A227738 Irregular table read by rows: each row n (n>=1) lists the positions where the runs of bits change between 0's and 1's in the binary expansion of n, when scanning it from the least significant to the most significant end.

1, 1, 2, 2, 2, 3, 1, 2, 3, 1, 3, 3, 3, 4, 1, 3, 4, 1, 2, 3, 4, 2, 3, 4, 2, 4, 1, 2, 4, 1, 4, 4, 4, 5, 1, 4, 5, 1, 2, 4, 5, 2, 4, 5, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 3, 4, 5, 3, 4, 5, 3, 5, 1, 3, 5, 1, 2, 3, 5, 2, 3, 5, 2, 5, 1, 2, 5, 1, 5, 5, 5, 6, 1, 5, 6, 1, 2, 5, 6

Offset

1,3

Comments

Row n has A005811(n) terms.

As a sequence, seems to have a particular fractal structure, probably allowing additional formulas.

Row n lists the positions of 1-bits in the binary expansion of the Gray code for n, A003188(n), when 1 is the rightmost position. A003188(17) = 25 = 11001_2 gives row 17: 1,4,5. - Alois P. Heinz, Feb 01 2023

Links

Antti Karttunen, The rows 1..1023 of the table, flattened

Wikipedia, Gray code

Formula

a(n) = A227188(A227737(n),A227740(n)).

Alternatively, if A227740(n) is 0, then a(n) = A227736(n), otherwise a(n) = a(n-1) + A227736(n). [Each row gives cumulative sums of the runlengths of binary representation of n]

Example

Table begins as:
  Row  n in    Terms on
   n   binary  that row
   1      1    1;
   2     10    1,2;
   3     11    2;
   4    100    2,3;
   5    101    1,2,3;
   6    110    1,3;
   7    111    3;
   8   1000    3,4;
   9   1001    1,3,4;
  10   1010    1,2,3,4;
  11   1011    2,3,4;
  12   1100    2,4;
  13   1101    1,2,4;
  14   1110    1,4;
  15   1111    4;
  16  10000    4,5;
etc.
The terms also give the partial sums of runlengths, when the binary expansion of n is scanned from the least significant to the most significant end.

Maple

T:= n-> (l-> seq(`if`(l[i]=1, i, [][]), i=1..nops(l)))(
                 Bits[Split](Bits[Xor](n, iquo(n, 2)))):
seq(T(n), n=1..50);  # Alois P. Heinz, Feb 01 2023

Mathematica

Table[Rest@FoldList[Plus, 0, Length/@Split[Reverse[IntegerDigits[n, 2]]]], {n, 34}]//Flatten (Wouter Meeussen, Aug 31 2013)

Prog

(Scheme, with Antti Karttunen's IntSeq-library)
(define (A227738 n) (A227188bi (A227737 n) (A227740 n))) ;; The Scheme-function for A227188bi has been given in A227188.
(definec (A227738 n) (if (zero? (A227740 n)) (A227736 n) (+ (A227738 (- n 1)) (A227736 n)))) ;; Alternative definition.

Crossrefs

Each row n (n>=1) contains the initial A005811(n) nonzero terms from the beginning of row n of A227188. A227192(n) gives the sum of terms on row n. A136480 gives the first column.

Cf. also A227188, A227736, A227739.

A318926 is a compressed version. If the order is reversed we get A101211 and A318927.

Cf. A003188, A360287.

Sequence in context: A256915 A348367 A209254 * A103960 A242626 A360387

Adjacent sequences: A227735 A227736 A227737 * A227739 A227740 A227741

Keyword

nonn,base,tabf

Author

Antti Karttunen, Jul 25 2013

Status

approved