A266072 -id:A266072 - cp's OEIS frontend

A271343 Triangle read by rows: T(n,k) = A196020(n,k) - A266537(n,k), n>=1, k>=1.

1, 1, 5, 1, 1, 0, 9, 3, 1, -2, 1, 13, 5, 0, 1, 0, 0, 17, 7, 3, 1, -6, 0, 1, 21, 9, 0, 0, 1, 0, 3, 0, 25, 11, 0, 0, 1, -10, 0, 3, 29, 13, 7, 0, 1, 1, 0, 0, 0, 0, 33, 15, 0, 0, 0, 1, -14, 3, 5, 0, 37, 17, 0, 0, 0, 1, 0, 0, -2, 3, 41, 19, 11, 0, 0, 1, 1, -18, 0, 7, 0, 0, 45, 21, 0, 0, 0, 0, 1, 0, 3, 0, 0, 0

Offset: 1

Omar E. Pol, Apr 06 2016

Gives an identity for A000593. Alternating sum of row n equals the sum of odd divisors of n, i.e., Sum_{k=1..A003056(n)} (-1)^(k-1)*T(n,k) = A000593(n).
Row n has length A003056(n) hence the column k starts in row A000217(k).
Since the odd-indexed rows of the triangle A266537 contain all zeros then odd-indexed rows of this triangle are the same as the odd-indexed rows of the triangle A196020.
If T(n,k) is the second odd number in the column k then T(n+1,k+1) = 1 is the first element in the column k+1.
Alternating row sums of A196020 give A000203.
Alternating row sums of A266537 give A146076.

			Triangle begins:
1;
1;
5,   1;
1,   0;
9,   3;
1,  -2,  1;
13,  5,  0;
1,   0,  0;
17,  7,  3;
1,  -6,  0,  1;
21,  9,  0,  0;
1,   0,  3,  0;
25, 11,  0,  0;
1, -10,  0,  3;
29, 13,  7,  0,  1;
1,   0,  0,  0,  0;
33, 15,  0,  0,  0;
1, -14,  3,  5,  0;
37, 17,  0,  0,  0;
1,   0,  0, -2,  3;
41, 19, 11,  0,  0,  1;
1, -18,  0,  7,  0,  0;
45, 21,  0,  0,  0,  0;
1,   0,  3,  0,  0,  0;
49, 23,  0,  0,  5,  0;
1, -22,  0,  9,  0,  0;
53, 25, 15,  0,  0,  3;
1,   0,  0, -6,  0,  0,  1;
...
For n = 18 the divisors of 18 are 1, 2, 3, 6, 9, 18 and the sum of odd divisors of 18 is 1 + 3 + 9 = 13. On the other hand, the 18th row of the triangle is 1, -14, 3, 5, 0, so the alternating row sum is 1 -(-14) + 3 - 5 + 0 = 13, equaling the sum of odd divisors of 18.

Column 1 is A266072.

Cf. A000217, A000593, A001227, A003056, A146076, A196020, A236104, A237593, A261699, A266537.

A266069 Decimal representation of the n-th iteration of the "Rule 3" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 4, 2, 121, 4, 2035, 8, 32743, 16, 524239, 32, 8388511, 64, 134217535, 128, 2147483263, 256, 34359737599, 512, 549755812351, 1024, 8796093019135, 2048, 140737488349183, 4096, 2251799813672959, 8192, 36028797018939391, 16384, 576460752303374335, 32768

Offset: 0

Robert Price, Dec 20 2015

Rule 35 also generates this sequence.

			From _Michael De Vlieger_, Dec 21 2015: (Start)
First 8 rows, replacing leading zeros with ".", the row converted to its binary, then decimal equivalent at right:
              1                =               1 ->     1
            1 0 0              =             100 ->     4
          . . . 1 0            =              10 ->     2
        1 1 1 1 0 0 1          =         1111001 ->   121
      . . . . . . 1 0 0        =             100 ->     4
    1 1 1 1 1 1 1 0 0 1 1      =     11111110011 ->  2035
  . . . . . . . . . 1 0 0 0    =            1000 ->     8
1 1 1 1 1 1 1 1 1 1 0 0 1 1 1  = 111111111100111 -> 32743
(End)

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Robert Price, Table of n, a(n) for n = 0..999
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

Cf. A263428, A266068, A266070, A266071, A081253, A266072, A247375, A266073, A266074.

rule = 3; rows = 30; Table[FromDigits[Table[Take[CellularAutomaton[rule,{{1},0}, rows-1, {All,All}][[k]], {rows-k+1, rows+k-1}], {k,1,rows}][[k]],2], {k,1,rows}]

print([2*4**n - 3*2**((n-1)//2) - 1 if n%2 else 2**(n//2) for n in range(30)]) # Karl V. Keller, Jr., Aug 25 2021

G.f.: (1+4*x-17*x^2+45*x^3+16*x^4-64*x^5) / ((1-x)*(1+x)*(1-4*x)*(1+4*x)*(1-2*x^2)). - Colin Barker, Dec 21 2015 and Apr 18 2019

a(n) = 2*4^n - 3*2^((n-1)/2) - 1 for odd n; a(n) = 2^(n/2) for even n. - Karl V. Keller, Jr., Aug 25 2021

cp's OEIS Frontend

A271343 Triangle read by rows: T(n,k) = A196020(n,k) - A266537(n,k), n>=1, k>=1.

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A266069 Decimal representation of the n-th iteration of the "Rule 3" elementary cellular automaton starting with a single ON (black) cell.

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