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Search: a083036 -id:a083036
Displaying 1-7 of 7 results found. page 1
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A083037 a(n)=2*A083036(n)-n. Also -A123737(n). +20
7
1, 0, -1, 0, 1, 0, 1, 2, 1, 0, 1, 0, -1, 0, 1, 0, -1, 0, -1, -2, -1, 0, -1, 0, 1, 0, -1, 0, 1, 0, 1, 2, 1, 0, 1, 0, -1, 0, 1, 0, 1, 2, 1, 0, 1, 2, 1, 2, 3, 2, 1, 2, 1, 0, 1, 2, 1, 0, 1, 0, -1, 0, 1, 0, 1, 2, 1, 0, 1, 0, -1, 0, 1, 0, -1, 0, -1, -2, -1, 0, -1, 0, 1, 0, -1, 0, 1, 0, 1, 2, 1, 0, 1, 0, -1, 0, 1, 0, -1, 0, -1, -2, -1, 0, -1, 0, 1, 0, -1, 0, -1, -2, -1, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,8
LINKS
Table of n, a(n) for n=1..114.
Kevin O'Bryant, Bruce Reznick and Monika Serbinowska, Almost alternating sums, Amer. Math. Monthly, Vol. 113 (October 2006), pp. 673-688.
FORMULA
O'Bryant, Reznick, & Serbinowska show that |a(n)| <= k log n + 1, with k = 1/(2 log (1 + sqrt(2))), and further a(n) > k log n + 0.78 infinitely often. - Charles R Greathouse IV, Feb 07 2013
PROG
(PARI) a(n)=-sum(i=1, n, (-1)^sqrtint(2*i^2)) \\ Charles R Greathouse IV, Feb 07 2013
CROSSREFS
Cf. A083035, A083036, A083038, A123737.
KEYWORD
sign
AUTHOR
Benoit Cloitre, Apr 17 2003
STATUS
approved
A083035 a(n) = floor(sqrt(2)*n)-2*floor(n/sqrt(2)). +10
14
1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Benoit Cloitre, Fractal walk starting at (0,0) with step of unit length turning right if a(n)=0 and left otherwise for n=1 up to 10^6
Sela Fried, The beauty of Beatty sequences.
FORMULA
a(n) = floor(n*sqrt(2)) mod 2. - T. D. Noe, Oct 11 2006
MATHEMATICA
Table[Floor[Sqrt[2]n]-2Floor[n/Sqrt[2]], {n, 120}] (* Harvey P. Dale, Mar 08 2017 *)
Table[Mod[Floor[Sqrt[2] n], 2], {n, 120}] (* IWABUCHI Yu(u)ki, May 01 2020 *)
PROG
(PARI) a(n)=sqrtint(2*n^2)%2 \\ Charles R Greathouse IV, Oct 14 2013
CROSSREFS
Cf. A083036, A083037, A083038, A001951 (floor(n*sqrt(2))).
KEYWORD
nonn,easy
AUTHOR
Benoit Cloitre, Apr 17 2003
STATUS
approved
A083038 A fractal sequence. +10
11
1, 1, 0, 0, 1, 1, 2, 4, 5, 5, 6, 6, 5, 5, 6, 6, 5, 5, 4, 2, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 2, 4, 5, 5, 6, 6, 5, 5, 6, 6, 7, 9, 10, 10, 11, 13, 14, 16, 19, 21, 22, 24, 25, 25, 26, 28, 29, 29, 30, 30, 29, 29, 30, 30, 31, 33, 34, 34, 35, 35, 34, 34, 35, 35, 34, 34, 33, 31, 30, 30, 29, 29, 30, 30 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,7
COMMENTS
Sequence presents fractal patterns.
LINKS
Table of n, a(n) for n=1..84.
Boris Gourevitch, Graph of a(n).
FORMULA
a(n)=sum(k=1, n, A083037(k))
a(2*A001109(n)-1)=A001109(n) - Benoit Cloitre, Dec 12 2003
CROSSREFS
Cf. A083035, A083036, A083037, A071992 (which presents similar fractal aspects).
KEYWORD
nonn
AUTHOR
Benoit Cloitre, Apr 17 2003
STATUS
approved
A085002 a(n) = floor(phi*n) - 2*floor(phi*n/2) where phi is the golden ratio. +10
7
1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
The lower Wythoff sequence (A000201) mod 2 (see formula section). - Michel Dekking, Feb 01 2021
A fractal sequence.
