Search: a083036 -id:a083036
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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
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OFFSET
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1,8
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LINKS
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Kevin O'Bryant, Bruce Reznick and Monika Serbinowska, Almost alternating sums, Amer. Math. Monthly, Vol. 113 (October 2006), pp. 673-688.
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FORMULA
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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
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PROG
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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A083035
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a(n) = floor(sqrt(2)*n)-2*floor(n/sqrt(2)).
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+10
14
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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
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OFFSET
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1,1
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LINKS
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FORMULA
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a(n) = floor(n*sqrt(2)) mod 2. - T. D. Noe, Oct 11 2006
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MATHEMATICA
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Table[Floor[Sqrt[2]n]-2Floor[n/Sqrt[2]], {n, 120}] (* Harvey P. Dale, Mar 08 2017 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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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
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OFFSET
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1,7
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COMMENTS
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Sequence presents fractal patterns.
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LINKS
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FORMULA
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A085002
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a(n) = floor(phi*n) - 2*floor(phi*n/2) where phi is the golden ratio.
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+10
7
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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
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OFFSET
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1,1
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COMMENTS
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A fractal sequence.
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)
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LINKS
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FORMULA
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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)
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MATHEMATICA
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Table[Floor[GoldenRatio n] - 2 Floor[GoldenRatio n/2], {n, 110}] (* Harvey P. Dale, Dec 11 2012 *)
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PROG
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(Python)
from math import isqrt
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CROSSREFS
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Characteristic function of A283766.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A085005
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A Von Koch curve related to the Golden ratio.
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+10
6
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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
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OFFSET
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1,2
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LINKS
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FORMULA
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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)
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PROG
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(PARI) a(n)=(-1)*sum(i=1, n, sum(j=1, i, (-1)^floor(j*(1+sqrt(5))/2)))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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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
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OFFSET
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1,2
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LINKS
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FORMULA
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MATHEMATICA
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Accumulate[Table[Floor[GoldenRatio*n]-2*Floor[GoldenRatio*n/2], {n, 110}]] (* Harvey P. Dale, Dec 11 2012 *)
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PROG
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(Scheme, with memoization-macro definec)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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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
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listen;
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OFFSET
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1,2
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LINKS
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FORMULA
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PROG
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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