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A007306 Denominators of Farey tree fractions (i.e., the Stern-Brocot subtree in the range [0,1]).
(Formerly M0437)
95
1, 1, 2, 3, 3, 4, 5, 5, 4, 5, 7, 8, 7, 7, 8, 7, 5, 6, 9, 11, 10, 11, 13, 12, 9, 9, 12, 13, 11, 10, 11, 9, 6, 7, 11, 14, 13, 15, 18, 17, 13, 14, 19, 21, 18, 17, 19, 16, 11, 11, 16, 19, 17, 18, 21, 19, 14, 13, 17, 18, 15, 13, 14, 11, 7, 8, 13, 17, 16, 19, 23, 22, 17, 19, 26, 29, 25, 24 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Also number of odd entries in n-th row of triangle of Stirling numbers of the second kind (A008277). - Benoit Cloitre, Feb 28 2004
Apparently (except for the first term) the number of odd entries in the alternated diagonals of Pascal's triangle at 45 degrees slope. - Javier Torres (adaycalledzero(AT)hotmail.com), Jul 26 2009
The Kn3 and Kn4 triangle sums, see A180662 for their definitions, of Sierpiński's triangle A047999 equal a(n+1). - Johannes W. Meijer, Jun 05 2011
From Yosu Yurramendi, Jun 23 2014: (Start)
If the terms (n>1) are written as an array:
2,
3, 3,
4, 5, 5, 4,
5, 7, 8, 7, 7, 8, 7, 5,
6, 9, 11, 10, 11, 13, 12, 9, 9, 12, 13, 11, 10, 11, 9, 6,
7, 11, 14, 13, 15, 18, 17, 13, 14, 19, 21, 18, 17, 19, 16, 11, 11, 16, 19,17,18,
then the sum of the k-th row is 2*3^(k-2), each column is an arithmetic progression. The differences of the arithmetic progressions give the sequence itself (a(2^(m+1)+1+k) - a(2^m+1+k) = a(k+1), m >= 1, 1 <= k <= 2^m), because a(n) = A002487(2*n-1) and A002487 has these properties. A071585 also has these properties. Each row is a palindrome: a(2^(m+1)+1-k) = a(2^m+k), m >= 0, 1 <= k <= 2^m.
If the terms (n>0) are written in this way:
1,
2, 3,
3, 4, 5, 5,
4, 5, 7, 8, 7, 7, 8, 7,
5, 6, 9, 11, 10, 11, 13, 12, 9, 9, 12, 13, 11, 10, 11, 9,
6, 7, 11, 14, 13, 15, 18, 17, 13, 14, 19, 21, 18, 17, 19, 16, 11, 11, 16, 19,
each column is an arithmetic progression and the steps also give the sequence itself (a(2^(m+1)+k) - a(2^m+k) = a(k), m >= 0, 0 <= k < 2^m). Moreover, by removing the first term of each column:
a(2^(m+1)+k) = A049448(2^m+k+1), m >= 0, 0 <= k < 2^m.
(End)
k > 1 occurs in this sequence phi(k) = A000010(k) times. - Franklin T. Adams-Watters, May 25 2015
Except for the initial 1, this is the odd bisection of A002487. The even bisection of A002487 is A002487 itself. - Franklin T. Adams-Watters, May 25 2015
For all m >= 0, max_{k=1..2^m} a(2^m+k) = A000045(m+3) (Fibonacci sequence). - Yosu Yurramendi, Jun 05 2016
For all n >= 2, max(m: a(2^m+k) = n, 1<=k<=2^m) = n-2. - Yosu Yurramendi, Jun 05 2016
a(2^m+1) = m+2, m >= 0; a(2^m+2) = 2m+1, m>=1; min_{m>=0, k=1..2^m} a(2^m+k) = m+2; min_{m>=2, k=2..2^m-1} a(2^m+k) = 2m+1. - Yosu Yurramendi, Jun 06 2016
a(2^(m+2) + 2^(m+1) - k) - a(2^(m+1) + 2^m-k) = 2*a(k+1), m >= 0, 0 <= k <= 2^m. - Yosu Yurramendi, Jun 09 2016
If the initial 1 is omitted, this is the number of nonzero entries in row n of the generalized Pascal triangle P_2, see A282714 [Leroy et al., 2017]. - N. J. A. Sloane, Mar 02 2017
Apparently, this sequence was introduced by Johann Gustav Hermes in 1894. His paper gives a strong connection between this sequence and the so-called "Gaussian brackets" ("Gauss'schen Klammer"). For an independent discussion about Gaussian brackets, see the relevant MathWorld article and the article by Herzberger (1943). Srinivasan (1958) gave another, more modern, explanation of the connection between this sequence and the Gaussian brackets. (Parenthetically, J. G. Hermes is the mathematician who completed or constructed the regular polygon with 65537 sides.) - Petros Hadjicostas, Sep 18 2019
REFERENCES
P. Bachmann, Niedere Zahlentheorie (1902, 1910), reprinted Chelsea, NY, 1968, vol. 2, p. 61.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 1, p. 158.
