A130665 -id:A130665 - cp's OEIS frontend

A147562 Number of "ON" cells at n-th stage in the "Ulam-Warburton" two-dimensional cellular automaton.

0, 1, 5, 9, 21, 25, 37, 49, 85, 89, 101, 113, 149, 161, 197, 233, 341, 345, 357, 369, 405, 417, 453, 489, 597, 609, 645, 681, 789, 825, 933, 1041, 1365, 1369, 1381, 1393, 1429, 1441, 1477, 1513, 1621, 1633, 1669, 1705, 1813, 1849, 1957, 2065, 2389, 2401, 2437, 2473

Offset: 0

N. J. A. Sloane, based on emails from Franklin T. Adams-Watters, R. J. Mathar and David W. Wilson, Apr 29 2009

Studied by Holladay and Ulam circa 1960. See Fig. 1 and Example 1 of the Ulam reference. - N. J. A. Sloane, Aug 02 2009.
Singmaster calls this the Ulam-Warburton cellular automaton. - N. J. A. Sloane, Aug 05 2009
On the infinite square grid, start with all cells OFF.
Turn a single cell to the ON state.
At each subsequent step, each cell with exactly one neighbor ON is turned ON, and everything that is already ON remains ON.
Here "neighbor" refers to the four adjacent cells in the X and Y directions.
Note that "neighbor" could equally well refer to the four adjacent cells in the diagonal directions, since the graph formed by Z^2 with "one-step rook" adjacencies is isomorphic to Z^2 with "one-step bishop" adjacencies.
Also toothpick sequence starting with a central X-toothpick followed by T-toothpicks (see A160170 and A160172). The sequence gives the number of polytoothpicks in the structure after n-th stage. - Omar E. Pol, Mar 28 2011
It appears that this sequence shares infinitely many terms with both A162795 and A169707, see Formula section and Example section. - Omar E. Pol, Feb 20 2015
It appears that the positive terms are also the odd terms (a bisection) of A151920. - Omar E. Pol, Mar 06 2015
Also, the number of active (ON, black) cells in the n-th stage of growth of two-dimensional cellular automaton defined by Wolfram's "Rule 558" or "Rule 686" based on the 5-celled von Neumann neighborhood. - Robert Price, May 10 2016
From Omar E. Pol, Mar 05 2019: (Start)
a(n) is also the total number of "hidden crosses" after 4*n stages in the toothpick structure of A139250, including the central cross, beginning to count the crosses when their nuclei are totally formed with 4 quadrilaterals.
a(n) is also the total number of "flowers with six petals" after 4*n stages in the toothpick structure of A323650.
Note that the location of the "nuclei of the hidden crosses" and the "flowers with six petals" in both toothpick structures is essentially the same as the location of the "ON" cells in the version "one-step bishop" of this sequence (see the illustration of initial terms, figure 2). (End)
This sequence has almost exactly the same graph as A187220, A162795, A169707 and A160164 which is twice A139250. - Omar E. Pol, Jun 18 2022

			If we label the generations of cells turned ON by consecutive numbers we get a rosetta cell pattern:
. . . . . . . . . . . . . . . . .
. . . . . . . . 4 . . . . . . . .
. . . . . . . 4 3 4 . . . . . . .
. . . . . . 4 . 2 . 4 . . . . . .
. . . . . 4 3 2 1 2 3 4 . . . . .
. . . . . . 4 . 2 . 4 . . . . . .
. . . . . . . 4 3 4 . . . . . . .
. . . . . . . . 4 . . . . . . . .
. . . . . . . . . . . . . . . . .
In the first generation, only the central "1" is ON, a(1)=1. In the next generation, we turn ON four "2", leading to a(2)=a(1)+4=5. In the third generation, four "3" are turned ON, a(3)=a(2)+4=9. In the fourth generation, each of the four wings allows three 4's to be turned ON, a(4)=a(3)+4*3=21.
From _Omar E. Pol_, Feb 18 2015: (Start)
Also, written as an irregular triangle T(j,k), j>=0, k>=1, in which the row lengths are the terms of A011782:
1;
5;
9,   21;
25,  37, 49, 85;
89, 101,113,149,161,197,233,341;
345,357,369,405,417,453,489,597,609,645,681,789,825,933,1041,1365;
...
The right border gives the positive terms of A002450.
(End)
It appears that T(j,k) = A162795(j,k) = A169707(j,k), if k is a power of 2, for example: it appears that the three mentioned triangles only share the elements from the columns 1, 2, 4, 8, 16, ... - _Omar E. Pol_, Feb 20 2015

S. Ulam, On some mathematical problems connected with patterns of growth of figures, pp. 215-224 of R. E. Bellman, ed., Mathematical Problems in the Biological Sciences, Proc. Sympos. Applied Math., Vol. 14, Amer. Math. Soc., 1962.
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 928.

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
David Applegate, The movie version
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
Steven R. Finch, Toothpicks and Live Cells, July 21, 2015. [Cached copy, with permission of the author]
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 31.
Bradley Klee, Log-periodic coloring of the first quadrant, over the chair tiling.
Omar E. Pol, Illustration of initial terms (Fig. 1: one-step rook - the current sequence), (Fig. 2: one-step bishop), (Fig. 3: overlapping squares), (Fig. 4: overlapping X-toothpicks), (2009), (Fig. 5: overlapping circles), (2010)
Omar E. Pol, Illustration of initial terms of A139250, A160120, A147562 (overlapping figures), (2009).
David Singmaster, On the cellular automaton of Ulam and Warburton, M500 Magazine of the Open University, #195 (December 2003), pp. 2-7. Also scanned annotated cached copy, included with permission.
N. J. A. Sloane, Illustration of terms 0 through 9
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
N. J. A. Sloane, Exciting Number Sequences (video of talk), Mar 05 2021
N. J. A. Sloane and Brady Haran, Terrific Toothpick Patterns, Numberphile video (2018).
Mike Warburton, Ulam-Warburton Automaton - Counting Cells with Quadratics, arXiv:1901.10565 [math.CO], 2019.
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 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata

Cf. A000120, A139250, A147582 (number turned ON at n-th step), A147610, A130665, A151920, A151917, A160120, A160164, A160410, A160414, A162795, A169707, A187220, A246331, A323650.

