A048713 -id:A048713 - cp's OEIS frontend

A048710 Family 1 "Rule 90 x Rule 150 Array" read by antidiagonals.

1, 5, 7, 17, 27, 21, 85, 119, 65, 107, 257, 427, 325, 455, 273, 1285, 1799, 1105, 1755, 1365, 1911, 4369, 6939, 5397, 7607, 4097, 6827, 5189, 21845, 30583, 16705, 27499, 20485, 28679, 17745, 28123, 65537

Offset: 0

Antti Karttunen, Mar 18 1999

Infinitely many one-dimensional cellular automaton rules (given in sequence A048705) occur in this array, as combinations of CA-rules "90" (generates rows) and "150" (generates columns).
No pattern occurs twice in such arrays.
Each row/column can be generated from its predecessor row/column with SHIFTXORADJ transformation, given in A048711.

			   1  5  17   85  257 1105 ... [ beginning of array ]
   7 27 119  427 1799 ...
  21 65 325 1105 5397 ...

Rows = A038183, A048711, A048713, columns = A038184, A048712, A048713, diagonal = A048709. Cf. A048720.

trinv := n -> floor((1+sqrt(1+8*n))/2); # Gives integral inverses of the triangular numbers

a(n) = rule150(rule90(1, (((trinv(n)-1)*(((1/2)*trinv(n))+1))-n)), (n-((trinv(n)*(trinv(n)-1))/2))).

A048711 2nd row of Family 1 "90 X 150 array": generations 0 .. n of Rule 90 starting from seed pattern 7.

Original entry on oeis.org

7, 27, 119, 427, 1799, 6939, 30583, 109227, 458759, 1769499, 7798903, 27984299, 117901063, 454761243, 2004318071, 7158278827, 30064771079, 115964117019, 511101108343, 1833951035819, 7726646167303

Offset: 0

Antti Karttunen

nonn

Also generated by applying one generation of "Rule 150" to each term of A038183 or by doing a transformation SHIFTXORADJ(A038183)

N. J. A. Sloane, Transforms: Maple implementation of binary eXclusive OR (XORnos).

A048713.

# Maple procedure for doing Shift XOR adjacent terms transformation:
SHIFTXORADJ := proc(a) local b,i:
if whattype(a) <> list then RETURN([ ]); fi: if nops(a) <= 1 then RETURN([ ]); fi: b := [ ]:
for i from 2 to nops(a) do b := [ op(b), XORnos((a[ i-1 ]*2),a[ i ]) ]: od: RETURN(b); end:

a(n) = product('((bit_i((n+1), i)*(2^(2^(i+1))))+1)', 'i'=0..floor_log_2(n+2)) + 2*product('((bit_i(n, i)*(2^(2^(i+1))))+1)', 'i'=0..floor_log_2(n+1));

cp's OEIS Frontend

A048710 Family 1 "Rule 90 x Rule 150 Array" read by antidiagonals.

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