A269707 -id:A269707 - cp's OEIS frontend

A346909 Continued fraction expansion of the constant whose decimal expansion is A269707.

3, 3, 3, 30, 330, 303000, 33003300000, 3030000030300000000000, 3300330000000000330033000000000000000000000, 30300000303000000000000000000000303000003030000000000000000000000000000000000000000000

Offset: 1

Amiram Eldar, Aug 06 2021

The next term has 171 digits and is too large to include in the Data section.

			3 + 1/(3 + 1/(3 + 1/(30 + 1/(330 + ... )))) = 3.30033000000000033... (A269707).

André Blanchard and Michel Mendès France, Symétrie et transcendance, Bull. Sc. Math., 2nd series, Vol. 106 (1982), pp. 325-335.

Amiram Eldar, Table of n, a(n) for n = 1..13
M. Mendes France and A. J. van der Poorten, Some explicit continued fraction expansions, Mathematika, Vol. 38, No. 1 (1991), pp. 1-9.

Cf. A269707.

a[1] = a[2] = 3; a[n_] := 3 * If[OddQ[n], 10^((4^((n - 3)/2) - 1)/3) * Product[1 + 10^(4^k), {k, 0, (n - 5)/2}], 10^((2*4^(n/2 - 2) + 1)/3) * Product[1 + 10^(2*4^k), {k, 0, n/2 - 3}]]; Array[a, 10]

a(n) = 3 * 10^((4^((n-3)/2)-1)/3) * Product_{k=0..(n-5)/2} (1 + 10^(4^k)), if n > 2 is odd, and 3 * 10^((2*4^(n/2-2)+1)/3) * Product_{k=0..n/2-3} (1 + 10^(2*4^k)), if n > 2 is even.

A151666 Number of partitions of n into distinct powers of 4.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, May 30 2009

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
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.]
Lukasz Merta, Composition inverses of the variations of the Baum-Sweet sequence, arXiv:1803.00292 [math.NT], 2018. See q(n) (with different offset) p. 9.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

For generating functions Prod_{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.

Cf. A039966, A151667, A000695, A269707.

a151666 n = fromEnum (n < 2 || m < 2 && a151666 n' == 1)
   where (n', m) = divMod n 4
-- Reinhard Zumkeller, Dec 03 2011

terms = 105;
kmax = Log[4, terms] // Ceiling;
CoefficientList[Product[1+x^(4^k), {k, 0, kmax}] + O[x]^(kmax terms), x][[1 ;; terms]] (* Jean-François Alcover, Jul 31 2018 *)

G.f.: Prod_{k >= 0 } (1+x^(4^k)). Exponents give A000695.

G.f. A(x) satisfies: A(x) = (1 + x) * A(x^4). - Ilya Gutkovskiy, Aug 12 2019

A269709 Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 14", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 6, 14, 34, 54, 86, 126, 202, 270, 350, 438, 562, 686, 846, 1030, 1322, 1582, 1854, 2134, 2450, 2766, 3118, 3494, 3978, 4438, 4934, 5454, 6082, 6710, 7446, 8254, 9386, 10414, 11454, 12502, 13586, 14670, 15790, 16934, 18186, 19414, 20678, 21966, 23362

Offset: 0

Robert Price, Mar 04 2016

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

Rules 46, 142 and 174 also generate this sequence.

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

Robert Price, Table of n, a(n) for n = 0..300
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
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata

Cf. A269707.

rule=14; stages=300;
ca=CellularAutomaton[{rule,{2,{{0,2,0},{2,1,2},{0,2,0}}},{1,1}},{{{1}},0},stages]; (* Start with single black cell *)
on=Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)
Table[Total[Part[on,Range[1,i]]],{i,1,Length[on]}] (* Sum at each stage *)

A269708 Number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 14", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 5, 20, 76, 292, 1132, 4420, 17356, 68452, 270892, 1074820, 4273036, 17013412, 67817452, 270561220, 1080119116

Offset: 0

Robert Price, Mar 04 2016

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

Rules 46, 142 and 174 also generate this sequence.

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

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
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata

Cf. A269707.

rule=14; stages=300;
ca=CellularAutomaton[{rule,{2,{{0,2,0},{2,1,2},{0,2,0}}},{1,1}},{{{1}},0},stages]; (* Start with single black cell *)
on=Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)
Part[on,2^Range[0,Log[2,stages]]] (* Extract relevant terms *)

Conjectures from Colin Barker, Mar 08 2016: (Start)
a(n) = 4*3^(n-2)+4^n for n>1.
a(n) = 7*a(n-1)-12*a(n-2) for n>3.
G.f.: (1-2*x-3*x^2-4*x^3) / ((1-3*x)*(1-4*x)).
(End)

a(9)-a(15) from Lars Blomberg, Apr 12 2016

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A346909 Continued fraction expansion of the constant whose decimal expansion is A269707.

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A151666 Number of partitions of n into distinct powers of 4.

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A269709 Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 14", based on the 5-celled von Neumann neighborhood.

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A269708 Number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 14", based on the 5-celled von Neumann neighborhood.

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