From Michel Dekking, Apr 24 2018: (Start)
Usually an integer sequence is called 'fractal' if it has the self-generating properties of a morphic sequence, i.e., the letter to letter projection of a fixed point of a morphism. Indeed, take the alphabet {1,2,...,8} and the morphism eta defined by
eta: 1->5, 2->7, 3->8, 4->6, 5->53, 6->71, 7->82, 8->64.
Then eta has fixed point
x = (5,3,8,6,4,7,1,6,8,2,5,71,6,4,7,5,3,...).
Let pi be the projection morphism
pi(1)=1, pi(2)=0, pi(3)=1, pi(4)=0, pi(5)=1, pi(6)=0, pi(7)=1, pi(8)=0.
Then pi(x) = (a(n)).
To prove this, one may use my paper "Iteration of maps by an automaton".
The two maps are phi_a and phi_b defined by
phi_a(0) = 1, phi_a(1) = 0, phi_b(0) = 0, phi_b(1) = 1.
The substitution is the Fibonacci substitution sigma given by
sigma(a) = b, sigma(b) = ba.
Since the first differences of the lower Wythoff sequence are given by the Fibonacci substitution 1->2, 2-> 21, it follows from replacing 1 with a and 2 with b that (a(n)) is generated by iterating the two maps phi_a and phi_b according to the fixed point babba...of sigma. The two maps phi_a and phi_b are commuting bijections on {a,b}, exactly as in the Example on page 85 of "Iteration of maps by an automaton". It follows as in that example that (a(n)) is generated by the projection of a fixed point of a substitution on an alphabet of 8 letters, and a simple computation as on that page yields the morphism eta. (End)
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..10946
B. Cloitre, Graph of A085005(n) for n=1 up to 3874 [archive.org link]
Benoit Cloitre, Fractal walk starting at (0,0) with step of unit length turning 90° right if a(n)=0 left otherwise for n=1 up to 10^6
Michel Dekking, Iteration of maps by an automaton, Discrete Mathematics 126 (1994), 81-86.
M. Schaefer, E. Sedgwick, and D. Štefankovič, Spiraling and folding: the word view, Algorithmica 60 (2011), 609-626. See section 4.
FORMULA
a(n) = A105774(n) mod 2 = A000201(n) mod 2. - Benoit Cloitre, May 10 2005
From Michel Dekking, Apr 24 2018: (Start)
Proof that this sequence is the parity sequence of the lower Wythoff sequence:
if n*phi/2 = M + e, with 0 < e < 1, then 2*floor(phi*n/2) = 2M, and
floor(phi*n) = floor(2M+2e) = 2M or 2M+1.
So floor(phi*n) - 2*floor(phi*n/2) = 0 if floor(phi*n) is even, and equals 1 if floor(phi*n) is odd. (End)
MATHEMATICA
Table[Floor[GoldenRatio n] - 2 Floor[GoldenRatio n/2], {n, 110}] (* Harvey P. Dale, Dec 11 2012 *)
PROG
(PARI) a(n)=(n+sqrtint(5*n^2))%4>1 \\ Charles R Greathouse IV, Feb 07 2013
(Scheme) (define (A085002 n) (A000035 (A105774 n))) ;; Antti Karttunen, Mar 17 2017
(Python)
from math import isqrt
def A085002(n): return ((n+isqrt(5*n**2))&2)>>1 # Chai Wah Wu, Aug 10 2022
CROSSREFS
Characteristic function of A283766.
Cf. A000035, A001622, A083035, A083036, A083037, A083038, A085003 (partial sums), A085004, A085005, A105774.
See also A171587.
KEYWORD
nonn,easy
AUTHOR
Benoit Cloitre, Jun 17 2003
STATUS
approved
A085005 A Von Koch curve related to the Golden ratio. +10
6
1, 3, 4, 4, 3, 3, 4, 4, 3, 1, 0, 0, 1, 1, 0, 0, 1, 3, 4, 4, 5, 7, 10, 12, 13, 13, 14, 16, 17, 17, 16, 16, 17, 19, 20, 20, 21, 23, 26, 28, 29, 31, 34, 38, 41, 43, 44, 46, 49, 51, 52, 52, 53, 55, 56, 56, 55, 55, 56, 58, 59, 59, 60, 62, 65, 67, 68, 68, 69, 71, 72, 72, 71, 71, 72, 72, 71 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Table of n, a(n) for n=1..77.
