A213707 -id:A213707 - cp's OEIS frontend

A071542 Number of steps to reach 0 starting with n and using the iterated process : x -> x - (number of 1's in binary representation of x).

0, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 7, 7, 8, 8, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 19, 20, 20, 20, 20, 21, 21, 21, 21, 22, 22, 23, 23, 24, 24, 24, 24, 25, 25, 25, 25

Offset: 0

Benoit Cloitre, Jun 02 2002

			17 (= 10001 in binary) -> 15 (= 1111) -> 11 (= 1011) -> 8 (= 1000) -> 7 (= 111) -> 4 (= 100) -> 3 (= 11) -> 1 -> 0, hence a(17)=8.

Antti Karttunen, Table of n, a(n) for n = 0..131072

Cf. A000120, A011371, A071542, A213706, A213707, A213708, A218254.
A179016 gives the unique infinite sequence whose successive terms are related by this iterated process (in reverse order). Also, it seems that for n>=0, a(A213708(n)) = a(A179016(n+1)) = n.
A213709(n) = a((2^(n+1))-1) - a((2^n)-1).

Table[-1 + Length@ NestWhileList[# - DigitCount[#, 2, 1] &, n, # > 0 &], {n, 0, 75}] (* Michael De Vlieger, Jul 16 2017 *)

for(n=1, 150, s=n; t=0; while(s!=0, t++; s=s-sum(i=1, length(binary(s)), component(binary(s), i))); if(s==0, print1(t, ", "); ); )

a(n)=my(k);while(n,n-=hammingweight(n);k++);k \\ Charles R Greathouse IV, Oct 30 2012
(MIT/GNU Scheme)
;; with memoizing definec-macro:
(definec (A071542 n) (if (zero? n) n (+ 1 (A071542 (- n (A000120 n)))))) ;; Antti Karttunen, Oct 24 2012

a(0)=0, a(n) = 1 + A071542(n - A000120(n)). - Antti Karttunen, Oct 24 2012

It seems that a(n) ~ C n/log(n) asymptotically with C = 1.4... (n = 10^6 gives C = 1.469..., n = 10^7 gives C = 1.4614...).

Starting offset changed to 0 with a(0) prepended as 0 by Antti Karttunen, Oct 24 2012

A218254 Irregular table, where row n (n >= 0) starts with n, the next term is n-A000120(n), and the successive terms are obtained by repeatedly subtracting the number of 1's in the previous term's binary expansion, until zero is reached, after which the next row starts with one larger n.

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 3, 1, 0, 4, 3, 1, 0, 5, 3, 1, 0, 6, 4, 3, 1, 0, 7, 4, 3, 1, 0, 8, 7, 4, 3, 1, 0, 9, 7, 4, 3, 1, 0, 10, 8, 7, 4, 3, 1, 0, 11, 8, 7, 4, 3, 1, 0, 12, 10, 8, 7, 4, 3, 1, 0, 13, 10, 8, 7, 4, 3, 1, 0, 14, 11, 8, 7, 4, 3, 1, 0, 15, 11, 8, 7, 4, 3, 1, 0

Offset: 0

Nico Brown, Oct 24 2012

			The n-th row (starting indexing from zero) in this irregular table consists of block of length A071542(n)+1: 1,2,3,3,4,4,5,5,... which always ends with zero, as:
0
1,0
2,1,0
3,1,0
4,3,1,0
5,3,1,0
6,4,3,1,0
7,4,3,1,0
The 17th term is 6, which in binary is 110. The 18th term is then 6-2=4.

Antti Karttunen, Rows 0..255, flattened

Cf. A218252, A218253. A213707 gives the positions of zeros (i.e. the ending index of each row). A071542, A000120.

The reversed tails of the rows converge towards A179016.

for(n=0,9,k=n;while(k, print1(k", "); k-=hammingweight(k)); print1("0, ")) \\ Charles R Greathouse IV, Oct 30 2012

A213706 Partial sums of A071542.

Original entry on oeis.org

0, 1, 3, 5, 8, 11, 15, 19, 24, 29, 35, 41, 48, 55, 62, 69, 77, 85, 94, 103, 113, 123, 133, 143, 154, 165, 176, 187, 199, 211, 223, 235, 248, 261, 275, 289, 304, 319, 334, 349, 365, 381, 397, 413, 430, 447, 464, 481, 499, 517, 535, 553, 572, 591, 610, 629, 649, 669, 689, 709, 730

Offset: 0

Antti Karttunen, Oct 24 2012

Antti Karttunen, Table of n, a(n) for n = 0..720

Used to compute A213707. a(n)+1 gives the length of the n-th row in the irregular table A218254.

a(0)=0, a(n) = a(n-1)+A071542(n)

A219647 Positions of zeros in A219649.

