cp's OEIS Frontend

A247649 Number of terms in expansion of f^n mod 2, where f = 1/x^2 + 1/x + 1 + x + x^2 mod 2.

%I A247649 #55 Feb 10 2023 12:02:27
%S A247649 1,5,5,7,5,17,7,19,5,25,17,19,7,31,19,25,5,25,25,35,17,61,19,71,7,35,
%T A247649 31,41,19,71,25,77,5,25,25,35,25,85,35,95,17,85,61,71,19,91,71,77,7,
%U A247649 35,35,49,31,107,41,121,19,95,71,85,25,113,77,103
%N A247649 Number of terms in expansion of f^n mod 2, where f = 1/x^2 + 1/x + 1 + x + x^2 mod 2.
%C A247649 This is the number of cells that are ON after n generations in a one-dimensional cellular automaton defined by the odd-neighbor rule where the neighborhood consists of 5 contiguous cells.
%C A247649 a(n) is also the number of odd entries in row n of A035343. - _Leon Rische_, Feb 02 2023
%H A247649 Chai Wah Wu, <a href="/A247649/b247649.txt">Table of n, a(n) for n = 0..10000</a>
%H A247649 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 A247649 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%F A247649 The values of a(n) for n in A247647 (or A247648) determine all the values, as follows. Parse the binary expansion of n into terms from A247647 separated by at least two zeros: m_1 0...0 m_2 0...0 m_3 ... m_r 0...0. Ignore any number (one or more) of trailing zeros. Then a(n) = a(m_1)*a(m_2)*...*a(m_r). For example, n = 37_10 = 100101_2 is parsed into 1.00.101, and so a(37) = a(1)*a(5) = 5*17 = 85. This is a generalization of the Run Length Transform.
%e A247649 The first few generations are:
%e A247649 ..........X..........
%e A247649 ........XXXXX........
%e A247649 ......X.X.X.X.X......
%e A247649 ....XX..X.X.X..XX.... (f^3)
%e A247649 ..X...X...X...X...X..
%e A247649 XXXX.XXX.XXX.XXX.XXXX
%e A247649 ...
%e A247649 f^3 mod 2 = x^6 + x^5 + x^2 + 1/x^2 + 1/x^5 + 1/x^6 + 1 has 7 terms, so a(3) = 7.
%e A247649 From _Omar E. Pol_, Mar 02 2015: (Start)
%e A247649 Also, written as an irregular triangle in which the row lengths are the terms of A011782, the sequence begins:
%e A247649   1;
%e A247649   5;
%e A247649   5, 7;
%e A247649   5,17, 7,19;
%e A247649   5,25,17,19, 7,31,19,25;
%e A247649   5,25,25,35,17,61,19,71, 7,35,31,41,19,71,25,77;
%e A247649   5,25,25,35,25,85,35,95,17,85,61,71,19,91,71,77,7,35,35,49,31,107,41,121,19, ...
%e A247649 (End)
%e A247649 It follows from the Generalized Run Length Transform result mentioned in the comments that in each row the first quarter of the terms (and no more) are equal to 5 times the beginning of the sequence itself. It cannot be said that the rows converge (in any meaningful sense) to five times the sequence. - _N. J. A. Sloane_, Mar 03 2015
%o A247649 (Python)
%o A247649 import sympy
%o A247649 from functools import reduce
%o A247649 from operator import mul
%o A247649 x = sympy.symbols('x')
%o A247649 f = 1/x**2+1/x+1+x+x**2
%o A247649 A247649_list, g = [1], 1
%o A247649 for n in range(1,1001):
%o A247649     s = [int(d,2) for d in bin(n)[2:].split('00') if d != '']
%o A247649     g = (g*f).expand(modulus=2)
%o A247649     if len(s) == 1:
%o A247649         A247649_list.append(g.subs(x,1))
%o A247649     else:
%o A247649         A247649_list.append(reduce(mul,(A247649_list[d] for d in s)))
%o A247649 # _Chai Wah Wu_, Sep 25 2014
%Y A247649 Cf. A071053, A247647, A247648, A253085, A255490.
%Y A247649 Partial sums are in A255654.
%K A247649 nonn
%O A247649 0,2
%A A247649 _N. J. A. Sloane_, Sep 25 2014