A136013 -id:A136013 - cp's OEIS frontend

A080277 Partial sums of A038712.

1, 4, 5, 12, 13, 16, 17, 32, 33, 36, 37, 44, 45, 48, 49, 80, 81, 84, 85, 92, 93, 96, 97, 112, 113, 116, 117, 124, 125, 128, 129, 192, 193, 196, 197, 204, 205, 208, 209, 224, 225, 228, 229, 236, 237, 240, 241, 272, 273, 276, 277, 284, 285, 288, 289, 304, 305, 308

Offset: 1

N. J. A. Sloane, Mar 19 2003

nonn

			From _Omar E. Pol_, Sep 10 2019: (Start)
Illustration of initial terms:
a(n) is also the total area (or the total number of cells) in first n regions of an infinite diagram of compositions (ordered partitions) of the positive integers, where the length of the n-th horizontal line segment is equal to A001511(n), the length of the n-th vertical line segment is equal to A006519(n), and area of the n-th region is equal to A038712(n), as shown below (first eight regions):
-----------------------------------
n  A038712(n)  a(n)       Diagram
-----------------------------------
.                         _ _ _ _
1      1         1       |_| | | |
2      3         4       |_ _| | |
3      1         5       |_|   | |
4      7        12       |_ _ _| |
5      1        13       |_| |   |
6      3        16       |_ _|   |
7      1        17       |_|     |
8     15        32       |_ _ _ _|
.
The above diagram represents the eight compositions of 4: [1,1,1,1],[2,1,1],[1,2,1],[3,1],[1,1,2],[2,2],[1,3],[4].
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..16383
M. J. Bannister, Z. Cheng, W. E. Devanny, and D. Eppstein, Superpatterns and universal point sets, 21st Int. Symp. Graph Drawing, 2013, arXiv:1308.0403 [cs.CG], 2013-2014.
M. J. Bannister, Z. Cheng, W. E. Devanny, and D. Eppstein, Superpatterns and universal point sets, Journal of Graph Algorithms and Applications 18(2) (2014), 177-209.
Klaus Brockhaus, Illustration of A038712 and A080277.
B. Dearden, J. Iiams, and J. Metzger, A Function Related to the Rumor Sequence Conjecture, J. Int. Seq. 14 (2011), #11.2.3.
Ralf Stephan, Some divide-and-conquer sequences with (relatively) simple ordinary generating functions, 2004.
Ralf Stephan, Table of generating functions. [ps file]
Ralf Stephan, Table of generating functions. [pdf file]

Cf. A001787, A038712, A050292, A080333, A120385.

A135481 a(n) = 2^A007814(n+1) - 1.

Original entry on oeis.org

0, 1, 0, 3, 0, 1, 0, 7, 0, 1, 0, 3, 0, 1, 0, 15, 0, 1, 0, 3, 0, 1, 0, 7, 0, 1, 0, 3, 0, 1, 0, 31, 0, 1, 0, 3, 0, 1, 0, 7, 0, 1, 0, 3, 0, 1, 0, 15, 0, 1, 0, 3, 0, 1, 0, 7, 0, 1, 0, 3, 0, 1, 0, 63, 0, 1, 0, 3, 0, 1, 0, 7, 0, 1, 0, 3, 0, 1, 0, 15, 0, 1, 0, 3, 0, 1, 0, 7, 0, 1, 0, 3, 0, 1, 0, 31, 0, 1, 0, 3, 0, 1, 0

Offset: 0

N. J. A. Sloane, based on a message from Guy Steele and Don Knuth, Mar 01 2008

This is Guy Steele's sequence GS(1, 6) (see A135416).

