118, A083420.

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 209; 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[1, i]], 2], {i, 1, stages - 1}]

Conjectures from Chai Wah Wu, Aug 02 2021: (Start)
a(n) = 5*a(n-2) - 4*a(n-4) for n > 3.
G.f.: (2*x^2 + 1)/(4*x^4 - 5*x^2 + 1). (End)

A331894 Positive numbers such that both their binary and negabinary representations are palindromic.

Original entry on oeis.org

0, 1, 3, 5, 7, 17, 21, 31, 51, 65, 85, 127, 195, 257, 273, 325, 341, 455, 511, 771, 819, 1025, 1105, 1285, 1365, 1799, 2047, 3075, 4097, 4161, 4369, 4433, 5125, 5189, 5397, 5461, 7175, 7967, 8191, 12291, 12483, 13107, 16385, 16705, 17425, 17745, 20485, 20805

Offset: 1

Amiram Eldar, Jan 30 2020

Numbers of the form 2^(2*m-1) - 1 (A083420) and 2^(2*m) + 1 (A052539) are terms.

			7 is a term since the binary representation of 7, 111, and the negabinary representation of 7, 11011, are both palindromic.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Intersection of A006995 and A331891.

Cf. A039724, A095309.

negabin[n_] := negabin[n] = If[n==0, 0, negabin[Quotient[n-1, -2]]*10 + Mod[n, 2]]; Select[Range[0, 2*10^4], And @@ (PalindromeQ /@ {IntegerDigits[#, 2], negabin[#]}) &]

A335921 Total height of all binary search trees with n internal nodes.

Original entry on oeis.org

0, 1, 4, 14, 50, 178, 644, 2347, 8624, 31908, 118768, 444308, 1669560, 6298280, 23842032, 90531032, 344702646, 1315726218, 5033357852, 19294463682, 74099098212, 285056401796, 1098314920968, 4237879802726, 16373796107092, 63341371265892, 245315823125496

Offset: 0

Alois P. Heinz, Jun 29 2020

nonn

Empty external nodes are counted in determining the height of a search tree.

			a(3) = 14 = 3 + 3 + 2 + 3 + 3:
.
          3         3        2        1         1
         / \       / \      / \      / \       / \
        2   o     1   o    1   3    o   3     o   2
       / \       / \      ( ) ( )      / \       / \
      1   o     o   2     o o o o     2   o     o   3
     / \           / \               / \           / \
    o   o         o   o             o   o         o   o
.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Binary search tree
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Cf. A083420, A316944, A335919, A335920, A335922.

g:= n-> `if`(n=0, 0, ilog2(n)+1):
b:= proc(n, h) option remember; `if`(n=0, 1, `if`(n<2^h,
      add(b(j-1, h-1)*b(n-j, h-1), j=1..n), 0))
    end:
T:= (n, k)-> b(n, k)-`if`(k>0, b(n, k-1), 0):
a:= n-> add(T(n, k)*k, k=g(n)..n):
seq(a(n), n=0..35);

g[n_] := If[n == 0, 0, Floor@Log2[n] + 1];
b[n_, h_] := b[n, h] = If[n == 0, 1, If[n < 2^h,
     Sum[b[j - 1, h - 1]*b[n - j, h - 1], {j, 1, n}], 0]];
T[n_, k_] := b[n, k] - If[k > 0, b[n, k - 1], 0];
a[n_] := Sum[T[n, k]*k, {k, g[n], n}];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Apr 26 2022, after Alois P. Heinz *)

a(n) = Sum_{k=0..n} k * A335919(n,k) = Sum_{k=0..n} k * A335920(n,k).

a(n) is odd <=> n in { A083420 }.

A118175 Binary representation of n-th iteration of the Rule 220 elementary cellular automaton starting with a single black cell.

Original entry on oeis.org

1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0

Offset: 0

Eric W. Weisstein, Apr 13 2006

From Franklin T. Adams-Watters, Jul 05 2009: (Start)
Divided into rows of length 2n, row n consists of n 1's followed by n 0's.
Characteristic function of A061885, 1-based characteristic function of A004201. (End)
From Wolfdieter Lang, Dec 05 2012: (Start)
The row lengths sequence is A005408 (the odd numbers). The sum of row No. n is given by A000027(n+1).
This table is the first difference table of the q-binomial (Gauss polynomial) coefficient table G(2;n,k) = [q^k]( [n+2,2]_q) (see table A008967): a(n,k) = G(2;n,k) - G(2;n-1,k). The o.g.f. for the row polynomials is therefore G2(q,z) = (1-z)/Product((1-q^j*z),j=0..2) = 1/((1-q*z)*(1-q^2*z)). Therefore, a(n,k) determines the number of partitions of k into precisely n parts, each <= 2. It determines also the number of partitions of k into at most 2 parts, each <= n but not <= (n-1), i.e., with part n present. See comments on A008967 regarding partitions.
From the o.g.f. G2(q,z) it should be clear that there are 0's for n > k and only 1's for k = n,...,2*n.
(End)
This sequence is also generated by Rule 252. - Robert Price, Jan 31 2016
a(n) is 1 if the nearest square to n is >= n, otherwise 0. - Branko Curgus, Apr 25 2017