A006046 Total number of odd entries in first n rows of Pascal's triangle: a(0) = 0, a(1) = 1, a(2k) = 3a(k), a(2k+1) = 2a(k) + a(k+1). a(n) = Sum_{i=0..n-1} 2^wt(i).

Original entry on oeis.org

0, 1, 3, 5, 9, 11, 15, 19, 27, 29, 33, 37, 45, 49, 57, 65, 81, 83, 87, 91, 99, 103, 111, 119, 135, 139, 147, 155, 171, 179, 195, 211, 243, 245, 249, 253, 261, 265, 273, 281, 297, 301, 309, 317, 333, 341, 357, 373, 405, 409, 417, 425, 441, 449, 465, 481, 513, 521

Offset: 0

Jeffrey Shallit

The graph has a blancmange or Takagi appearance. For the asymptotics, see the references by Flajolet with "Mellin" in the title. - N. J. A. Sloane, Mar 11 2021
The following alternative construction of this sequence is due to Thomas Nordhaus, Oct 31 2000: For each n >= 0 let f_n be the piecewise linear function given by the points (k /(2^n), a(k) / 3^n), k = 0, 1, ..., 2^n. f_n is a monotonic map from the interval [0,1] into itself, f_n(0) = 0, f_n(1) = 1. This sequence of functions converges uniformly. But the limiting function is not differentiable on a dense subset of this interval.
I submitted a problem to the Amer. Math. Monthly about an infinite family of non-convex sequences that solve a recurrence that involves minimization: a(1) = 1; a(n) = max { ua(k) + a(n-k) | 1 <= k <= n/2 }, for n > 1; here u is any real-valued constant >= 1. The case u=2 gives the present sequence. Cf. A130665 - A130667. - Don Knuth, Jun 18 2007
a(n) = sum of (n-1)-th row terms of triangle A166556. - Gary W. Adamson, Oct 17 2009
From Gary W. Adamson, Dec 06 2009: (Start)
Let M = an infinite lower triangular matrix with (1, 3, 2, 0, 0, 0, ...) in every column shifted down twice:
  1;
  3;
  2; 1;
  0, 3;
  0, 2, 1;
  0, 0, 3;
  0, 0, 2, 1;
  0, 0, 0, 3;
  0, 0, 0, 2, 1;
  ...
This sequence starting with "1" = lim_{n->infinity} M^n, the left-shifted vector considered as a sequence. (End)
a(n) is also the sum of all entries in rows 0 to n of Sierpiński's triangle A047999. - Reinhard Zumkeller, Apr 09 2012
The production matrix of Dec 06 2009 is equivalent to the following: Let p(x) = (1 + 3x + 2x^2). The sequence = P(x) * p(x^2) * p(x^4) * p(x^8) * .... The sequence divided by its aerated variant = (1, 3, 2, 0, 0, 0, ...). - Gary W. Adamson, Aug 26 2016
Also the total number of ON cells, rows 1 through n, for cellular automaton Rule 90 (Cf. A001316, A038183, also Mathworld Link). - Bradley Klee, Dec 22 2018

S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 2.16.
Flajolet, Philippe, and Mordecai Golin. "Mellin transforms and asymptotics." Acta Informatica 31.7 (1994): 673-696.
Flajolet, Philippe, Mireille Régnier, and Robert Sedgewick. "Some uses of the Mellin integral transform in the analysis of algorithms." in Combinatorial algorithms on words. Springer, 1985. 241-254.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 0..16383 (first 1000 terms from T. D. Noe)
L. Carlitz, The number of binomial coefficients divisible by a fixed power of a prime, Rend. Circ. Mat. Palermo (2) 16 (1967), pp. 299-320.
K.-N. Chang and S.-C. Tsai, Exact solution of a minimal recurrence, Inform. Process. Lett. 75 (2000), 61-64.
Prerona Chatterjee, Kshitij Gajjar, and Anamay Tengse, Transparency Beyond VNP in the Monotone Setting, arXiv:2202.13103 [cs.CC], 2022.
S. R. Finch, P. Sebah, and Z.-Q. Bai, Odd Entries in Pascal's Trinomial Triangle, arXiv:0802.2654 [math.NT], 2008.
N. J. Fine, Binomial coefficients modulo a prime, Amer. Math. Monthly 54:10 (1947), pp. 89-92.
Philippe Flajolet, Peter Grabner, Peter Kirschenhofer, Helmut Prodinger, and Robert F. Tichy, Mellin Transforms And Asymptotics: Digital Sums, Theoret. Computer Sci. 23 (1994), 291-314.
Philippe Flajolet, Xavier Gourdon, and Philippe Dumas, Mellin transforms and asymptotics: harmonic sums Special volume on mathematical analysis of algorithms. Theoret. Comput. Sci. 144 (1995), no. 1-2, 3-58.
Philippe Flajolet and Robert Sedgewick, Mellin transforms and asymptotics: Finite differences and Rice's integrals, Theoretical Computer Science 144.1-2 (1995): 101-124.
P. J. Grabner and H.-K. Hwang, Digital sums and divide-and-conquer recurrences: Fourier expansions and absolute convergence, Constructive Approximation, Jan. 2005, Volume 21, Issue 2, pp 149-179.
H. Harborth, Number of Odd Binomial Coefficients, Proc. Amer. Math. Soc. 62, 19-22, 1977.
H. Harborth, Number of Odd Binomial Coefficients, Proc. Amer. Math. Soc. 62.1 (1977), 19-22. (Annotated scanned copy)
F. T. Howard, The number of binomial coefficients divisible by a fixed power of 2, Proceedings of the American Mathematical Society, Vol. 29:2 (Jul 1971), pp. 236-242.
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, pp. 6, 27, 29-31.
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Periodic minimum in the count of binomial coefficients not divisible by a prime, arXiv:2408.06817 [math.NT], 2024. See p. 1.
Akhlesh Lakhtakia and Russell Messier, Self-similar sequences and chaos from Gauss sums, Computers & graphics 13.1 (1989): 59-62.
Akhlesh Lakhtakia and Russell Messier, Self-similar sequences and chaos from Gauss sums, Computers & Graphics 13.1 (1989), 59-60. (Annotated scanned copy)
A. Lakhtakia et al., Fractal sequences derived from the self-similar extensions of the Sierpinski gasket, J. Phys. A 21 (1988), 1925-1928.
Giuseppe Lancia and Paolo Serafini, Computational Complexity and ILP Models for Pattern Problems in the Logical Analysis of Data, Algorithms (2021) Vol. 14, No. 8, 235.
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Ralf Stephan, Divide-and-conquer generating functions. I. Elementary sequences, arXiv:math/0307027 [math.CO], 2003.
K. B. Stolarsky, Power and exponential sums of digital sums related to binomial coefficient parity, SIAM J. Appl. Math., 32 (1977), 717-730. See B(n). - _N. J. A. Sloane_, Apr 05 2014
Eric Weisstein's World of Mathematics, Pascal's Triangle
Eric Weisstein's World of Mathematics, Rule 90
Eric Weisstein's World of Mathematics, Stolarsky-Harborth Constant
Index entries for triangles and arrays related to Pascal's triangle