Antti Karttunen, Table of n, a(n) for n = 0..16384

Cf. A006519, A007814, A135416, A140670, A136013 (partial sums).

using IntegerSequences
A135481List(len) = [Bits("CNIMP", n+1, n) for n in 0:len]
println(A135481List(100))  # Peter Luschny, Sep 25 2021

Maple
```
GS(1,6,200); # see A135416
```

Table[BitAnd[i, BitNot[i+1]], {i, 0, 200}] (* Peter Luschny, Jun 01 2011 *)

a(n) = 2^valuation(n+1, 2)-1; \\ Michel Marcus, Nov 19 2017

a(n) = bitand(bitneg(n+1), n); \\ Ruud H.G. van Tol, Apr 05 2023

def A135481(n): return ~(n+1)&n # Chai Wah Wu, Jul 06 2022

a(n) = A006519(n+1) - 1. - R. J. Mathar, Feb 10 2016

a(0) = 0 prepended by Andrey Zabolotskiy, Oct 08 2019, based on Lothar Esser's contribution

A006520 Partial sums of A006519.

Original entry on oeis.org

1, 3, 4, 8, 9, 11, 12, 20, 21, 23, 24, 28, 29, 31, 32, 48, 49, 51, 52, 56, 57, 59, 60, 68, 69, 71, 72, 76, 77, 79, 80, 112, 113, 115, 116, 120, 121, 123, 124, 132, 133, 135, 136, 140, 141, 143, 144, 160, 161, 163, 164, 168, 169, 171, 172, 180, 181, 183, 184, 188, 189

Offset: 1

N. J. A. Sloane, Simon Plouffe

The subsequence of primes in this partial sum begins: 3, 11, 23, 29, 31, 59, 71, 79, 113, 163, 181. The subsequence of powers in this partial sum begins: 1, 4, 8, 9, 32, 49, 121, 144, 169. - Jonathan Vos Post, Feb 18 2010

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..1001 [Jul 23 2013; offset adapted by _Georg Fischer_, Jan 27 2020]
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 44.
Victor Meally, Letter to N. J. A. Sloane, May 1975.
Ralf Stephan, Some divide-and-conquer sequences with (relatively) simple ordinary generating functions.
Ralf Stephan, Table of generating functions.

First differences of A022560.

Cf. A001620, A001787, A006519, A136013, A159699.

Table[ 2^IntegerExponent[n, 2], {n, 1, 70}] // Accumulate (* Jean-François Alcover, May 14 2013 *)

PARI
```
a(n)=sum(i=1,n,2^valuation(i,2))
```

def A006520(n): return sum(i&-i for i in range(1,n+1)) # Chai Wah Wu, Jul 14 2022

a(n)/(n*log(n)) is bounded. - Benoit Cloitre, Dec 17 2002
G.f.: (1/(x*(1-x))) * (x/(1-x) + Sum_{k>=1} 2^(k-1)*x^2^k/(1-x^2^k)). - Ralf Stephan, Apr 17 2003
a(n) = b(n+1), with b(2n) = 2b(n) + n, b(2n+1) = 2b(n) + n + 1. - Ralf Stephan, Sep 07 2003
a(2^k-1) = k*2^(k-1) = A001787(k) for any k > 0. - Rémy Sigrist, Jan 21 2021
a(n) ~ (1/(2*log(2)))*n*log(n) + (3/4 + (gamma-1)/(2*log(2)))*n, where gamma is Euler's constant (A001620). - Amiram Eldar, Nov 15 2022
a(n) = A136013(n) + n = A159699(n) - n. - Alan Michael Gómez Calderón, Apr 13 2025

More terms from Benoit Cloitre, Dec 17 2002

Offset changed to 1 by N. J. A. Sloane, Oct 18 2019

A333979 Array read by antidiagonals, n >= 0, k >= 2: T(n,k) is the "digital derivative" of n in base k; if the base k representation of n is Sum_{j>=0} d_jk^j, then T(n,k) = Sum_{j>=1} d_jj*k^(j-1).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4, 0, 0, 0, 0, 1, 4, 0, 0, 0, 0, 1, 1, 5, 0, 0, 0, 0, 0, 1, 2, 5, 0, 0, 0, 0, 0, 1, 1, 2, 12, 0, 0, 0, 0, 0, 0, 1, 1, 2, 12, 0, 0, 0, 0, 0, 0, 1, 1, 2, 6, 13, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 6, 13

