A059555 -id:A059555 - cp's OEIS frontend

A186539 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f(i)=i^2 and g(j)=-2+3j^2. Complement of A186540.

1, 3, 4, 6, 7, 9, 11, 12, 14, 15, 17, 18, 20, 22, 23, 25, 26, 28, 29, 31, 33, 34, 36, 37, 39, 41, 42, 44, 45, 47, 48, 50, 52, 53, 55, 56, 58, 59, 61, 63, 64, 66, 67, 69, 70, 72, 74, 75, 77, 78, 80, 82, 83, 85, 86, 88, 89, 91, 93, 94, 96, 97, 99, 100, 102, 104, 105, 107, 108, 110, 111, 113, 115, 116, 118, 119, 121, 123, 124, 126, 127, 129, 130, 132, 134, 135, 137, 138, 140, 141, 143, 145, 146, 148, 149, 151, 153, 154, 156, 157

Offset: 1

Clark Kimberling, Feb 23 2011

nonn

See A186219 for a discussion of adjusted joint rank sequences.

Differs from A059555 at n=97, 123, 194, 220, 246, ... - R. J. Mathar, May 18 2020

			First, write
1..4..9..16..25..36..49.... (i^2)
.......10....25.....46.. (-2+3j^2)
Then replace each number by its rank, where ties are settled by ranking i^2 before -2+3j^2:
a=(1,3,4,6,7,9,11,12,14,15,17,18,..)=A186539
b=(2,5,8,10,13,16,19,21,24,27,30...)=A186540.

Cf. A186219, A186540, A186541, A186542.

(* adjusted joint rank sequences a and b, using general formula for ranking ui^2+vi+w and xj^2+yj+z *)
d = 1/2; u = 1; v = 0; w = 0; x = 3; y = 0; z = -2;
h[n_] := -y + (4 x (u*n^2 + v*n + w - z - d) + y^2)^(1/2);
a[n_] := n + Floor[h[n]/(2 x)];
k[n_] := -v + (4 u (x*n^2 + y*n + z - w + d) + v^2)^(1/2);
b[n_] := n + Floor[k[n]/(2 u)];
Table[a[n], {n, 1, 100}]  (* A186539 *)
Table[b[n], {n, 1, 100}]  (* A186540 *)

a(n)=n+floor(sqrt((1/3)n^2+1/24))=A186539(n).

b(n)=n+floor(sqrt(3n^2-3/2))=A186540(n).

A059556 Beatty sequence for 1 + 1/gamma.

Original entry on oeis.org

2, 5, 8, 10, 13, 16, 19, 21, 24, 27, 30, 32, 35, 38, 40, 43, 46, 49, 51, 54, 57, 60, 62, 65, 68, 71, 73, 76, 79, 81, 84, 87, 90, 92, 95, 98, 101, 103, 106, 109, 112, 114, 117, 120, 122, 125, 128, 131, 133, 136, 139, 142, 144, 147, 150, 153, 155, 158, 161, 163, 166

Offset: 1

Mitch Harris, Jan 22 2001

Differs from A054088 at indices 56, 71, 112, 127, 142, 168, 183 etc. - R. J. Mathar, Oct 05 2008

Let r = gamma (the Euler constant, 0.5772...). When {k*r, k >= 1} is jointly ranked with the positive integers, A059555(n) is the position of n and A059556(n) is the position of n*r. - Clark Kimberling, Oct 21 2014

Harry J. Smith, Table of n, a(n) for n = 1..2000
Aviezri S. Fraenkel, Jonathan Levitt, Michael Shimshoni, Characterization of the set of values f(n)=[n alpha], n=1,2,..., Discrete Math. 2 (1972), no.4, 335-345.
Eric Weisstein's World of Mathematics, Beatty Sequence.
Index entries for sequences related to Beatty sequences

Beatty complement is A059555.

t = N[Table[k*EulerGamma, {k, 1, 200}]]; u = Union[Range[200], t]
Flatten[Table[Flatten[Position[u, n]], {n, 1, 100}]]  (* A059556 *)
Flatten[Table[Flatten[Position[u, t[[n]]]], {n, 1, 100}]] (* A059555 *)
(* Clark Kimberling, Oct 21 2014 *)

{ default(realprecision, 100); b=1 + 1/Euler; for (n = 1, 2000, write("b059556.txt", n, " ", floor(n*b)); ) } \\ Harry J. Smith, Jun 28 2009

A250253 First row of spectral array W(gamma+1).

Original entry on oeis.org

1, 2, 3, 5, 7, 13, 20, 35, 55, 95, 149, 259, 408, 707, 1115, 1931, 3045, 5276, 8321, 14416, 22737, 39391, 62128, 107634, 169762, 294105, 463867, 803628, 1267494, 2195877, 3463371, 6000134, 9463505, 16395094

Offset: 1

Colin Barker, Nov 15 2014

Gamma is Euler's (or Euler-Mascheroni) constant.

A. Fraenkel and C. Kimberling, Generalized Wythoff arrays, shuffles and interspersions, Discrete Mathematics 126 (1994) 137-149.

Cf. A001620 (Gamma), A059555 (Corresponding Beatty sequence), A250254, A250255.

\\ Row i of the generalized Wythoff array W(h),
\\   where h is an irrational number between 1 and 2,
\\   and m is the number of terms in the vectors a and b.
row(h, i, m) = {
  if(h<=1 || h>=2, print("Invalid value for h"); return);
  my(
    a=vector(m, n, floor(n*h)),
    b=vector(m, n, floor(n*h/(h-1))),
    w=[a[a[i]], b[a[i]]],
    j=3
  );
  while(1,
    if(j%2==1,
      if(w[j-1]<=#a, w=concat(w, a[w[j-1]]), return(w))
    ,
      if(w[j-2]<=#b, w=concat(w, b[w[j-2]]), return(w))
    );
    j++
  )
}
allocatemem(10^9)
row(Euler+1, 1, 10^7)

cp's OEIS Frontend

A186539 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f(i)=i^2 and g(j)=-2+3j^2. Complement of A186540.

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A059556 Beatty sequence for 1 + 1/gamma.

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A250253 First row of spectral array W(gamma+1).

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