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A255066 The trunk of number-of-runs beanstalk (A255056) with reversed subsections. 8
0, 2, 6, 4, 14, 12, 10, 30, 28, 26, 22, 18, 62, 60, 58, 54, 50, 46, 42, 36, 32, 126, 124, 122, 118, 114, 110, 106, 100, 96, 94, 90, 84, 78, 74, 68, 64, 254, 252, 250, 246, 242, 238, 234, 228, 224, 222, 218, 212, 206, 202, 196, 192, 190, 186, 180, 174, 168, 162, 156, 152, 148, 142, 138, 132, 128, 510 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

This can be viewed as an irregular table: after the initial zero on row 0, start each row n with term x = (2^(n+1))-2 and subtract repeatedly the number of runs in binary representation of x to get successive x's, until the number that has already been listed (which is always (2^n)-2) is encountered, which is not listed second time, but instead, the current row is finished [and thus containing only terms of equal binary length, A000523(n) on row n]. The next row then starts with (2^(n+2))-2, with the same process repeated.

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..16142; Rows 0..16, flattened

FORMULA

a(0) = 0, a(1) = 2, a(2) = 6; and for n > 2, a(n) = A004755(A004755(A236840(a(n-1)))) if A236840(a(n-1))+2 is power of 2, otherwise just A236840(a(n-1)) [where A004755(x) adds one 1-bit to the left of the most significant bit of x].

In other words, for n > 2, let k = A236840(a(n-1)). Then, if k+2 is not a power of 2, a(n) = k, otherwise a(n) = k + (6 * (2^A000523(k))).

Other identities. For all n >= 0:

a(n) = A255056(A255122(n)).

EXAMPLE

Rows 0 - 5 of the array:

0;

2;

6, 4;

14, 12, 10;

30, 28, 26, 22, 18;

62, 60, 58, 54, 50, 46, 42, 36, 32;

After row 0, the length of row n is given by A255071(n).

PROG

(Scheme, with memoization-macro definec)

(definec (A255066 n) (cond ((< n 2) (+ n n)) ((= n 2) 6) ((A236840 (A255066 (- n 1))) => (lambda (next) (if (pow2? (+ 2 next)) (A004755 (A004755 next)) next)))))

(define (pow2? n) (and (> n 0) (zero? (A004198bi n (- n 1))))) ;; A004198bi implements bitwise and. See A004198, A209229.

CROSSREFS

Cf. A255067 (same seq, terms divided by 2).

Cf. A255071 (gives row lengths).

Cf. A000523, A004755, A005811, A236840, A255056, A255122.

Analogous sequences: A218616, A230416.

Sequence in context: A286712 A268716 A204985 * A184364 A226569 A111807

Adjacent sequences:  A255063 A255064 A255065 * A255067 A255068 A255069

KEYWORD

nonn,base,tabf

AUTHOR

Antti Karttunen, Feb 14 2015

STATUS

approved

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Last modified July 21 06:55 EDT 2019. Contains 325192 sequences. (Running on oeis4.)