J. C. Lagarias, Number Theory and Dynamical Systems, pp. 35-72 of S. A. Burr, ed., The Unreasonable Effectiveness of Number Theory, Proc. Sympos. Appl. Math., 46 (1992). Amer. Math. Soc.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Suayb S. Arslan, Asymptotically MDS Array BP-XOR Codes, arXiv:1709.07949 [cs.IT], 2017.
Alexander Bogomolny, Stern-Brocot tree.
M. Herzberger, Gaussian optics and Gaussian brackets, Journal of the Optical Society of America 33(12) (1943), 651-655. [This paper gives a clear description of Gaussian brackets that are related to this sequence as explained by Hermes (1894).]
Jennifer Lansing, On the Stern sequence and a related sequence, Ph.D. dissertation in Mathematics, University of Illinois at Urbana-Champaign, 2014. [This doctoral dissertation discusses the so-called Stern sequence on which Hermes' papers are based (according to Srinivasan (1958)).]
Julien Leroy, Michel Rigo, and Manon Stipulanti, Counting the number of non-zero coefficients in rows of generalized Pascal triangles, Discrete Mathematics 340 (2017), 862-881.
Julien Leroy, Michel Rigo, and Manon Stipulanti, Counting subword occurrences in base-b expansions, Integers (2018) 18A, Article #A13.
G. Melançon, Lyndon factorization of sturmian words, Discr. Math. 210 (2000), 137-149.
N. J. A. Sloane, Stern-Brocot or Farey Tree.
M. S. Srinivasan, The enumeration of positive rational numbers, Proc. Indian Acad. Sci. Sect. A 47 (1958), 12-24.
M. Stern, Über eine zahlentheoretische Function, Journal für die reine und angewandte Mathematik 55 (1858), 193-220. [According to Srinivasan (1958), Hermes's (1894) paper, where this sequence is introduced, is based on Stern's sequence.]
Manon Stipulanti, Convergence of Pascal-Like Triangles in Parry-Bertrand Numeration Systems, arXiv:1801.03287 [math.CO], 2018.
Javier Torres Suarez, Number theory - geometric connection (part 2) (YouTube video that mentions this sequence - link sent by Pacha Nambi, Aug 26 2009).
Eric Weisstein's World of Mathematics, Gaussian brackets; they are related to this sequence.
Wikipedia, Johann Gustav Hermes. [He is the person who introduced this sequence and the person who completed or constructed a regular polygon with 65537 sides.]
FORMULA
Recurrence: a(0) to a(8) are 1, 1, 2, 3, 3, 4, 5, 5, 4; thereafter a(n) = a(n-2^p) + a(2^(p+1)-n+1), where 2^p < n <= 2^(p+1). [J. Hermes, Math. Ann., 1894; quoted by Dickson, Vol. 1, p. 158] - N. J. A. Sloane, Mar 24 2019
For n > 0, a(n) = A002487(n-1) + A002487(n) = A002487(2*n-1).
