This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A071053 #125 Jun 18 2025 23:32:20
%S A071053 1,3,3,5,3,9,5,11,3,9,9,15,5,15,11,21,3,9,9,15,9,27,15,33,5,15,15,25,
%T A071053 11,33,21,43,3,9,9,15,9,27,15,33,9,27,27,45,15,45,33,63,5,15,15,25,15,
%U A071053 45,25,55,11,33,33,55,21,63,43,85,3,9,9,15,9,27,15,33,9,27,27
%N A071053 Number of ON cells at n-th generation of 1-D CA defined by Rule 150, starting with a single ON cell at generation 0.
%C A071053 Number of 1's in n-th row of triangle in A071036.
%C A071053 Number of odd coefficients in (x^2+x+1)^n. - _Benoit Cloitre_, Sep 05 2003. This result was given in Wolfram (1983). - _N. J. A. Sloane_, Feb 17 2015
%C A071053 This is also the odd-rule cellular automaton defined by OddRule 007 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link). - _N. J. A. Sloane_, Feb 25 2015
%C A071053 This is the Run Length Transform of S(n) = Jacobsthal(n+2) (cf. A001045). The Run Length Transform of a sequence {S(n), n>=0} is defined to be the sequence {T(n), n>=0} given by T(n) = Product_i S(i), where i runs through the lengths of runs of 1's in the binary expansion of n. E.g., 19 is 10011 in binary, which has two runs of 1's, of lengths 1 and 2. So T(19) = S(1)*S(2). T(0)=1 (the empty product). - _N. J. A. Sloane_, Sep 05 2014
%D A071053 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; Chapter 3.
%H A071053 N. J. A. Sloane, <a href="/A071053/b071053.txt">Table of n, a(n) for n = 0..16383</a> (first 1024 terms from T. D. Noe)
%H A071053 Joerg Arndt, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2015-March/014530.html">A071053 (number of odd terms in expansion of (1+x+x^2)^n)</a>, SeqFan Mailing List, Mar 09 2015.
%H A071053 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 A071053 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 A071053 S. R. Finch, P. Sebah and Z.-Q. Bai, <a href="http://arXiv.org/abs/0802.2654">Odd Entries in Pascal's Trinomial Triangle</a>, arXiv:0802.2654 [math.NT], 2008.
%H A071053 Janko Gravner and Alexander E. Holroyd, <a href="https://dx.doi.org/10.1214/14-AOP918">Percolation and Disorder-Resistance in Cellular Automata</a>, Annals of Probability, volume 43, number 4, 2015, pages 1731-1776. Also <a href="https://arxiv.org/abs/1304.7301">arxiv:1304.7301</a> [math.PR], 2013-2015 and <a href="http://aeholroyd.org/papers/resistance.pdf">second author's copy</a>. See Fig. 1.1 (left).
%H A071053 S. Kropf, S. Wagner, <a href="https://arxiv.org/abs/1605.03654">q-Quasiadditive functions</a>, arXiv:1605.03654 [math.CO], 2016.
%H A071053 N. Pitsianis, P. Tsalides, G. L. Bleris, A. Thanailakis & H. C. Card, <a href="https://www.researchgate.net/publication/226210273_Deterministic_one-dimensional_cellular_automata">Deterministic one-dimensional cellular automata</a>, Journal of Statistical Physics, 56(1-2), 99-112, 1989. [Discusses Rule 150]
%H A071053 T. Sillke and Helmut Postl, <a href="http://www.mathematik.uni-bielefeld.de/~sillke/PUZZLES/trinomials">Odd trinomials: t(n) = (1+x+x^2)^n</a>
%H A071053 T. Sillke and Helmut Postl, <a href="/A071053/a071053.txt">Odd trinomials: t(n) = (1+x+x^2)^n</a> [Cached copy, with permission]
%H A071053 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 A071053 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 A071053 S. Wolfram, <a href="http://dx.doi.org/10.1103/RevModPhys.55.601">Statistical mechanics of cellular automata</a>, Rev. Mod. Phys., 55 (1983), 601--644.
%H A071053 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%F A071053 a(n) = Product_{i in row n of A245562} A001045(i+2) [Sillke]. For example, a(11) = A001045(3)*A001045(4) = 3*5 = 15. - _N. J. A. Sloane_, Aug 10 2014
%F A071053 Floor((a(n)-1)/4) mod 2 = A020987(n). - _Ralf Stephan_, Mar 18 2004
%F A071053 a(2*n) = a(n); a(2*n+1) = a(n) + 2*a(floor(n/2)). - _Peter J. Taylor_, Mar 26 2020
%F A071053 Sum_{k = 0..2^n-1} a(k) = A087206(n). - _Linhua Zou_, Jun 13 2025
%e A071053 May be arranged into blocks of sizes 1,1,2,4,8,16,...:
%e A071053 1,
%e A071053 3,
%e A071053 3, 5,
%e A071053 3, 9, 5, 11,
%e A071053 3, 9, 9, 15, 5, 15, 11, 21,
%e A071053 3, 9, 9, 15, 9, 27, 15, 33, 5, 15, 15, 25, 11, 33, 21, 43,
%e A071053 3, 9, 9, 15, 9, 27, 15, 33, 9, 27, 27, 45, 15, 45, 33, 63, 5, 15, 15, 25, 15, 45, 25, 55, 11, 33, 33, 55, 21, 63, 43, 85,
%e A071053 3, 9, 9, 15, 9, 27, 15, 33, 9, 27, 27, ...
