A283233 -id:A283233 - cp's OEIS frontend

A050140 a(n) = 2floor(nphi)-n, where phi = (1+sqrt(5))/2.

1, 4, 5, 8, 11, 12, 15, 16, 19, 22, 23, 26, 29, 30, 33, 34, 37, 40, 41, 44, 45, 48, 51, 52, 55, 58, 59, 62, 63, 66, 69, 70, 73, 76, 77, 80, 81, 84, 87, 88, 91, 92, 95, 98, 99, 102, 105, 106, 109, 110, 113, 116, 117, 120, 121, 124, 127, 128, 131, 134, 135, 138, 139

Offset: 1

Clark Kimberling

nonn

Old name was a(n) = last number in repeating block in continued fraction for n*phi.

Alfred Brousseau, Fibonacci and Related Number Theoretic Tables, The Fibonacci Association, 1972, 101-103.

Clark Kimberling, Table of n, a(n) for n = 1..1000
J.-P. Allouche and F. M. Dekking, Generalized Beatty sequences and complementary triples, arXiv:1809.03424 [math.NT], 2018.

Cf. A000201, A001622, A005206, A050141, A005614, A001350.

[-n + 2*Floor(n*(1+Sqrt(5))/2): n in [1..50]]; // G. C. Greubel, Oct 15 2017

Table[-n+2Floor[n*GoldenRatio],{n,1,100}]

for(n=1,50, print1(-n + 2*floor(n*(1+sqrt(5))/2), ", ")) \\ G. C. Greubel, Oct 15 2017

def A050140(n): return (n+isqrt(5*n**2)&-2)-n # Chai Wah Wu, Aug 25 2022

a(n) = -n + 2*floor(n*phi) = A283233(n)-n.
a(n) = floor(n*phi) + floor(n*sigma) where phi = (sqrt(5)+1)/2 and sigma = (sqrt(5)-1)/2.
a(n) = last number in repeating block in continued fraction for n*phi.

Formula and more terms from Vladeta Jovovic, Nov 23 2001

Name changed by Michel Dekking, Dec 27 2017

A283234 2*A001950.

Original entry on oeis.org

4, 10, 14, 20, 26, 30, 36, 40, 46, 52, 56, 62, 68, 72, 78, 82, 88, 94, 98, 104, 108, 114, 120, 124, 130, 136, 140, 146, 150, 156, 162, 166, 172, 178, 182, 188, 192, 198, 204, 208, 214, 218, 224, 230, 234, 240, 246, 250, 256, 260, 266, 272, 276, 282, 286, 292

Offset: 1

Clark Kimberling, Mar 03 2017

This is one of three sequences that partition the positive integers.  In general, suppose that r, s, t are positive real numbers for which the sets {i/r: i>=1}, {j/s: j>=1}, {k/t: k>=1} are pairwise disjoint.  Let a(n) be the rank of n/r when all the numbers in the three sets are jointly ranked.  Define b(n) and c(n) as the ranks of n/s and n/t.  It is easy to prove that
a(n)=n+[ns/r]+[nt/r],
b(n)=n+[nr/s]+[nt/s],
c(n)=n+[nr/t]+[ns/t], where [ ]=floor.
Taking r=1, s=(-1+sqrt(5))/2, t=(1+sqrt(5))/2 gives
a=A283233, b=A283234, c=A005843.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A000201, A283233, A005843.

r = 1; s = (-1 + 5^(1/2))/2; t = (1 + 5^(1/2))/2;
a[n_] := n + Floor[n*s/r] + Floor[n*t/r];
b[n_] := n + Floor[n*r/s] + Floor[n*t/s];
c[n_] := n + Floor[n*r/t] + Floor[n*s/t]
Table[a[n], {n, 1, 120}]  (* A283233 *)
Table[b[n], {n, 1, 120}]  (* A283234 *)
Table[c[n], {n, 1, 120}]  (* A005408 *)

from math import isqrt
def A283234(n): return ((n+isqrt(5*n**2))&-2)+(n<<1) # Chai Wah Wu, Aug 10 2022

a(n) = 2*floor(n*s), where r = (-1+sqrt(5))/2.

A287802 Positions of 0 in A287801; complement of A287803.

Original entry on oeis.org

2, 3, 4, 5, 8, 9, 11, 12, 13, 14, 17, 18, 19, 20, 23, 24, 26, 27, 28, 29, 32, 33, 35, 36, 37, 38, 41, 42, 43, 44, 47, 48, 50, 51, 52, 53, 56, 57, 58, 59, 62, 63, 65, 66, 67, 68, 71, 72, 74, 75, 76, 77, 80, 81, 82, 83, 86, 87, 89, 90, 91, 92, 95, 96, 98, 99

Offset: 1

Clark Kimberling, Jun 03 2017

Let d(n) = 3n/2 - a(n).  Then d(n) is in {-1/2, 0, 1/2, 1} for n >= 1.  Indeed,
d(n) = - 1/2 if n is in A075317; d(n) = 1/2 if n is in A075318;
d(n) = 0 if n is in A283234; d(n) = 1 if n is in A283233.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A075317, A075318, A283233, A283234, A287801, A287803.

s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0}}] &, {0}, 10] (* A003849 *)
w = StringJoin[Map[ToString, s]]
w1 = StringReplace[w, {"0" -> "100", "1" -> "001"}]
st = ToCharacterCode[w1] - 48    (* A287801 *)
Flatten[Position[st, 0]]  (* A287802 *)
Flatten[Position[st, 1]]  (* A287803 *)

cp's OEIS Frontend

A050140 a(n) = 2floor(nphi)-n, where phi = (1+sqrt(5))/2.

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A283234 2*A001950.

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A287802 Positions of 0 in A287801; complement of A287803.

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cp's OEIS Frontend

A050140 a(n) = 2*floor(n*phi)-n, where phi = (1+sqrt(5))/2.

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Magma

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A283234 2*A001950.

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A287802 Positions of 0 in A287801; complement of A287803.

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Mathematica

A050140 a(n) = 2floor(nphi)-n, where phi = (1+sqrt(5))/2.