Partial sums of A001316.
See A130665 for Sum 3^wt(n).
a(n) = A074330(n-1) + 1 for n >= 2. A080978(n) = 2*a(n) + 1. Cf. A080263.
Cf. also A159912, A166556, A038183, A048896, A360189.
Sequences of form a(n) = r*a(ceiling(n/2)) + s*a(floor(n/2)), a(1)=1, for (r,s) = (1,1), (1,2), (2,1), (1,3), (2,2), (3,1), (1,4), (2,3), (3,2), (4,1): A000027, A006046, A064194, A130665, A073121, A268524, A116520, A268525, A268526, A268527.

a006046 = sum . concat . (`take` a047999_tabl)
-- Reinhard Zumkeller, Apr 09 2012

[0] cat [n le 1 select 1 else 2*Self(Floor(n/2)) + Self(Floor(Ceiling(n/2))): n in [1..60]]; // Vincenzo Librandi, Aug 30 2016

f:=proc(n) option remember;
if n <= 1 then n elif n mod 2 = 0 then 3*f(n/2)
else 2*f((n-1)/2)+f((n+1)/2); fi; end;
[seq(f(n),n=0..130)]; # N. J. A. Sloane, Jul 29 2014

f[n_] := Sum[ Mod[ Binomial[n, k], 2], {k, 0, n} ]; Table[ Sum[ f[k], {k, 0, n} ], {n, 0, 100} ]
Join[{0},Accumulate[Count[#,?OddQ]&/@Table[Binomial[n,k],{n,0,60},{k,0,n}]]] (* _Harvey P. Dale, Dec 10 2014 *)
FoldList[Plus, 0, Total /@ CellularAutomaton[90, Join[Table[0, {#}], {1}, Table[0, {#}]], #]][[2 ;; -1]] &@50 (* Bradley Klee, Dec 23 2018 *)
Join[{0}, Accumulate[2^DigitCount[Range[0, 127], 2, 1]]] (* Paolo Xausa, Oct 24 2024 *)
Join[{0}, Accumulate[2^Nest[Join[#, #+1]&, {0}, 7]]] (* Paolo Xausa, Oct 24 2024, after IWABUCHI Yu(u)ki in A000120 *)

A006046(n)={ n<2 & return(n); A006046(n\2)*3+if(n%2,1<M. F. Hasler, May 03 2009

a(n) = if(!n, 0, my(r=0, t=1); forstep(i=logint(n, 2), 0, -1, r*=3; if(bittest(n, i), r+=t; t*=2)); r); \\ Ruud H.G. van Tol, Jul 06 2024

from functools import lru_cache
@lru_cache(maxsize=None)
def A006046(n):return n if n<=1 else 2*A006046((n-1)//2)+A006046((n+1)//2)if n%2 else 3*A006046(n//2) # Guillermo Hernández, Dec 31 2023

from math import prod
def A006046(n):
    d = list(map(lambda x:int(x)+1,bin(n)[:1:-1]))
    return sum((b-1)*prod(d[a:])*3**a for a, b in enumerate(d))>>1 # Chai Wah Wu, Aug 13 2025

a(n) = Sum_{k=0..n-1} 2^A000120(k). - Paul Barry, Jan 05 2005; simplified by N. J. A. Sloane, Apr 05 2014
For asymptotics see Stolarsky (1977). - N. J. A. Sloane, Apr 05 2014
a(n) = a(n-1) + A001316(n-1). a(2^n) = 3^n. - Henry Bottomley, Apr 05 2001
a(n) = n^(log_2(3))*G(log_2(n)) where G(x) is a function of period 1 defined by its Fourier series. - Benoit Cloitre, Aug 16 2002; formula modified by S. R. Finch, Dec 31 2007
G.f.: (x/(1-x))*Product_{k>=0} (1 + 2*x^2^k). - Ralf Stephan, Jun 01 2003; corrected by Herbert S. Wilf, Jun 16 2005
a(1) = 1, a(n) = 2*a(floor(n/2)) + a(ceiling(n/2)).
a(n) = 3*a(floor(n/2)) + (n mod 2)*2^A000120(n-1). - M. F. Hasler, May 03 2009
a(n) = Sum_{k=0..floor(log_2(n))} 2^k * A360189(n-1,k). - Alois P. Heinz, Mar 06 2023

More terms from James Sellers, Aug 21 2000

Definition expanded by N. J. A. Sloane, Feb 16 2016

A048883 a(n) = 3^wt(n), where wt(n) = A000120(n).