			The table a(n,k) begins:
  n\k 0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 ...
  0:  1
  1:  0  1  1
  2:  0  0  1  1  1
  3:  0  0  0  1  1  1  1
  4:  0  0  0  0  1  1  1  1  1
  5:  0  0  0  0  0  1  1  1  1  1  1
  6:  0  0  0  0  0  0  1  1  1  1  1  1  1
  7:  0  0  0  0  0  0  0  1  1  1  1  1  1  1  1
  8:  0  0  0  0  0  0  0  0  1  1  1  1  1  1  1  1  1
  9:  0  0  0  0  0  0  0  0  0  1  1  1  1  1  1  1  1  1  1
... Reformatted and extended by _Wolfdieter Lang_, Dec 05 2012
Partition examples: a(n,k) = 0 if n>k because the maximal number of parts of a partition of k is k. a(n,n) = 1, n >= 1, because only the partition 1^n has n parts, and 1 <= 2.
  a(2,3) = 1 because the only partition of 3 with 2 parts, each <= 2, is 1,2. Also, the only partition of 3 with at most 2 parts, each <= 2, and a part 2 present is also 1,2.
  a(5,7) =1 because the only 5-part partition of 7 with maximal part 2 is 1^3,2^3. Also, the only partition of 7 with at most 2 parts, each <= 5, which a part 5 present is 2,5.

Robert Price, Table of n, a(n) for n = 0..9999
Eric Weisstein's World of Mathematics, Rule 220
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
Stephen Wolfram, A New Kind of Science
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

Cf. A083420, A219238.

Table[1 - Ceiling[Sqrt[n]] + Round[Sqrt[n]], {n, 1, 257}] (* Branko Curgus, Apr 26 2017 *)
Table[{Array[1&,n],Array[0&,n]},{n,1,5}]//Flatten (* Wolfgang Hintze, Jul 28 2017 *)

from math import isqrt
def A118175(n): return 1+int(n-(m:=isqrt(n+1))*(m+1)>=0)-int(m**2!=n+1) # Chai Wah Wu, Jul 30 2022

a(n) = 1 - A079813(n+1). - Philippe Deléham, Jan 02 2012
a(n) = 1 - ceiling(sqrt(n+1)) + round(sqrt(n+1)). - Branko Curgus, Apr 27 2017 [Corrected by Ridouane Oudra, Dec 01 2019]
G.f.: x/(1 - x)*( Sum_{n >= 1} x^(n^2-n)*(1-x^n)) = 1/(2-2*x)* ( x + x^(3/4)*EllipticTheta(2,0,x) - x*EllipticTheta(3,0,x) ). - Wolfgang Hintze, Jul 28 2017
a(n) = floor(sqrt(n+1)+1/2) - floor(sqrt(n)) = round(sqrt(n+1)) - floor(sqrt(n)). - Ridouane Oudra, Dec 01 2019

A132679 Starting with a(1)=1 and a(2)=2: if m is a term then also 4m and 4m+3.

Original entry on oeis.org

1, 2, 4, 7, 8, 11, 16, 19, 28, 31, 32, 35, 44, 47, 64, 67, 76, 79, 112, 115, 124, 127, 128, 131, 140, 143, 176, 179, 188, 191, 256, 259, 268, 271, 304, 307, 316, 319, 448, 451, 460, 463, 496, 499, 508, 511, 512, 515, 524, 527, 560, 563, 572, 575, 704, 707, 716

Offset: 1

Reinhard Zumkeller, Aug 26 2007

nonn

Subsequence of A000069; A010060(a(n))=1; A000120(a(n)) mod 2 = 1;

A000079, A083420, A002042, A002089 are subsequences.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Odious Number
Index entries for sequences related to binary expansion of n

Cf. A179888. [From Reinhard Zumkeller, Jul 31 2010]

import Data.Set (fromList, insert, deleteFindMin)
a132679 n = a132679_list !! (n-1)
a132679_list = f $ fromList [1,2] where
   f s = m : f (insert (4*m) $ insert (4*m+3) s') where
       (m,s') = deleteFindMin s
-- Reinhard Zumkeller, Oct 29 2011

Union[Nest[Flatten[{#,4#,4#+3}]&,{1,2},4]] (* Harvey P. Dale, Aug 02 2013 *)

A225825 a(2n)=A001896(n). a(2n+1)=(-1)^n*A110501(n+1).