a(0) = 1; a(n) = Sum_{k=0..n-1} C(n-1+k, n-1-k) mod 2, n > 0. - Benoit Cloitre, Jun 20 2003
a(n+1) = Sum_{k=0..n} binomial(2*n-k, k) mod 2; a(n) = 0^n + Sum_{k=0..n-1} binomial(2(n-1)-k, k) mod 2. - Paul Barry, Dec 11 2004
a(n) = Sum_{k=0..n} C(n+k,2*k) mod 2. - Paul Barry, Jun 12 2006
a(0) = a(1) = 1; a(n) = a(A003602(n-1)) + a(A003602(n)), n > 1. - Alessandro De Luca, May 08 2014
a(n) = A007305(n+(2^m-1)), m=A029837(n), n=1,2,3,... . - Yosu Yurramendi, Jul 04 2014
a(n) = A007305(2^(m+1)-n) - A007305(2^m-n), m >= (A029837(n)+1), n=1,2,3,... - Yosu Yurramendi, Jul 05 2014
a(2^m) = m+1, a(2^m+1) = m+2 for m >= 0. - Yosu Yurramendi, Jan 01 2015
a(n+2) = A007305(n+2) + A047679(n) n >= 0. - Yosu Yurramendi, Mar 30 2016
a(2^m+2^r+k) = a(2^r+k)(m-r+1) - a(k), m >= 2, 0 <= r <= m-1, 0 <= k < 2^r. Example: a(73) = a(2^6+2^3+1) = a(2^3+1)*(6-3+1) - a(1) = 5*4 - 1 = 19 . - Yosu Yurramendi, Jul 19 2016
From Antti Karttunen, Mar 21 2017 & Apr 12 2017: (Start)
For n > 0, a(n) = A001222(A277324(n-1)) = A001222(A260443(n-1)*A260443(n)).
The following decompositions hold for all n > 0:
a(n) = A277328(n-1) + A284009(n-1).
a(n) = A283986(n) + A283988(n) = A283987(n) + 2*A283988(n).
a(n) = 2*A284265(n-1) + A284266(n-1).
a(n) = A284267(n-1) + A284268(n-1).
a(n) = A284565(n-1) + A284566(n-1).
a(n) = A285106(n-1) + A285108(n-1) = A285107(n-1) + 2*A285108(n-1). (End)
a(A059893(n)) = a(n+1) for n > 0. - Yosu Yurramendi, May 30 2017
a(n) = A287731(n) + A287732(n) for n > 0. - I. V. Serov, Jun 09 2017
a(n) = A287896(n) + A288002(n) for n > 1.
a(n) = A287896(n-1) + A002487(n-1) - A288002(n-1) for n > 1.
a(n) = a(n-1) + A002487(n-1) - 2*A288002(n-1) for n > 1. - I. V. Serov, Jun 14 2017
From Yosu Yurramendi, May 14 2019: (Start)
For m >= 0, M >= m, 0 <= k < 2^m,
a((2^(m+1) + A119608(2^m+k+1))*2^(M-m) - A000035(2^m+k)) =
a((2^(m+2) - A119608(2^m+k+1))*2^(M-m) - A000035(2^m+k)-1) =
a(2^(M+2) - (2^m+k)) = a(2^(M+1) + (2^m+k) + 1) =
a(2^m+k+1)*(M-m) + a(2^(m+1)+2^m+k+1). (End)
a(k) = sqrt(A007305(2^(m+1)+k)*A047679(2^(m+1)+k-2) - A007305(2^m+k)*A047679(2^m+k-2)), m >= 0, 0 <= k < 2^m. - Yosu Yurramendi, Jun 09 2019
G.f.: 1 + x * (1 + x) * Product_{k>=0} (1 + x^(2^k) + x^(2^(k+1))). - Ilya Gutkovskiy, Jul 19 2019
Conjecture: a(n) = a(n-1) + b(n-1) - 2*(a(n-1) mod b(n-1)) for n > 1 with a(0) = a(1) = 1 where b(n) = a(n) - b(n-1) for n > 1 with b(1) = 1. - Mikhail Kurkov, Mar 13 2022
EXAMPLE
[ 0/1; 1/1; ] 1/2; 1/3, 2/3; 1/4, 2/5, 3/5, 3/4; 1/5, 2/7, 3/8, 3/7, 4/7, 5/8, 5/7, 4/5; ...