Original entry on oeis.org

1, 3, 3, 9, 3, 9, 9, 27, 3, 9, 9, 27, 9, 27, 27, 81, 3, 9, 9, 27, 9, 27, 27, 81, 9, 27, 27, 81, 27, 81, 81, 243, 3, 9, 9, 27, 9, 27, 27, 81, 9, 27, 27, 81, 27, 81, 81, 243, 9, 27, 27, 81, 27, 81, 81, 243, 27, 81, 81, 243, 81, 243, 243, 729, 3, 9, 9, 27, 9, 27, 27, 81, 9, 27, 27, 81, 27, 81

Offset: 0

John W. Layman

Or, a(n)=number of 1's ("live" cells) at stage n of a 2-dimensional cellular automata evolving by the rule: 1 if NE+NW+S=1, else 0.
This is the odd-rule cellular automaton defined by OddRule 013 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link). - N. J. A. Sloane, Feb 25 2015
Or, start with S=[1]; replace S by [S, 3*S]; repeat ad infinitum.
Fixed point of the morphism 1 -> 13, 3 -> 39, 9 -> 9(27), ... = 3^k -> 3^k 3^(k+1), ... starting from a(0) = 1; 1 -> 13 -> 1339 -> = 1339399(27) -> 1339399(27)399(27)9(27)(27)(81) -> ..., . - Robert G. Wilson v, Jan 24 2006
Equals row sums of triangle A166453 (the square of Sierpiński's gasket, A047999). - Gary W. Adamson, Oct 13 2009
First bisection of A169697=1,5,3,19,3,. a(2n+2)+a(2n+3)=12,12,36,=12*A147610 ? Distribution of terms (in A000244): A011782=1,A000079 for first array, A000079 for second. - Paul Curtz, Apr 20 2010
a(A000225(n)) = A000244(n) and a(m) != A000244(n) for m < A000225(n). - Reinhard Zumkeller, Nov 14 2011
This sequence pertains to phenotype Punnett square mathematics. Start with X=1. Each hybrid cross involves the equation X:3X. Therefore, the ratio in the first (mono) hybrid cross is X=1:3X=3(1) or 3; or 3:1. When you move up to the next hybridization level, replace the previous cross ratio with X. X now represents 2 numbers-1:3. Therefore, the ratio in the second (di) hybrid cross is X=(1:3):3X=[3(1):3(3)] or (3:9). Put it together and you get 1:3:3:9. Each time you move up a hybridization level, replace the previous ratio with X, and use the same equation-X:3X to get its ratio. - John Michael Feuk, Dec 10 2011
Number of odd values in the n-th layer of Pascal's tetrahedron (see A268240). - Caden Le, Mar 03 2025
a(x*y) <= a(x)^A000120(y). - Joe Amos, Mar 28 2025

			From _Omar E. Pol_, Jun 07 2009: (Start)
Triangle begins:
  1;
  3;
  3,9;
  3,9,9,27;
  3,9,9,27,9,27,27,81;
  3,9,9,27,9,27,27,81,9,27,27,81,27,81,81,243;
  3,9,9,27,9,27,27,81,9,27,27,81,27,81,81,243,9,27,27,81,27,81,81,243,27,...
Or
  1;
  3,3;
  9,3,9,9;
  27,3,9,9,27,9,27,27;
  81,3,9,9,27,9,27,27,81,9,27,27,81,27,81,81;
  243,3,9,9,27,9,27,27,81,9,27,27,81,27,81,81,243,9,27,27,81,27,81,81,243,27...
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
David Applegate, The movie version
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata, arXiv:1503.01796 [math.CO], 2015; see also the Accompanying Maple Package.
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, Odd-Rule Cellular Automata on the Square Grid, arXiv:1503.04249 [math.CO], 2015.
Po-Yi Huang and Wen-Fong Ke, Sequences Derived from The Symmetric Powers of {1, 2, ..., k}, arXiv:2307.07733 [math.CO], 2023.
Tanya Khovanova, There are no coincidences, arXiv preprint 1410.2193 [math.CO], 2014.
Tanya Khovanova and Joshua Xiong, Nim Fractals, arXiv:1405.594291 [math.CO] (2014), p. 10. and J. Int. Seq. 17 (2014) # 14.7.8.
T. Pisanski and T. W. Tucker, Growth in Repeated Truncations of Maps, Atti. Sem. Mat. Fis. Univ. Modena, Vol. 49 (2001), 167-176. (preprint)
Omar E. Pol, Illustration of initial terms: Fig. 1. Neighbors of the vertices, Fig. 2. Overlapping squares, Fig. 3. One-step bishop, (Nov 06 2009).
N. J. A. Sloane, Illustration of a(15) = 81 corresponding to number of ON cells in Odd-rule 013 CA at generation 15
N. J. A. Sloane, On the No. of ON Cells in Cellular Automata, Video of talk in Doron Zeilberger's Experimental Math Seminar at Rutgers University, Feb. 05 2015: Part 1, Part 2
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Ralf Stephan, Divide-and-conquer generating functions. I. Elementary sequences, arXiv:math/0307027 [math.CO], 2003.
Index entries for sequences related to cellular automata
Index entries for sequences that are fixed points of mappings

For generating functions Product_{k>=0} (1+a*x^(b^k)) for the following values of (a,b) see: (1,2) A000012 and A000027, (1,3) A039966 and A005836, (1,4) A151666 and A000695, (1,5) A151667 and A033042, (2,2) A001316, (2,3) A151668, (2,4) A151669, (2,5) A151670, (3,2) A048883, (3,3) A117940, (3,4) A151665, (3,5) A151671, (4,2) A102376, (4,3) A151672, (4,4) A151673, (4,5) A151674.
A generalization of A001316. Cf. A102376.
Partial sums give A130665. - David Applegate, Jun 11 2009
Cf. A000079, A122018, A166453, A268240.

