cp's OEIS Frontend

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

%I A048883 #144 Apr 02 2025 15:03:46
%S A048883 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,
%T A048883 27,81,81,243,3,9,9,27,9,27,27,81,9,27,27,81,27,81,81,243,9,27,27,81,
%U A048883 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
%N A048883 a(n) = 3^wt(n), where wt(n) = A000120(n).
%C A048883 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.
%C A048883 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
%C A048883 Or, start with S=[1]; replace S by [S, 3*S]; repeat ad infinitum.
%C A048883 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
%C A048883 Equals row sums of triangle A166453 (the square of Sierpiński's gasket, A047999). - _Gary W. Adamson_, Oct 13 2009
%C A048883 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
%C A048883 a(A000225(n)) = A000244(n) and a(m) != A000244(n) for m < A000225(n). - _Reinhard Zumkeller_, Nov 14 2011
%C A048883 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
%C A048883 Number of odd values in the n-th layer of Pascal's tetrahedron (see A268240). - _Caden Le_, Mar 03 2025
%C A048883 a(x*y) <= a(x)^A000120(y). - _Joe Amos_, Mar 28 2025
%H A048883 Reinhard Zumkeller, <a href="/A048883/b048883.txt">Table of n, a(n) for n = 0..10000</a>
%H A048883 David Applegate, <a href="/A139250/a139250.anim.html">The movie version</a>
%H A048883 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="/A000695/a000695_1.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>, 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.]
%H A048883 Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, <a href="http://arxiv.org/abs/1503.01796">A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata</a>, arXiv:1503.01796 [math.CO], 2015; see also the <a href="http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/CAcount.html">Accompanying Maple Package</a>.
%H A048883 Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, <a href="http://arxiv.org/abs/1503.04249">Odd-Rule Cellular Automata on the Square Grid</a>, arXiv:1503.04249 [math.CO], 2015.
%H A048883 Po-Yi Huang and Wen-Fong Ke, <a href="https://arxiv.org/abs/2307.07733">Sequences Derived from The Symmetric Powers of {1, 2, ..., k}</a>, arXiv:2307.07733 [math.CO], 2023.
%H A048883 Tanya Khovanova, <a href="http://arxiv.org/abs/1410.2193">There are no coincidences</a>, arXiv preprint 1410.2193 [math.CO], 2014.
%H A048883 Tanya Khovanova and Joshua Xiong, <a href="http://arxiv.org/abs/1405.5942">Nim Fractals</a>, arXiv:1405.594291 [math.CO] (2014), p. 10. and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Khovanova/khova6.html">J. Int. Seq. 17 (2014) # 14.7.8</a>.
%H A048883 T. Pisanski and T. W. Tucker, <a href="https://citeseerx.ist.psu.edu/pdf/3c773fe830967d425448196d25d0d70e182d0c5f">Growth in Repeated Truncations of Maps</a>, Atti. Sem. Mat. Fis. Univ. Modena, Vol. 49 (2001), 167-176. (<a href="https://web.archive.org/web/20130522102010/http://www.imfm.si/preprinti/PDF/00696.pdf">preprint</a>)
%H A048883 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polca012.jpg">Illustration of initial terms: Fig. 1. Neighbors of the vertices</a>, <a href="http://www.polprimos.com/imagenespub/polca014.jpg">Fig. 2. Overlapping squares</a>, <a href="http://www.polprimos.com/imagenespub/polca016.jpg">Fig. 3. One-step bishop</a>, (Nov 06 2009).
%H A048883 N. J. A. Sloane, <a href="/A048883/a048883.png">Illustration of a(15) = 81 corresponding to number of ON cells in Odd-rule 013 CA at generation 15</a>
%H A048883 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: <a href="https://vimeo.com/119073818">Part 1</a>, <a href="https://vimeo.com/119073819">Part 2</a>
%H A048883 N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%H A048883 N. J. A. Sloane, <a href="http://arxiv.org/abs/1503.01168">On the Number of ON Cells in Cellular Automata</a>, arXiv:1503.01168 [math.CO], 2015.
%H A048883 Ralf Stephan, <a href="https://arxiv.org/abs/math/0307027">Divide-and-conquer generating functions. I. Elementary sequences</a>, arXiv:math/0307027 [math.CO], 2003.
%H A048883 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A048883 <a href="/index/Fi#FIXEDPOINTS">Index entries for sequences that are fixed points of mappings</a>
%F A048883 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_
%F A048883 a(n) = 3*a(n/2) if n is even, otherwise a(n) = a((n+1)/2).
%F A048883 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)*...
%F A048883 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
%F A048883 a(0)=1, a(n) = 3*a(n-A053644(n)) for n > 0. - _Joe Slater_, Jan 31 2016
%F A048883 G.f. A(x) satisfies: A(x) = (1 + 3*x) * A(x^2). - _Ilya Gutkovskiy_, Jul 09 2019
%e A048883 From _Omar E. Pol_, Jun 07 2009: (Start)
%e A048883 Triangle begins:
%e A048883   1;
%e A048883   3;
%e A048883   3,9;
%e A048883   3,9,9,27;
%e A048883   3,9,9,27,9,27,27,81;
%e A048883   3,9,9,27,9,27,27,81,9,27,27,81,27,81,81,243;
%e A048883   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,...
%e A048883 Or
%e A048883   1;
%e A048883   3,3;
%e A048883   9,3,9,9;
%e A048883   27,3,9,9,27,9,27,27;
%e A048883   81,3,9,9,27,9,27,27,81,9,27,27,81,27,81,81;
%e A048883   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...
%e A048883 (End)
%t A048883 Nest[ Join[#, 3#] &, {1}, 6] (* _Robert G. Wilson v_, Jan 24 2006 and modified Jul 27 2014*)
%t A048883 a[n_] := 3^DigitCount[n, 2, 1]; Array[a, 80, 0] (* _Jean-François Alcover_, Nov 15 2017 *)
%o A048883 (PARI) a(n)=n=binary(n);3^sum(i=1,#n,n[i])
%o A048883 (Haskell)
%o A048883 a048883 = a000244 . a000120  -- _Reinhard Zumkeller_, Nov 14 2011
%Y A048883 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.
%Y A048883 A generalization of A001316. Cf. A102376.
%Y A048883 Partial sums give A130665. - _David Applegate_, Jun 11 2009
%Y A048883 Cf. A000079, A122018, A166453, A268240.
%K A048883 nonn,nice,easy,hear
%O A048883 0,2
%A A048883 _John W. Layman_
%E A048883 Corrected by _Ralf Stephan_, Jun 19 2003
%E A048883 Entry revised by _N. J. A. Sloane_, May 30 2009
%E A048883 Offset changed to 0, Jun 11 2009