Offset: 0

Pontus von Brömssen, Sep 04 2020

			Array begins:
  n\k|  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16
  ---|---------------------------------------------
   0 |  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
   1 |  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
   2 |  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0
   3 |  1  1  0  0  0  0  0  0  0  0  0  0  0  0  0
   4 |  4  1  1  0  0  0  0  0  0  0  0  0  0  0  0
   5 |  4  1  1  1  0  0  0  0  0  0  0  0  0  0  0
   6 |  5  2  1  1  1  0  0  0  0  0  0  0  0  0  0
   7 |  5  2  1  1  1  1  0  0  0  0  0  0  0  0  0
   8 | 12  2  2  1  1  1  1  0  0  0  0  0  0  0  0
   9 | 12  6  2  1  1  1  1  1  0  0  0  0  0  0  0
  10 | 13  6  2  2  1  1  1  1  1  0  0  0  0  0  0
  11 | 13  6  2  2  1  1  1  1  1  1  0  0  0  0  0
  12 | 16  7  3  2  2  1  1  1  1  1  1  0  0  0  0
  13 | 16  7  3  2  2  1  1  1  1  1  1  1  0  0  0
  14 | 17  7  3  2  2  2  1  1  1  1  1  1  1  0  0
  15 | 17  8  3  3  2  2  1  1  1  1  1  1  1  1  0
  16 | 32  8  8  3  2  2  2  1  1  1  1  1  1  1  1
64 = 2*3^3 + 1*3^2 + 0*3^1 + 1*3^0, so T(64,3) = 2*3*3^2 + 1*2*3^1 + 0*1*3^0 = 60. Alternatively, using the formula T(n,k) = floor(n/k) + k*T(floor(n/k),k), we get T(64,3) = 21 + 3*T(21,3) = 21 + 3*(7 + 3*T(7,3)) = 42 + 9*(2 + 3*T(2,3)) = 60.

Pontus von Brömssen, Antidiagonals n = 0..99, flattened
M. J. Bannister, Z. Cheng, W. E. Devanny, and D. Eppstein, Superpatterns and universal point sets, Journal of Graph Algorithms and Applications 18(2) (2014), 177-209.

Cf. A136013 (column k=2), A080277 (every second term of column k=2), A080333 (every third term of column k=3).

Cf. A007824, A055129, A286561.

import sympy
def A333979(n,k):
  d=sympy.ntheory.factor_.digits(n,k)
  return sum(j*d[-j-1]*k**(j-1) for j in range(1,len(d)-1))

# Second program (faster)
def A333979(n,k):
  return n//k+k*A333979(n//k,k) if n>=k else 0

T(n,k) = floor(n/k) + k*T(floor(n/k),k). Proof: With n = Sum_{j>=0} d_j*k^j we have floor(n/k) + k*T(floor(n/k),k) = Sum_{j>=1} (d_j*k^(j-1) + k*d_j*(j-1)*k^(j-2)) = Sum_{j>=1} d_j*j*k^(j-1) = T(n,k).
T(n,k) = T(n-1,k) + A055129(A286561(n,k),k). Proof: Let n = Sum_{j>=0} d_j*k^j and pick v so that d_j = 0 for j < v and d_v > 0 (so v = A286561(n,k)). Then n - 1 = sum_{j>=0} e_j*k^j, where e_j = k - 1 for j < v, e_v = d_v - 1, and e_j = d_j for j > v. We get T(n,k) - T(n-1,k) = Sum_{j>=1} j*(d_j-e_j)*k^(j-1) = v*k^(v-1) - (k-1)*Sum_{1<=jA055129(A286561(n,k),k).
For fixed k, T(n,k) ~ n*log(n)/(k*log(k)). (The proof for k = 2 by Bannister et al. (p. 182) can be adapted to general k.)
T(n,k) = Sum_{j>=0} k^j*floor(n/k**(j+1)).

A007824 a(n) = f(a(n-1)), with f(m) = Sum ib(i)2^(i-1), m = Sum b(i)*2^i, and starting value 16.

