A237449 -id:A237449 - cp's OEIS frontend

A236840 n minus number of runs in the binary expansion of n: a(n) = n - A005811(n).

0, 0, 0, 2, 2, 2, 4, 6, 6, 6, 6, 8, 10, 10, 12, 14, 14, 14, 14, 16, 16, 16, 18, 20, 22, 22, 22, 24, 26, 26, 28, 30, 30, 30, 30, 32, 32, 32, 34, 36, 36, 36, 36, 38, 40, 40, 42, 44, 46, 46, 46, 48, 48, 48, 50, 52, 54, 54, 54, 56, 58, 58, 60, 62, 62, 62, 62, 64, 64, 64

Offset: 0

Antti Karttunen, Apr 18 2014

All terms are even. Used by the "number-of-runs beanstalk" sequence A255056 and many of its associated sequences.

Antti Karttunen, Table of n, a(n) for n = 0..8192

Cf. A091067 (the positions of records), A106836 (run lengths).
Cf. A255070 (terms divided by 2).
Cf. A005811, A000120, A003188.
Cf. A255056, A255066, A255061, A255062, A255071, A255072, A255058, A255327.
Other subtracting maps: A011371, A178503, A219641, A219651, A227190, A227191, A237449, A244320, A244234, A255131.

A236840 := proc(n) local i, b; if n=0 then 0 else b := convert(n, base, 2); select(i -> (b[i-1]<>b[i]), [$2..nops(b)]); n-1-nops(%) fi end: seq(A236840(i), i=0..69); # Peter Luschny, Apr 19 2014

a[n_] := n - Length@ Split[IntegerDigits[n, 2]]; a[0] = 0; Array[a, 100, 0] (* Amiram Eldar, Jul 16 2023 *)

Scheme
```
(define (A236840 n)  (- n (A005811 n)))
```

a(n) = n - A005811(n) = n - A000120(A003188(n)).

a(n) = 2*A255070(n).

A236855 a(n) is the sum of digits in A239903(n).

Original entry on oeis.org

0, 1, 1, 2, 3, 1, 2, 2, 3, 4, 3, 4, 5, 6, 1, 2, 2, 3, 4, 2, 3, 3, 4, 5, 4, 5, 6, 7, 3, 4, 4, 5, 6, 5, 6, 7, 8, 6, 7, 8, 9, 10, 1, 2, 2, 3, 4, 2, 3, 3, 4, 5, 4, 5, 6, 7, 2, 3, 3, 4, 5, 3, 4, 4, 5, 6, 5, 6, 7, 8, 4, 5, 5, 6, 7, 6, 7, 8, 9, 7, 8, 9, 10, 11, 3, 4

Offset: 0

Antti Karttunen, Apr 18 2014

			As the 0th Catalan String is empty, indicated by A239903(0)=0, a(0)=0.
As the 18th Catalan String is [1,0,1,2] (A239903(18)=1012), a(18) = 1+0+1+2 = 4.
Note that although the range of validity of A239903 is inherently limited by the decimal representation employed, it doesn't matter here: We have a(58785) = 55, as the corresponding 58785th Catalan String is [1,2,3,4,5,6,7,8,9,10], even though A239903 cannot represent that unambiguously.

Antti Karttunen, Table of n, a(n) for n = 0..16796

Cf. A239903, A014420, A237449, A236859, A009766, A071155, A081291, A034968, A060130.

A236855list[m_] := With[{r = 2*Range[2, m]-1}, Reverse[Map[Total[r-#] &, Select[Subsets[Range[2, 2*m-1], {m-1}], Min[r-#] >= 0 &]]]];
A236855list[6] (* Generates C(6) terms *) (* Paolo Xausa, Feb 19 2024 *)

(define (A236855 n) (apply + (A239903raw n)))
(define (A239903raw n) (if (zero? n) (list) (let loop ((n n) (row (- (A081288 n) 1)) (col (- (A081288 n) 2)) (srow (- (A081288 n) 2)) (catstring (list 0))) (cond ((or (zero? row) (negative? col)) (reverse! (cdr catstring))) ((> (A009766tr row col) n) (loop n srow (- col 1) (- srow 1) (cons 0 catstring))) (else (loop (- n (A009766tr row col)) (+ row 1) col srow (cons (+ 1 (car catstring)) (cdr catstring))))))))
;; Alternative definition:
(define (A236855 n) (let ((x (A071155 (A081291 n)))) (- (A034968 x) (A060130 x))))

a(n) = A034968(x) - A060130(x), where x = A071155(A081291(n)).
For up to n = A000108(11)-2 = 58784, a(n) = A007953(A239903(n)).
Catalan numbers, A000108, give the positions of ones, and the n-th triangular number occurs for the first time at the position immediately before that, i.e., a(A001453(n)) = A000217(n-1).
For each n, a(n) >= A000217(A236859(n)).

A244320 a(n) = n - A014420(n).

Original entry on oeis.org

0, 0, 1, 1, 2, 4, 4, 5, 5, 6, 8, 8, 9, 9, 13, 13, 14, 14, 15, 17, 17, 18, 18, 19, 21, 21, 22, 22, 26, 26, 27, 27, 28, 30, 30, 31, 31, 32, 34, 34, 35, 35, 41, 41, 42, 42, 43, 45, 45, 46, 46, 47, 49, 49, 50, 50, 54, 54, 55, 55, 56, 58, 58, 59, 59, 60, 62, 62, 63, 63, 67, 67, 68, 68, 69

Offset: 0

Antti Karttunen, Jul 02 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 0..4862

Cf. A014419, A014420, A244234, A237449.

Scheme
```
(define (A244320 n) (- n (A014420 n)))
```

a(n) = n - A014420(n).

A244234 a(n) = n - A244232(n).

Original entry on oeis.org

0, 0, 1, 1, 1, 4, 4, 5, 5, 5, 6, 6, 6, 9, 13, 13, 14, 14, 14, 17, 17, 18, 18, 18, 19, 19, 19, 22, 22, 23, 23, 23, 24, 24, 24, 27, 31, 31, 32, 32, 32, 35, 41, 41, 42, 42, 42, 45, 45, 46, 46, 46, 47, 47, 47, 50, 54, 54, 55, 55, 55, 58, 58, 59, 59, 59, 60, 60, 60

Offset: 0

Antti Karttunen, Jun 25 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 0..4862

Cf. A244159, A244232, A244320, A237449.

Scheme
```
(define (A244234 n) (- n (A244232 n)))
```

cp's OEIS Frontend

A236840 n minus number of runs in the binary expansion of n: a(n) = n - A005811(n).

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A236855 a(n) is the sum of digits in A239903(n).

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A244320 a(n) = n - A014420(n).

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A244234 a(n) = n - A244232(n).

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