a048883 = a000244 . a000120  -- Reinhard Zumkeller, Nov 14 2011

Nest[ Join[#, 3#] &, {1}, 6] (* Robert G. Wilson v, Jan 24 2006 and modified Jul 27 2014*)
a[n_] := 3^DigitCount[n, 2, 1]; Array[a, 80, 0] (* Jean-François Alcover, Nov 15 2017 *)

PARI
```
a(n)=n=binary(n);3^sum(i=1,#n,n[i])
```

a(n) = Product_{k=0..log_2(n)} 3^b(n,k), where b(n,k) = coefficient of 2^k in binary expansion of n (offset 0). - Paul D. Hanna
a(n) = 3*a(n/2) if n is even, otherwise a(n) = a((n+1)/2).
G.f.: Product_{k>=0} (1+3*x^(2^k)). The generalization k^A000120 has generating function (1 + kx)*(1 + kx^2)*(1 + kx^4)*...
a(n+1) = Sum_{i=0..n} (binomial(n, i) mod 2) * Sum_{j=0..i} (binomial(i, j) mod 2). - Benoit Cloitre, Nov 16 2003
a(0)=1, a(n) = 3*a(n-A053644(n)) for n > 0. - Joe Slater, Jan 31 2016
G.f. A(x) satisfies: A(x) = (1 + 3*x) * A(x^2). - Ilya Gutkovskiy, Jul 09 2019

Corrected by Ralf Stephan, Jun 19 2003
Entry revised by N. J. A. Sloane, May 30 2009
Offset changed to 0, Jun 11 2009

A116520 a(0) = 0, a(1) = 1; a(n) = max { 4*a(k) + a(n-k) | 1 <= k <= n/2 }, for n > 1.

Original entry on oeis.org

0, 1, 5, 9, 25, 29, 45, 61, 125, 129, 145, 161, 225, 241, 305, 369, 625, 629, 645, 661, 725, 741, 805, 869, 1125, 1141, 1205, 1269, 1525, 1589, 1845, 2101, 3125, 3129, 3145, 3161, 3225, 3241, 3305, 3369, 3625, 3641, 3705, 3769, 4025, 4089, 4345, 4601, 5625

Offset: 0

Roger L. Bagula, Mar 15 2006

Equivalently, a(n) = r*a(ceiling(n/2)) + s*a(floor(n/2)), a(0)=0, a(1)=1, for (r,s) = (1,4). - N. J. A. Sloane, Feb 16 2016
A 5-divide version of A084230.
Zero together with the partial sums of A102376. - Omar E. Pol, May 05 2010
Also, total number of cubic ON cells after n generations in a three-dimensional cellular automaton in which A102376(n-1) gives the number of cubic ON cells in the n-th level of the structure starting from the top. An ON cell remains ON forever. The structure looks like an irregular stepped pyramid, with n >= 1. - Omar E. Pol, Feb 13 2015
From Gary W. Adamson, Aug 27 2016: (Start)
The formula of Mar 26 2010 is equivalent to lim_{k->infinity} M^k of the following production matrix M:
  1, 0, 0, 0, 0, 0, ...
  5, 0, 0, 0, 0, 0, ...
  4, 1, 0, 0, 0, 0, ...
  0, 5, 0, 0, 0, 0, ...
  0, 4, 1, 0, 0, 0, ...
  0, 0, 5, 0, 0, 0, ...
  0, 0, 4, 1, 0, 0, ...
  0, 0, 0, 5, 0, 0, ...
  ...
The sequence with offset 1 divided by its aerated variant is (1, 5, 4, 0, 0, 0, ...). (End)

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
K.-N. Chang and S.-C. Tsai, Exact solution of a minimal recurrence, Inform. Process. Lett. 75 (2000), 61-64.
H. Harborth, Number of Odd Binomial Coefficients, Proc. Amer. Math. Soc. 62, 19-22, 1977.
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, pp. 27, 32.
D. E. Knuth, Problem 11320, The American Mathematical Monthly, Vol. 114, No. 9 (Nov., 2007), p. 835.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Eric Weisstein's World of Mathematics, Stolarsky-Harborth Constant
Index entries for sequences related to cellular automata

Cf. A000120, A006046, A077465, A130665, A130667, A102376, A360189.

Sequences of the form a(n) = r*a(ceiling(n/2)) + s*a(floor(n/2)), a(1)=1, for (r,s) = (1,1), (1,2), (2,1), (1,3), (2,2), (3,1), (1,4), (2,3), (3,2), (4,1): A000027, A006046, A064194, A130665, A073121, A268524, A116520, A268525, A268526, A268527.

import Data.List (transpose)
a116520 n = a116520_list !! n
a116520_list = 0 : zs where
   zs = 1 : (concat $ transpose
                      [zipWith (+) vs zs, zipWith (+) vs $ tail zs])
      where vs = map (* 4) zs
-- Reinhard Zumkeller, Apr 18 2012

a:=proc(n) if n=0 then 0 elif n=1 then 1 elif n mod 2 = 0 then 5*a(n/2) else 4*a((n-1)/2)+a((n+1)/2) fi end: seq(a(n),n=0..52);

b[0] := 0 b[1] := 1 b[n_?EvenQ] := b[n] = 5*b[n/2] b[n_?OddQ] := b[n] = 4*b[(n - 1)/2] + b[(n + 1)/2] a = Table[b[n], {n, 1, 25}]

a(0) = 1, a(1) = 1; thereafter a(2n) = 5a(n) and a(2n+1) = 4a(n) + a(n+1).
Let r(x) = (1 + 5x + 4x^2). Then (1 + 5x + 9x^2 + 25x^3 + ...) = r(x) * r(x^2) * r(x^4) * r(x^8) * ... . - Gary W. Adamson, Mar 26 2010
a(n) = Sum_{k=0..n-1} 4^wt(k), where wt = A000120. - Mike Warburton, Mar 14 2019
a(n) = Sum_{k=0..floor(log_2(n))} 4^k*A360189(n-1,k). - Alois P. Heinz, Mar 06 2023

Edited by N. J. A. Sloane, Apr 16 2006, Jul 02 2008

A160410 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (see Comments for precise definition).

