OFFSET
0,4
COMMENTS
The graph of this sequence is related to the Takagi (blancmange) curve: see Lagarias (2012), Section 9, especially Theorem 9.1. [Corrected by Laura Monroe, Oct 21 2020]
Theorem: a(n) is the cardinality of the set { 1<= m <= n, ((n-m) mod 2^floor(log_2(m)+1)) < 2^floor(log_2(m)) }. See links.
From Laura Monroe, Jun 11 2020: (Start)
Consider a full balanced binary tree with n unlabeled leaves such that for each internal node, the number of leaf descendants of the two children differs by at most 1. Call a tree with this even distribution of leaves "pairwise".
Apply labels to the internal nodes, where an internal node is labeled S if its two children have the same number of leaf descendants, and D if its two children have a different number of leaf descendants, and call this an SD-tree. (For a pairwise tree, this is equivalent to saying that a node is an S-node iff it has an even number of leaf descendants.)
a(n) is then the number of S-nodes on a pairwise SD-tree with n+1 leaves.
This is proved in Props. 17 and 18 of the Monroe et al. article in the links.
One example of such a tree is the summation tree generated by a pairwise summation on n+1 summands (see example below). Another example is the tree representing a neutral single-elimination tournament on n+1 teams, as in A096351.
(End)
From Laura Monroe, Oct 23 2020: (Start)
Subtracting a(n) from n gives a sequence of dilations of increasing length on the dyadic rational points of the Takagi function. The number of points in each dilation is 2^k and the scale of each dilation in both the x and y directions is 2^k, where k = floor(log_2(n+1)).
2^(a(n)) is the number of tree automorphisms on the pairwise (i.e., divide-and-conquer) tree with n+1 leaves.
(End)
LINKS
Laura Monroe, Table of n, a(n) for n = 0..16383 (first 10000 terms from N. J. A. Sloane)
Thomas Baruchel, Flattening Karatsuba's recursion tree into a single summation, arXiv:1902.08982 [math.NT], 2019. See (5) conjecture of theorem.
Thomas Baruchel, Properties of the cumulated deficient binary digit sum, arXiv:1908.02250 [math.NT], 2019. See 2.4 proof of theorem.
Thomas Baruchel, Flattening Karatsuba's Recursion Tree into a Single Summation, SN Computer Science (2020) Vol. 1, Article No. 48.
Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, Preprint 2016.
Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585.
Jeffrey C. Lagarias, The Takagi function and its properties, arXiv:1112.4205 [math.CA], 2011-2012.
Jeffrey C. Lagarias, The Takagi function and its properties, In Functions in number theory and their probabilistic aspects, 153--189, RIMS Kôkyûroku Bessatsu, B34, Res. Inst. Math. Sci. (RIMS), Kyoto, 2012. MR3014845.
Laura Monroe, A Few Identities of the Takagi Function on Dyadic Rationals, arXiv:2111.05996 [math.CO], 2021.
Laura Monroe, Takagi Function Identities on Dyadic Rationals, J. Int. Seq (2024) Vol. 27, Art. 24.2.7.
Laura Monroe and Vanessa Job, Computationally Inequivalent Summations and Their Parenthetic Forms, arXiv:2005.05387 [cs.DM], 2020. See Prop. 26, the alternate proof.
FORMULA
From N. J. A. Sloane, Mar 11 2016: (Start)
a(0)=0; thereafter a(2*n) = a(n) + a(n-1), a(2*n+1) = 2*a(n) + 1.
G.f.: (1/(1-x)^2) * Sum_{k >= 0} x^(2^k)*(1-x^(2^k))/(1+x^(2^k)).
a(2^k-1) = 2^k-1, a(3*2^k-1) = 2^(k+1)-1, a(5*2^k-1) = 3*2^k-1, etc.
(End)
From Laura Monroe, Jun 11 2020: (Start)
a(n-1) = Sum_{i=0..floor(log_2(n))} (((floor(n/(2^i))+1) mod 2)*(2^i)+(-1)^((floor(n/(2^i))+1) mod 2)*(n mod (2^i))), for n>=1.
This is an explicit formula for this sequence, and is O(log(n)). This formula is proven in Prop. 18, in the Monroe et al. reference in the links. (End)
From Laura Monroe, Oct 23 2020: (Start)
a(n) = n - A296062(n).
a(n+1) = (n+1) - (2^k)*tau(x/(2^k)), where tau is the Takagi function and n+1 = (2^k)+x with x < 2^k. (End)
EXAMPLE
From Laura Monroe, Jun 11 2020: (Start)
For n=2, the pairwise summation on 2+1=3 summands takes the form ((a+b)+c). The corresponding summation tree and SD-tree look like:
+ D
/ \ / \
+ c S c
/ \ / \
a b a b
and exactly 1 internal node has an even number of leaf descendants, hence is an S-node.
For n=3, the pairwise summation on 3+1=4 summands takes the form ((a+b)+(c+d)). The corresponding summation tree and SD-tree look like:
+ S
/ \ / \
+ + S S
/| |\ /| |\
a b c d a b c d
and exactly 3 internal nodes have an even number of leaf descendants, hence are S-nodes.
(End)
MAPLE
a000120 := proc(n) add(i, i=convert(n, base, 2)) end:
a023416 := proc(n) if n = 0 then 1; else add(1-e, e=convert(n, base, 2)) ; end if; end proc:
a268289:=proc(n) option remember; global a000120, a023416;
if n=0 then 0 else a268289(n-1)+a000120(n)-a023416(n); fi; end;
[seq(a268289(n), n=0..132)];
# N. J. A. Sloane, Mar 07 2016
# second Maple program:
a:= proc(n) option remember; `if`(n<0, 0,
a(n-1)+add(2*i-1, i=Bits[Split](n)))
end:
seq(a(n), n=0..60); # Alois P. Heinz, Jan 18 2022
MATHEMATICA
Join[{0}, Table[DigitCount[n, 2, 1] - DigitCount[n, 2, 0], {n, 1, 100}] // Accumulate] (* Jean-François Alcover, Oct 24 2016 *)
PROG
(Python)
def A268289(n): return (sum(i.bit_count() for i in range(1, n+1))<<1)-1-(n+1)*(m:=(n+1).bit_length())+(1<<m) # Chai Wah Wu, Mar 01 2023
(PARI) a(n) = if (n==0, 0, if (n%2, 2*a((n-1)/2)+1, a(n/2) + a(n/2-1))); \\ Michel Marcus, Jun 16 2020
(PARI) a(n) = my(v=binary(n+1), s=-1); for(i=1, #v, v[i]=if(v[i], s++, s--; 1)); fromdigits(v, 2); \\ Kevin Ryde, Jun 16 2020
(C)
int a(int n) {
int m=n+1;
int result=0;
int i=0;
while (n) {
int ith_bit_set = m&(1<<i);
if (ith_bit_set) result += (m%(1<<i));
else result += (1<<i) - (m%(1<<i));
i++;
n >>= 1;
}
return result;
}
/* Laura Monroe, Jun 17 2020 */
(Julia)
function A268289List(len)
A = zeros(Int, len)
for n in 1:len-1
a, b, c = n, n & 1, 1
while (a >>= 1) != 0
b += a & 1
c += 1
end
A[n+1] = A[n] + <<(b, 1) - c
end
A
end; println(A268289List(61)) # Peter Luschny, Jun 22 2020
CROSSREFS
AUTHOR
Mark Moore, Jan 30 2016
EXTENSIONS
Simplified definition following a suggestion from Michel Marcus. Corrected start, added more terms. - N. J. A. Sloane, Mar 07 2016
STATUS
approved