Benoit Cloitre, Plot of a(n) for n=1 up to 1362772
FORMULA
a(n)=(-1)*sum(i=1, n, sum(j=1, i, (-1)^floor(j*(1+sqrt(5))/2)))
a(n) = 2*sum(k = 1, n, sum(i = 1, k, b(i)))-n*(n+1)/2, where b(k) = floor(phi*k)-2*floor(phi*k/2)
PROG
(PARI) a(n)=(-1)*sum(i=1, n, sum(j=1, i, (-1)^floor(j*(1+sqrt(5))/2)))
CROSSREFS
Cf. A083035, A083036, A083037, A083038, A085002, A085003, A085004.
Cf. A094200, A094201.
KEYWORD
nonn
AUTHOR
Benoit Cloitre, Jun 17 2003
STATUS
approved
A085003 Partial sums of A085002. +10
5
1, 2, 2, 2, 2, 3, 4, 4, 4, 4, 5, 6, 7, 7, 7, 8, 9, 10, 10, 10, 11, 12, 13, 13, 13, 13, 14, 15, 15, 15, 15, 16, 17, 18, 18, 18, 19, 20, 21, 21, 21, 22, 23, 24, 24, 24, 24, 25, 26, 26, 26, 26, 27, 28, 28, 28, 28, 29, 30, 31, 31, 31, 32, 33, 34, 34, 34, 34, 35, 36, 36, 36, 36, 37, 38 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..10946
B. Cloitre, Graph of A085005(n) for n=1 up to 3874 [archive.org link]
FORMULA
a(n)=sum(k=1, n, A085002(k)).
a(A283766(n)) = n for all n >= 1. - Antti Karttunen, Mar 17 2017
MATHEMATICA
Accumulate[Table[Floor[GoldenRatio*n]-2*Floor[GoldenRatio*n/2], {n, 110}]] (* Harvey P. Dale, Dec 11 2012 *)
PROG
(PARI) sum(k=1, n, (k+sqrtint(5*k^2))%4>1) \\ Charles R Greathouse IV, Feb 07 2013
(Scheme, with memoization-macro definec)
(definec (A085003 n) (if (= 1 n) n (+ (A085002 n) (A085003 (- n 1)))))
;; Antti Karttunen, Mar 17 2017
CROSSREFS
Partial sums of A085002. A left inverse of A283766.
Cf. A083035, A083036, A083037, A083038, A085004, A085005.
KEYWORD
nonn
AUTHOR
Benoit Cloitre, Jun 17 2003
STATUS
approved
A085004 a(n)=2*A085003(n)-n. +10
3
1, 2, 1, 0, -1, 0, 1, 0, -1, -2, -1, 0, 1, 0, -1, 0, 1, 2, 1, 0, 1, 2, 3, 2, 1, 0, 1, 2, 1, 0, -1, 0, 1, 2, 1, 0, 1, 2, 3, 2, 1, 2, 3, 4, 3, 2, 1, 2, 3, 2, 1, 0, 1, 2, 1, 0, -1, 0, 1, 2, 1, 0, 1, 2, 3, 2, 1, 0, 1, 2, 1, 0, -1, 0, 1, 0, -1, -2, -1, 0, 1, 0, -1, 0, 1, 2, 1, 0, -1, 0, 1, 0, -1, -2, -1, 0, -1, -2, -3, -2, -1, 0, -1, -2, -1, 0, 1, 0, -1, 0, 1, 2, 1, 0, -1 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Table of n, a(n) for n=1..115.
FORMULA
|a(n+1)-a(n)| = 1. - Charles R Greathouse IV, Feb 07 2013
PROG
(PARI) 2*sum(k=1, n, (k+sqrtint(5*k^2))%4>1)-n \\ Charles R Greathouse IV, Feb 07 2013
CROSSREFS
Cf. A083035, A083036, A083037, A083038, A085003, A085005.
KEYWORD
sign
AUTHOR
Benoit Cloitre, Jun 17 2003
STATUS
approved
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Last modified August 30 11:14 EDT 2024. Contains 375543 sequences. (Running on oeis4.)