Original entry on oeis.org

0, 4, 16, 28, 64, 76, 112, 148, 256, 268, 304, 340, 448, 484, 592, 700, 1024, 1036, 1072, 1108, 1216, 1252, 1360, 1468, 1792, 1828, 1936, 2044, 2368, 2476, 2800, 3124, 4096, 4108, 4144, 4180, 4288, 4324, 4432, 4540, 4864, 4900, 5008, 5116, 5440, 5548, 5872, 6196

Offset: 0

Omar E. Pol, May 20 2009

On the infinite square grid, we consider cells to be the squares, and we start at round 0 with all cells in the OFF state, so a(0) = 0.
At round 1, we turn ON four cells, forming a square.
The rule for n > 1: A cell in turned ON iff exactly one of its four vertices is a corner vertex of the set of ON cells. So in each generation every exposed vertex turns on three new cells.
Therefore:
At Round 2, we turn ON twelve cells around the square.
At round 3, we turn ON twelve other cells. Three cells around of every corner of the square.
And so on.
For the first differences see the entry A161411.
Shows a fractal behavior similar to the toothpick sequence A139250.
A very similar sequence is A160414, which uses the same rule but with a(1) = 1, not 4.
When n=2^k then the polygon formed by ON cells is a square with side length 2^(k+1).
a(n) is also the area of the figure of A147562 after n generations if A147562 is drawn as overlapping squares. - Omar E. Pol, Nov 08 2009
From Omar E. Pol, Mar 28 2011: (Start)
Also, toothpick sequence starting with four toothpicks centered at (0,0) as a cross.
Rule: Each exposed endpoint of the toothpicks of the old generation must be touched by the endpoints of three toothpicks of new generation. (Note that these three toothpicks looks like a T-toothpick, see A160172.)
The sequence gives the number of toothpicks after n stages. A161411 gives the number of toothpicks added at the n-th stage.
(End)

			From _Omar E. Pol_, Sep 24 2015: (Start)
With the positive terms written as an irregular triangle in which the row lengths are the terms of A011782 the sequence begins:
    4;
   16;
   28,  64;
   76, 112, 148, 256;
  268, 304, 340, 448, 484, 592, 700, 1024;
  ...
Right border gives the elements of A000302 greater than 1.
This triangle T(n,k) shares with the triangle A256534 the terms of the column k, if k is a power of 2, for example, both triangles share the following terms: 4, 16, 28, 64, 76, 112, 256, 268, 304, 448, 1024, etc.
.
Illustration of initial terms, for n = 1..10:
.       _ _ _ _                         _ _ _ _
.      |  _ _  |                       |  _ _  |
.      | |  _|_|_ _ _ _ _ _ _ _ _ _ _ _|_|_  | |
.      | |_|  _ _     _ _     _ _     _ _  |_| |
.      |_ _| |  _|_ _|_  |   |  _|_ _|_  | |_ _|
.          | |_|  _ _  |_|   |_|  _ _  |_| |
.          |   | |  _|_|_ _ _ _|_|_  | |   |
.          |  _| |_|  _ _     _ _  |_| |_  |
.          | | |_ _| |  _|_ _|_  | |_ _| | |
.          | |_ _| | |_|  _ _  |_| | |_ _| |
.          |       |   | |   | |   |       |
.          |  _ _  |  _| |_ _| |_  |  _ _  |
.          | |  _|_| | |_ _ _ _| | |_|_  | |
.          | |_|  _| |_ _|   |_ _| |_  |_| |
.          |   | | |_ _ _ _ _ _ _ _| | |   |
.          |  _| |_ _| |_     _| |_ _| |_  |
.       _ _| | |_ _ _ _| |   | |_ _ _ _| | |_ _
.      |  _| |_ _|   |_ _|   |_ _|   |_ _| |_  |
.      | | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |
.      | |_ _| |                       | |_ _| |
.      |_ _ _ _|                       |_ _ _ _|
.
After 10 generations there are 304 ON cells, so a(10) = 304.
(End)

Michael De Vlieger, Table of n, a(n) for n = 0..1000
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.],
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 31.
Omar E. Pol, Illustration of initial terms (2009)
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata

Cf. A000079, A000302, A048883, A139250, A147562, A160118, A160412, A160414, A161411, A160717, A160720, A160727, A256530, A256534.

RasterGraphics[state_?MatrixQ,colors_Integer:2,opts___]:=
Graphics[Raster[Reverse[1-state/(colors -1)]],
AspectRatio ->(AspectRatio/.{opts}/.AspectRatio ->Automatic),
Frame ->True, FrameTicks ->None, GridLines ->None];
rule=1340761804646523638425234105559798690663900360577570370705802859623\
705267234688669629039040624964794287326910250673678735142700520276191850\
5902735959769690
Show[GraphicsArray[Map[RasterGraphics,CellularAutomaton[{rule, {2,
{{4,2,1}, {32,16,8}, {256,128,64}}}, {1,1}}, {{{1,1}, {1,1}}, 0}, 9,-10]]]];
ca=CellularAutomaton[{rule,{2,{{4,2,1},{32,16,8},{256,128,64}}},{1,
1}},{{{1,1},{1,1}},0},99,-100];
Table[Total[ca[[i]],2],{i,1,Length[ca]}]
(* John W. Layman, Sep 01 2009; Sep 02 2009 *)
a[n_] := 4*Sum[3^DigitCount[k, 2, 1], {k, 0, n-1}];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Nov 17 2017, after N. J. A. Sloane *)

A160410(n)=sum(i=0,n-1,3^norml2(binary(i)))<<2 \\ M. F. Hasler, Dec 04 2012

Equals 4*A130665. This provides an explicit formula for a(n). - N. J. A. Sloane, Jul 13 2009

a(2^k) = (2*(2^k))^2 for k>=0.