Original entry on oeis.org

0, 2, 5, 9, 13, 18, 23, 29, 36, 43, 51, 59, 67, 76, 85, 95, 105, 115, 126, 137, 148, 160, 172, 185, 198, 211, 225, 239, 253, 268, 283, 298, 314, 330, 347, 364, 382, 400, 418, 437, 456, 475, 495, 515, 535, 556, 577, 599, 621, 643, 666, 689, 712, 735, 759, 784

Offset: 0

Antti Karttunen, Nov 24 2012

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10946

Analogous sequence for binary system: A213707, for factorial number system: A219657.

Cf. A219646, A219649.

Scheme
```
(define (A219647 n) (+ n (A219646 n)))
```

a(n) = n+A219646(n).

A219657 Positions of zeros in A219659.

Original entry on oeis.org

0, 2, 5, 8, 12, 16, 21, 26, 32, 38, 44, 50, 57, 64, 71, 78, 86, 94, 102, 110, 119, 128, 137, 146, 156, 166, 177, 188, 199, 210, 222, 234, 246, 258, 271, 284, 297, 310, 324, 338, 352, 366, 381, 396, 411, 426, 441, 456, 472, 488, 504, 520, 537, 554, 571, 588, 606

Offset: 0

Antti Karttunen, Nov 25 2012

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10080

Cf. A219656, A219659. Analogous sequence for binary system: A213707, for Zeckendorf expansion: A219647.

Scheme
```
(define (A219657 n) (+ n (A219656 n)))
```

a(n) = n + A219656(n).

A271911 Number of ways to choose three distinct points from a 2 X n grid so that they form an isosceles triangle.

Original entry on oeis.org

0, 4, 10, 16, 24, 32, 42, 52, 64, 76, 90, 104, 120, 136, 154, 172, 192, 212, 234, 256, 280, 304, 330, 356, 384, 412, 442, 472, 504, 536, 570, 604, 640, 676, 714, 752, 792, 832, 874, 916, 960, 1004, 1050, 1096, 1144, 1192, 1242, 1292, 1344, 1396, 1450, 1504

Offset: 1

N. J. A. Sloane, Apr 24 2016

			n=3: Label the points
1 2 3
4 5 6
There are 8 small isosceles triangles like 124 plus 135 and 246, so a(3) = 10.

Chai Wah Wu, Counting the number of isosceles triangles in rectangular regular grids, arXiv:1605.00180 [math.CO], 2016.
Index entries for linear recurrences with constant coefficients, signature (2, 0, -2, 1).

Row 2 of A271910.
Cf. A186434, A187452.
Same start as, but totally different from, 2*A213707.

LinearRecurrence[{2,0,-2,1},{0,4,10,16},60] (* Harvey P. Dale, May 10 2018 *)

Conjectured g.f.: 2*x*(2*x^2-x-2)/((x+1)*(x-1)^3). It would be nice to have a proof!
Conjectures from Colin Barker, Apr 24 2016: (Start)
a(n) = (-1+(-1)^n+16*n+2*n^2)/4, or equivalently, a(n) = (n^2+8*n)/2 if n even, (n^2+8*n-1)/2 if n odd.
a(n) = 2*a(n-1)-2*a(n-3)+a(n-4) for n>4. (End)
The conjectured g.f. and recurrence are true. See paper in links. - Chai Wah Wu, May 07 2016
a(n) = round(n*(n/2+3)) - 4. - Bill McEachen, Aug 10 2025

More terms from Harvey P. Dale, May 10 2018

cp's OEIS Frontend

A071542 Number of steps to reach 0 starting with n and using the iterated process : x -> x - (number of 1's in binary representation of x).

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A218254 Irregular table, where row n (n >= 0) starts with n, the next term is n-A000120(n), and the successive terms are obtained by repeatedly subtracting the number of 1's in the previous term's binary expansion, until zero is reached, after which the next row starts with one larger n.

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A213706 Partial sums of A071542.

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A219647 Positions of zeros in A219649.

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A219657 Positions of zeros in A219659.

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A271911 Number of ways to choose three distinct points from a 2 X n grid so that they form an isosceles triangle.

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