Original entry on oeis.org

1, 1, -1, -1, 7, 3, -31, -17, 127, 155, -2555, -2073, 1414477, 38227, -57337, -929569, 118518239, 28820619, -5749691557, -1109652905, 91546277357, 51943281731, -1792042792463, -2905151042481, 1982765468311237, 191329672483963, -286994504449393, -14655626154768697, 3187598676787461083, 1291885088448017715, -4625594554880206790555

Offset: 0

Paul Curtz, Jul 30 2013

sign

a(n) is the numerators of numbers derived from Bernoulli and Genocchi numbers. The denominators b(n) are the Clausen numbers A141056.
The numbers are
BERGEN(n) = 1, 1/2, -1/6, -1/2, 7/30, 3/2, -31/42, -17/2, 127/30, 155/2,..
Difference table:
1,       1/2,     -1/6,     -1/2,      7/30,        3/2,    -31/42,...
-1/2,   -2/3,     -1/3,    11/15,     19/15,     -47/21,   -163/21,...
-1/6,    1/3,    16/15,     8/15,  -368/105,    -116/21,  2152/105,...
1/2,   11/15,    -8/15, -424/105,  -212/105,   2732/105,  4204/105,...
7/30, -19/15, -368/195,  212/105,  2944/105,   1472/105,...
-3/2, -47/21,   116/21, 2732/105, -1472/105, -70240/231, -35120/231,... .
a(n) is an autosequence. Its inverse binomial transform is the sequence signed. Its main diagonal is the double of the first upper diagonal.
a(n) is divisible by A051716(n+1).
Denominators of the main diagonal: A181131(n). Checked by Jean-François Alcover for the first 25 terms.
The numerators of the main diagonal:
1, -2, 16, -424, 2944, -70240, 70873856, -212648576, 98650550272,...
(thanks to Jean-François Alcover) are divisible by 2^n.

Cf. A083420.

A225825 := proc(n)
    local nhalf ;
    nhalf := floor(n/2) ;
    if type(n,'even') then
        A001896(nhalf) ;
    else
        (-1)^nhalf*A110501(nhalf+1) ;
    end if;
end proc; # R. J. Mathar, Oct 28 2013

a[0] = 1; a[n_] := Numerator[BernoulliB[n, 1/2] - (n+1)*EulerE[n, 0]]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Aug 01 2013 *)

c(n)=(0 followed by -A036968(n+1)) = 0, 1, 0, -1, 0, 3,... .

a(n) = A157779(n) + c(n).

More terms from Jean-François Alcover, Aug 01 2013

Definition corrected by R. J. Mathar, Oct 28 2013

A267888 Decimal representation of the n-th iteration of the "Rule 237" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 2, 31, 127, 511, 2047, 8191, 32767, 131071, 524287, 2097151, 8388607, 33554431, 134217727, 536870911, 2147483647, 8589934591, 34359738367, 137438953471, 549755813887, 2199023255551, 8796093022207, 35184372088831, 140737488355327, 562949953421311

Offset: 0

Robert Price, Jan 21 2016

Seems to differ from A083420 only at a(1). - R. J. Mathar, Jun 24 2025

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Robert Price, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

Cf. A267870.

rule=237; rows=20; ca=CellularAutomaton[rule,{{1},0},rows-1,{All,All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]],{rows-k+1,rows+k-1}],{k,1,rows}]; (* Truncated list of each row *) Table[FromDigits[catri[[k]],2],{k,1,rows}]   (* Decimal Representation of Rows *)

Conjectures from Colin Barker, Jan 22 2016 and Apr 17 2019: (Start)
a(n) = 5*a(n-1)-4*a(n-2) for n>3.
G.f.: (1-3*x+25*x^2-20*x^3) / ((1-x)*(1-4*x)).
(End)
a(n) = 2^(2*n+1) - 1 for n>1. - Colin Barker, Nov 25 2016

cp's OEIS Frontend

A156760 5*4^n-1.

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A132680 Number of ones in binary representation of odious numbers.

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A279872 Decimal representation of the x-axis, from the left edge to the origin, (and also from the origin to the right edge) of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 209", based on the 5-celled von Neumann neighborhood.

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A331894 Positive numbers such that both their binary and negabinary representations are palindromic.

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A335921 Total height of all binary search trees with n internal nodes.

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A118175 Binary representation of n-th iteration of the Rule 220 elementary cellular automaton starting with a single black cell.

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A132679 Starting with a(1)=1 and a(2)=2: if m is a term then also 4*m and 4*m+3.

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A132679 Starting with a(1)=1 and a(2)=2: if m is a term then also 4m and 4m+3.