Edited by David Applegate and N. J. A. Sloane, Jul 13 2009

A151920 a(n) = (Sum_{i=1..n+1} 3^wt(i))/3, where wt() = A000120().

Original entry on oeis.org

1, 2, 5, 6, 9, 12, 21, 22, 25, 28, 37, 40, 49, 58, 85, 86, 89, 92, 101, 104, 113, 122, 149, 152, 161, 170, 197, 206, 233, 260, 341, 342, 345, 348, 357, 360, 369, 378, 405, 408, 417, 426, 453, 462, 489, 516, 597, 600, 609, 618, 645, 654, 681, 708, 789, 798, 825, 852, 933, 960

Offset: 0

N. J. A. Sloane, Aug 05 2009, Aug 06 2009

Partial sums of A147610 (but with offset changed to 0).

It appears that the first bisection gives the positive terms of A147562. - Omar E. Pol, Mar 07 2015

			n=3: (3^1+3^1+3^2+3^1)/3 = 18/3 = 6.
n=18: the binary expansion of 18+1 is 10011, i.e., 19 = 2^4 + 2^1 + 2^0.
The exponents of these powers of 2 (4, 1 and 0) reoccur as exponents in the powers of 4: a(19) = 3^0 * [(4^4 - 1) / 3 + 1] + 3^1 * [(4^1 - 1) / 3 + 1] + 3^2 * [(4^0 - 1)/3 + 1] = 1 * 86 + 3 * 2 + 9 * 1 = 101. - _David A. Corneth_, Mar 21 2015

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 31.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata

Cf. A000120, A130665, A147562, A147610, A153000, A160410, A160414.

Cf. A139250, A151917, A152998.

t = Nest[Join[#, # + 1] &, {0}, 14]; Table[Sum[3^t[[i + 1]], {i, 1, n}]/3, {n, 60}] (* Michael De Vlieger, Mar 21 2015 *)

a(n) = sum(i=1, n+1, 3^hammingweight(i))/3; \\ Michel Marcus, Mar 07 2015

a(n)={b=binary(n+1);t=#b;e=-1;sum(i=1,#b,e+=(b[i]==1);(b[i]==1)*3^e*((4^(#b-i)-1)/3+1))} \\ David A. Corneth, Mar 21 2015

a(n) = (A147562(n+2) - 1)/4 = (A151917(n+2) - 1)/2. - Omar E. Pol, Mar 13 2011
a(n) = (A130665(n+1) - 1)/3. - Omar E. Pol, Mar 07 2015
a(n) = a(n-1) + 3^A000120(n+1)/3. - David A. Corneth, Mar 21 2015

A360189 Triangle T(n,k), n>=0, 0<=k<=floor(log_2(n+1)), read by rows: T(n,k) = number of nonnegative integers <= n having binary weight k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 1, 3, 1, 1, 3, 2, 1, 3, 3, 1, 3, 3, 1, 1, 4, 3, 1, 1, 4, 4, 1, 1, 4, 5, 1, 1, 4, 5, 2, 1, 4, 6, 2, 1, 4, 6, 3, 1, 4, 6, 4, 1, 4, 6, 4, 1, 1, 5, 6, 4, 1, 1, 5, 7, 4, 1, 1, 5, 8, 4, 1, 1, 5, 8, 5, 1, 1, 5, 9, 5, 1, 1, 5, 9, 6, 1, 1, 5, 9, 7, 1

Offset: 0

Alois P. Heinz, Mar 04 2023

T(n,k) is defined for all n >= 0 and k >= 0. Terms that are not in the triangle are zero.

			T(6,2) = 3: 3, 5, 6, or in binary: 11_2, 101_2, 110_2.
T(15,3) = 4: 7, 11, 13, 14, or in binary: 111_2, 1011_2, 1101_2, 1110_2.
Triangle T(n,k) begins:
  1;
  1, 1;
  1, 2;
  1, 2, 1;
  1, 3, 1;
  1, 3, 2;
  1, 3, 3;
  1, 3, 3, 1;
  1, 4, 3, 1;
  1, 4, 4, 1;
  1, 4, 5, 1;
  1, 4, 5, 2;
  1, 4, 6, 2;
  1, 4, 6, 3;
  1, 4, 6, 4;
  1, 4, 6, 4, 1;
  ...

Alois P. Heinz, Rows n = 0..2^12-1, flattened
Peter J. Taylor, Closed form for Sum_{k=0..n} [wt(k) = m] where wt(n) is the binary weight of n, answer to question on MathOverflow (2024).
Wikipedia, Iverson bracket

Columns k=0-2 give: A000012, A029837(n+1) = A113473(n) for n>0, A340068(n+1).
Last elements of rows give A090996(n+1).
Cf. A000027, A000120, A000225, A000788, A006046, A007318, A116520, A116522, A116525, A116526, A130665, A130667, A161342, A231500, A231501, A231502, A272020, A361257.

b:= proc(n) option remember; `if`(n<0, 0,
      b(n-1)+x^add(i, i=Bits[Split](n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n)):
seq(T(n), n=0..23);

T(n,k) = my(v1); v1 = Vecrev(binary(n+1)); v1 = Vecrev(select(x->(x>0),v1,1)); sum(j=0, min(k,#v1-1), binomial(v1[j+1]-1,k-j)) \\ Mikhail Kurkov, Nov 27 2024