%e A071053 ... - _N. J. A. Sloane_, Sep 05 2014
%e A071053 .
%e A071053 From _Omar E. Pol_, Mar 15 2015: (Start)
%e A071053 Apart from the initial 1, the sequence can be written also as an irregular tetrahedron T(s,r,k) = A001045(r+2) * a(k), s>=1, 1<=r<=s, 0<=k<=(A011782(s-r)-1) as shown below (see also _Joerg Arndt_'s equivalent program):
%e A071053 3;
%e A071053 ..
%e A071053 3;
%e A071053 5;
%e A071053 .......
%e A071053 3, 9;
%e A071053 5;
%e A071053 11;
%e A071053 ...............
%e A071053 3, 9, 9, 15;
%e A071053 5, 15;
%e A071053 11;
%e A071053 21;
%e A071053 ...............................
%e A071053 3, 9, 9, 15, 9, 27, 15, 33;
%e A071053 5, 15, 15, 25;
%e A071053 11, 33;
%e A071053 21;
%e A071053 43;
%e A071053 ..............................................................
%e A071053 3, 9, 9, 15, 9, 27, 15, 33, 9, 27, 27, 45, 15, 45, 33, 63;
%e A071053 5, 15, 15, 25, 15, 45, 25, 55;
%e A071053 11, 33, 33, 55;
%e A071053 21, 63;
%e A071053 43;
%e A071053 85;
%e A071053 ...
%e A071053 Note that every row r is equal to A001045(r+2) times the beginning of the sequence itself, thus in 3D every column contains the same number.
%e A071053 (End)
%t A071053 a[n_] := Total[CoefficientList[(x^2 + x + 1)^n, x, Modulus -> 2]];
%t A071053 Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Aug 05 2018 *)
%o A071053 (PARI)
%o A071053 b(n) = { (2^n - (-1)^n) / 3; } \\ A001045
%o A071053 a(n)=
%o A071053 {
%o A071053 if ( n==0, return(1) );
%o A071053 \\ Use a( 2^k * t ) = a(t)
%o A071053 n \= 2^valuation(n,2);
%o A071053 if ( n==1, return(3) ); \\ Use a(2^k) == 3
%o A071053 \\ now n is odd
%o A071053 my ( v1 = valuation(n+1, 2) );
%o A071053 \\ Use a( 2^k - 1 ) = A001045( 2 + k ):
%o A071053 if ( n == 2^v1 - 1 , return( b( v1 + 2 ) ) );
%o A071053 my( k2 = 1, k = 0 );
%o A071053 while ( k2 < n, k2 <<= 1; k+=1 );
%o A071053 if ( k2 > n, k2 >>= 1; k-=1 );
%o A071053 my( t = n - k2 );
%o A071053 \\ here n == 2^k + 1 where k maximal
%o A071053 \\ Use the following:
%o A071053 \\ a( 2^k + t ) = 3 * a(t) if t <= 2^(k-1)
%o A071053 \\ a( 2^k + 2^(k-1) + t ) = 5 * a(t) if t <= 2^(k-2)
%o A071053 \\ a( 2^k + 2^(k-1) + 2^(k-2) + t ) = 11* a(t) if t <= 2^(k-3)
%o A071053 \\ ... etc. ...
%o A071053 \\ a( 2^k + ... + 2^(k-s) + t ) = A001045(s+2) * a(t) if t <= 2^((k-1)-s)
%o A071053 my ( s=1 );
%o A071053 while ( 1 ,
%o A071053 k2 >>= 1;
%o A071053 if ( t <= k2 , return( b(s+2) * a(t) ) );
%o A071053 t -= k2;
%o A071053 s += 1;
%o A071053 );
%o A071053 }
%o A071053 \\ _Joerg Arndt_, Mar 15 2015, from SeqFan Mailing List, Mar 09 2015
%Y A071053 Cf. A038184, A118110, A071036, A001045, A253102, A020987, A246035, A245562, A087206.
%K A071053 nonn,tabf
%O A071053 0,2
%A A071053 _Hans Havermann_, May 26 2002
%E A071053 Entry revised by _N. J. A. Sloane_, Aug 13 2014