MAPLE
A007306 := proc(n): if n=0 then 1 else A002487(2*n-1) fi: end: A002487 := proc(m) option remember: local a, b, n; a := 1; b := 0; n := m; while n>0 do if type(n, odd) then b := a + b else a := a + b end if; n := floor(n/2); end do; b; end proc: seq(A007306(n), n=0..77); # Johannes W. Meijer, Jun 05 2011
MATHEMATICA
a[0] = 1; a[n_] := Sum[ Mod[ Binomial[n+k-1, 2k] , 2], {k, 0, n}]; Table[a[n], {n, 0, 77}] (* Jean-François Alcover, Dec 16 2011, after Paul Barry *)
a[0] = 0; a[1] = 1;
Flatten[{1, Table[a[2*n] = a[n]; a[2*n + 1] = a[n] + a[n + 1], {n, 0, 50}]}] (* Horst H. Manninger, Jun 09 2021 *)
PROG
(PARI) {a(n) = if( n<1, n==0, n--; sum( k=0, n, binomial( n+k, n-k)%2))};
(PARI) {a(n) = my(m); if( n<2, n>=0, m = 2^length( binary( n-1)); a(n - m/2) + a(m-n+1))}; /* Michael Somos, May 30 2005 */
(Sage)
@CachedFunction
def a(n):
return a((odd_part(n-1)+1)/2)+a((odd_part(n)+1)/2) if n>1 else 1
[a(n) for n in (0..77)] # after Alessandro De Luca, Peter Luschny, May 20 2014
(Sage)
def A007306(n):
if n == 0: return 1
M = [1, 1]
for b in (n-1).bits():
M[b] = M[0] + M[1]
return M[1]
print([A007306(n) for n in (0..77)]) # Peter Luschny, Nov 28 2017
(R)
maxrow <- 6 # by choice
a <- c(1, 2)
for(m in 0:maxrow) for(k in 1:2^m){
a[2^(m+1)+k ] <- a[2^m+k] + a[k]
a[2^(m+1)-k+1] <- a[2^m+k]
}
a
# Yosu Yurramendi, Jan 05 2015
(R)
# Given n, compute directly a(n)
# by taking into account the binary representation of n-1
# aa <- function(n){
b <- as.numeric(intToBits(n))
l <- sum(b)
m <- which(b == 1)-1
d <- 1
if(l > 1) for(j in 1:(l-1)) d[j] <- m[j+1]-m[j]+1
f <- c(1, m[1]+2) # In A002487: f <- c(0, 1)
if(l > 1) for(j in 3:(l+1)) f[j] <- d[j-2]*f[j-1]-f[j-2]
return(f[l+1])
}
# a(0) = 1, a(1) = 1, a(n) = aa(n-1) n > 1
#
# Example
n <- 73
aa(n-1)
#
# Yosu Yurramendi, Dec 15 2016
(Scheme) (define (A007306 n) (if (zero? n) 1 (A002487 (+ n n -1)))) ;; Code for A002487 given in that entry. - Antti Karttunen, Mar 21 2017
(Python)
from sympy import binomial
def a(n):
return 1 if n<1 else sum(binomial(n + k - 1, 2*k) % 2 for k in range(n + 1))
print([a(n) for n in range(101)]) # Indranil Ghosh, Mar 22 2017
(Python)
from functools import reduce
def A007306(n): return sum(reduce(lambda x, y:(x[0], sum(x)) if int(y) else (sum(x), x[1]), bin((n<<1)-1)[-1:2:-1], (1, 0))) if n else 1 # Chai Wah Wu, May 18 2023
(Magma) [1] cat [&+[Binomial(n+k, 2*k) mod 2: k in [0..n]]: n in [0..80]]; // Vincenzo Librandi, Jun 10 2019
CROSSREFS
For numerators see A007305.
See also A282714.
Sequence in context: A338587 A338580 A278149 * A286544 A238690 A332299
KEYWORD
nonn,frac,tabf,nice,changed
AUTHOR
EXTENSIONS
Formula fixed and extended by Franklin T. Adams-Watters, Jul 07 2009
Incorrect Maple program removed by Johannes W. Meijer, Jun 05 2011
STATUS
approved

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Last modified April 24 17:45 EDT 2024. Contains 371962 sequences. (Running on oeis4.)