T(n,k) = T(n-1,k) + [A000120(n) = k] where [] is the Iverson bracket and T(n,k) = 0 for n<0.
T(2^n-1,k) = A007318(n,k) = binomial(n,k).
T(n,floor(log_2(n+1))) = A090996(n+1).
Sum_{k>=0} T(n,k) = n+1.
Sum_{k>=0} k * T(n,k) = A000788(n).
Sum_{k>=0} k^2 * T(n,k) = A231500(n).
Sum_{k>=0} k^3 * T(n,k) = A231501(n).
Sum_{k>=0} k^4 * T(n,k) = A231502(n).
Sum_{k>=0} 2^k * T(n,k) = A006046(n+1).
Sum_{k>=0} 3^k * T(n,k) = A130665(n).
Sum_{k>=0} 4^k * T(n,k) = A116520(n+1).
Sum_{k>=0} 5^k * T(n,k) = A130667(n+1).
Sum_{k>=0} 6^k * T(n,k) = A116522(n+1).
Sum_{k>=0} 7^k * T(n,k) = A161342(n+1).
Sum_{k>=0} 8^k * T(n,k) = A116526(n+1).
Sum_{k>=0} 10^k * T(n,k) = A116525(n+1).
Sum_{k>=0} n^k * T(n,k) = A361257(n).
T(n,k) = Sum_{j=0..min(k, A000120(n+1)-1)} binomial(A272020(n+1,j+1)-1,k-j) for n >= 0, k >= 0 (see Peter J. Taylor link). - Mikhail Kurkov, Nov 27 2024

A064194 a(2n) = 3a(n), a(2n+1) = 2a(n+1)+a(n), with a(1) = 1.

Original entry on oeis.org

1, 3, 7, 9, 17, 21, 25, 27, 43, 51, 59, 63, 71, 75, 79, 81, 113, 129, 145, 153, 169, 177, 185, 189, 205, 213, 221, 225, 233, 237, 241, 243, 307, 339, 371, 387, 419, 435, 451, 459, 491, 507, 523, 531, 547, 555, 563, 567, 599, 615, 631, 639, 655, 663, 671, 675

Offset: 1

Guillaume Hanrot and Paul Zimmermann, Sep 21 2001

Number of ring multiplications needed to multiply two degree-n polynomials using Karatsuba's algorithm.
Number of gates in the AND/OR problem (see Chang/Tsai reference).
a(n) is also the number of odd elements in the n X n symmetric Pascal matrix. - Stefano Spezia, Nov 14 2022

A. A. Karatsuba and Y. P. Ofman, Multiplication of multiplace numbers by automata. Dokl. Akad. Nauk SSSR 145, 2, 293-294 (1962).

N. J. A. Sloane, Table of n, a(n) for n = 1..10000
K.-N. Chang and S.-C. Tsai, Exact solution of a minimal recurrence, Inform. Process. Lett. 75 (2000), 61-64.
P. J. Grabner and H.-K. Hwang, Digital sums and divide-and-conquer recurrences: Fourier expansions and absolute convergence, Constructive Approximation, Jan. 2005, Volume 21, Issue 2, pp. 149-179.
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, pp. 27-28.

Cf. A023416, A267584, A047999 (Sierpinski triangle).
Cf. also A268514.
Sequences of form a(n)=r*a(ceil(n/2))+s*a(floor(n/2)), a(1)=1, for (r,s) = (1,1), (1,2), (2,1), (1,3), (2,2), (3,1), (1,4), (2,3), (3,2), (4,1): A000027, A006046, A064194, A130665, A073121, A268524, A116520, A268525, A268526, A268527.

[n le 1 select 1 else Self(Floor(n/2)) + 2*Self(Ceiling(n/2)): n in [1..60]]; // Vincenzo Librandi, Aug 30 2016

f:=proc(n) option remember; if n=1 then 1 elif n mod 2 = 0 then 3*f(n/2) else 2*f((n+1)/2)+f((n-1)/2); fi; end; [seq(f(n),n=1..60)]; # N. J. A. Sloane, Jan 17 2016

a[n_] := a[n] = If[EvenQ[n], 3 a[n/2], 2 a[# + 1] + a[#] &[(n - 1)/2]]; a[1] = 1; Array[a, 56] (* Michael De Vlieger, Oct 29 2022 *)

a(n) = sum(i=0, n-1, sum(j=0, n-1, binomial(i+j, i) % 2)); \\ Michel Marcus, Aug 25 2013

Partial sums of the sequence { b(1)=1, b(n)=2^(e0(n-1)+1) } (essentially A267584), where e0(n)=A023416(n) is the number of zeros in the binary expansion of n. [Chang/Tsai] - Ralf Stephan, Jul 29 2003
a(1) = 1, a(n) = a(floor(n/2)) + 2*a(ceiling(n/2)), n > 1.
a(n+1) = Sum_{0<=i, j<=n} (binomial(i+j, i) mod 2). - Benoit Cloitre, Mar 07 2005
In particular, a(2^k)=3^k, a(3*2^k)=7*3^k. - N. J. A. Sloane, Jan 18 2016
a(n) = 2*A268514(n-1) + 1. - N. J. A. Sloane, Feb 07 2016

Edited with clearer definition by N. J. A. Sloane, Jan 18 2016

cp's OEIS Frontend

A173066 a(n) = A130665(n-1) - A160120(n).

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A173067 a(n) = A130665(n-1) - A160715(n).

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A147562 Number of "ON" cells at n-th stage in the "Ulam-Warburton" two-dimensional cellular automaton.

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A006046 Total number of odd entries in first n rows of Pascal's triangle: a(0) = 0, a(1) = 1, a(2k) = 3*a(k), a(2k+1) = 2*a(k) + a(k+1). a(n) = Sum_{i=0..n-1} 2^wt(i).

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A048883 a(n) = 3^wt(n), where wt(n) = A000120(n).

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A116520 a(0) = 0, a(1) = 1; a(n) = max { 4*a(k) + a(n-k) | 1 <= k <= n/2 }, for n > 1.

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A160410 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (see Comments for precise definition).

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A006046 Total number of odd entries in first n rows of Pascal's triangle: a(0) = 0, a(1) = 1, a(2k) = 3a(k), a(2k+1) = 2a(k) + a(k+1). a(n) = Sum_{i=0..n-1} 2^wt(i).

A064194 a(2n) = 3a(n), a(2n+1) = 2a(n+1)+a(n), with a(1) = 1.