Original entry on oeis.org

16, 32, 80, 224, 720, 2976, 15120, 92448, 704592, 6455520, 69518032, 897158048, 12700875536, 203746961696, 3674124288080, 74060581880032, 1692860294726352, 42020514867170208, 1144626896435067664, 33222330682700081440

Offset: 0

Ralph Buchholz [ ralph(AT)defcen.gov.au ], Leisa Condie

nonn

Previous name was : For N=Sum a(i).2^i, a(i)=0,1, set D(N)/D(2)=Sum i.a(i).2^(i-1); sequence gives n-th derivative of 16.

Actually, it appears that f(n) for n=0, 1, 2, ... gives 0, 0, 1, 1, 4, 4, 5, 5, 12, 12, 13, 13, 16, 16, 17, 17, 32, 32, 33, 33, 36, ... that is A136013. - Michel Marcus, Jul 18 2013 [This now follows from the comments in A136013. - Pontus von Brömssen, Sep 06 2020]

Cf. A007823, A136013.

D(n) = {b = binary(n); lb = #b; rb = vector(lb, i, b[lb-i+1]); return(sum (i=0, lb-1, i*2^(i-1)*rb[i+1]));}
lista(nn) = {m = 16; for (n=0, nn, print1(m, ", "); m = D(m););} \\ Michel Marcus, Jul 18 2013

a(n) = 16*A007823(n). - Pontus von Brömssen, Sep 06 2020

Edited by Michel Marcus, Jul 18 2013

A368595 Alternating sum of A006519.

Original entry on oeis.org

-1, 1, 0, 4, 3, 5, 4, 12, 11, 13, 12, 16, 15, 17, 16, 32, 31, 33, 32, 36, 35, 37, 36, 44, 43, 45, 44, 48, 47, 49, 48, 80, 79, 81, 80, 84, 83, 85, 84, 92, 91, 93, 92, 96, 95, 97, 96, 112, 111, 113, 112, 116, 115, 117, 116, 124, 123, 125, 124, 128, 127, 129, 128

Offset: 1

Jeffrey Shallit, Dec 31 2023

a(n) <= (n/2)*log_2 n, with equality at powers of 2.

Cf. A006519. A006520 (all positive signs), A136013.

Cf. A093347 (with powers of 3).

a[1]=-1;a[n_]:=If[OddQ[n],a[n-1]-2^IntegerExponent[n,2],a[n-1]+2^IntegerExponent[n,2]];Table[a[n],{n,63}] (* James C. McMahon, Dec 31 2023 *)

a(n) = fromdigits(Vec(Pol(binary(n))'),2) - bitand(n,1); \\ Kevin Ryde, Jan 01 2024

def A368595(n): return sum(map(lambda x:(x[0]+1)*(1<Chai Wah Wu, Jan 01 2024

a(n) = Sum_{i=1..n} (-1)^i*A006519(i).

a(n) = A136013(n) - (n mod 2). - Kevin Ryde, Jan 01 2024

cp's OEIS Frontend

A080277 Partial sums of A038712.

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A135481 a(n) = 2^A007814(n+1) - 1.

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A006520 Partial sums of A006519.

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A333979 Array read by antidiagonals, n >= 0, k >= 2: T(n,k) is the "digital derivative" of n in base k; if the base k representation of n is Sum_{j>=0} d_j*k^j, then T(n,k) = Sum_{j>=1} d_j*j*k^(j-1).

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A007824 a(n) = f(a(n-1)), with f(m) = Sum i*b(i)*2^(i-1), m = Sum b(i)*2^i, and starting value 16.

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A368595 Alternating sum of A006519.

Views

Author

Keywords

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Mathematica

PARI

Python

Formula

A333979 Array read by antidiagonals, n >= 0, k >= 2: T(n,k) is the "digital derivative" of n in base k; if the base k representation of n is Sum_{j>=0} d_jk^j, then T(n,k) = Sum_{j>=1} d_jj*k^(j-1).

A007824 a(n) = f(a(n-1)), with f(m) = Sum ib(i)2^(i-1), m = Sum b(i)*2^i, and starting value 16.