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.
{0}~Join~Table[1 + BitXor[#, Floor[#/2]] &[BitXor @@ Table[Floor[n/2^m], {m, 0, Floor[Log[2, n]]}] - 1], {n, 81}] (* Michael De Vlieger, Feb 29 2016, after Jean-François Alcover at A006068 and Robert G. Wilson v at A003188 *)
a003188(n)=bitxor(n, n>>1);
a006068(n)= {
my( s=1, ns );
while ( 1,
ns = n >> s;
if ( 0==ns, break() );
n = bitxor(n, ns);
s <<= 1;
);
return (n);
} \\ by Joerg Arndt
a(n)=if(n==0, 0, 1 + a003188(a006068(n) - 1)); \\ Indranil Ghosh, Jun 07 2017
def a003188(n): return n^(n>>1)
def a006068(n):
s=1
while True:
ns=n>>s
if ns==0: break
n=n^ns
s<<=1
return n
def a(n): return 0 if n==0 else 1 + a003188(a006068(n) - 1) # Indranil Ghosh, Jun 07 2017
(define (A268718 n) (if (zero? n) n (A105081 (A006068 n))))
A003188(n) = bitxor(n, floor(n/2)); A006068(n) = if(n<2, n, {my(m = A006068(floor(n/2))); 2*m + (n%2 + m%2)%2}); A268717(n) = if(n<1, 0, A003188(1 + A006068(n - 1))); for(n=0, 100, print1(if(n<2, n, A268717(1 + A268717(1 + A268717(n - 2)))),", ")) \\ Indranil Ghosh, Mar 31 2017
def A003188(n): return n^(n//2) def A006068(n): if n<2: return n else: m=A006068(n//2) return 2*m + (n%2 + m%2)%2 def A268717(n): return 0 if n<1 else A003188(1 + A006068(n - 1)) def a(n): return A268717(1 + A268717(1 + A268717(n - 2))) if n>1 else n print([a(n) for n in range(101)]) # Indranil Ghosh, Mar 31 2017
(define (A268823 n) (if (<= n 1) n (A268717 (+ 1 (A268717 (+ 1 (A268717 (- n 2))))))))
The top left [0 .. 16] x [0 .. 19] section of the array: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19 0, 1, 4, 2, 6, 8, 3, 7, 10, 12, 15, 11, 5, 13, 16, 14, 18, 20, 23, 19 0, 1, 3, 2, 9, 8, 5, 4, 13, 12, 17, 16, 7, 6, 15, 14, 21, 20, 25, 24 0, 1, 3, 2, 9, 5, 4, 6, 13, 17, 16, 18, 10, 8, 15, 7, 21, 25, 24, 26 0, 1, 3, 2, 6, 7, 4, 5, 18, 19, 16, 17, 10, 11, 8, 9, 26, 27, 24, 25 0, 1, 3, 2, 8, 6, 4, 5, 20, 18, 9, 17, 7, 11, 10, 12, 28, 26, 33, 25 0, 1, 3, 2, 7, 6, 4, 5, 19, 18, 11, 10, 9, 8, 13, 12, 27, 26, 35, 34 0, 1, 3, 2, 7, 6, 4, 5, 19, 11, 14, 12, 8, 10, 13, 9, 27, 35, 38, 36 0, 1, 3, 2, 7, 6, 4, 5, 12, 13, 14, 15, 8, 9, 10, 11, 36, 37, 38, 39 0, 1, 3, 2, 7, 6, 4, 5, 14, 16, 11, 15, 8, 9, 12, 10, 38, 40, 35, 39 0, 1, 3, 2, 7, 6, 4, 5, 17, 16, 13, 12, 8, 9, 11, 10, 41, 40, 37, 36 0, 1, 3, 2, 7, 6, 4, 5, 17, 13, 12, 14, 8, 9, 11, 10, 41, 37, 36, 38 0, 1, 3, 2, 7, 6, 4, 5, 14, 15, 12, 13, 8, 9, 11, 10, 38, 39, 36, 37 0, 1, 3, 2, 7, 6, 4, 5, 16, 14, 12, 13, 8, 9, 11, 10, 40, 38, 21, 37 0, 1, 3, 2, 7, 6, 4, 5, 15, 14, 12, 13, 8, 9, 11, 10, 39, 38, 23, 22 0, 1, 3, 2, 7, 6, 4, 5, 15, 14, 12, 13, 8, 9, 11, 10, 39, 23, 26, 24 0, 1, 3, 2, 7, 6, 4, 5, 15, 14, 12, 13, 8, 9, 11, 10, 24, 25, 26, 27
def a003188(n): return n^(n>>1)
def a006068(n):
s=1
while True:
ns=n>>s
if ns==0: break
n=n^ns
s<<=1
return n
def a278618(n): return 0 if n==0 else 1 + a003188(a006068(n) - 1)
def A(r, c): return c if r==0 else 0 if c==0 else 1 + A(r - 1, a278618(c) - 1)
for r in range(21): print([A(c, r - c) for c in range(r + 1)]) # Indranil Ghosh, Jun 07 2017
(define (A268830 n) (A268830bi (A002262 n) (A025581 n))) ;; o=0: Square array of shifted powers of A268718. (define (A268830bi row col) (cond ((zero? row) col) ((zero? col) 0) (else (+ 1 (A268830bi (- row 1) (- (A268718 col) 1)))))) (define (A268830bi row col) (cond ((zero? row) col) ((zero? col) 0) (else (+ 1 (A268830bi (- row 1) (A003188 (+ -1 (A006068 col))))))))
Join[{0, 1}, 1 + 4 Select[Range[1, 251, 2], OddQ[Total[IntegerDigits[#, 2]]]&]] (* Jean-François Alcover, Mar 15 2016 *)
def A268673(n): return (((m:=n-2)<<4)+(13 if m.bit_count()&1 else 5)) if n>1 else n # Chai Wah Wu, Mar 03 2023
(define (A268673 n) (if (<= n 1) n (+ 1 (* 4 (A092246 (- n 1))))))
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