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284, 287, 289, 292, 294, 296, 299, 301, 304, 306, 309, 311, 313, 316, 318, 321, 323, 325, 328, 330, 333, 335, 337, 340, 342, 345, 347, 350, 352, 354, 357, 359, 362, 364, 366, 369, 371, 374, 376, 379, 381, 383, 386, 388, 391, 393, 395, 398, 400
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OFFSET
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1,1
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COMMENTS
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This is the third of four sequences that partition the positive integers. Starting with a general overview, suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers. Let u' and v' be their complements, and assume that the following four sequences are infinite:
(1) u ^ v = intersection of u and v (in increasing order);
(2) u ^ v';
(3) u' ^ v;
(4) u' ^ v'.
Every positive integer is in exactly one of the four sequences.
Assume that if w is any of the sequences u, v, u', v', then lim_{n->oo) w(n)/n exists and defines the (limiting) density of w. For w = u,v,u',v', denote the densities by r,s,r',s'. Then the densities of sequences (1)-(4) exist, and
1/(r*r') + 1/(r*s') + 1/(s*s') + 1/(s*r') = 1.
For A356219, u, v, u', v', are the Beatty sequences given by u(n) = floor(n*sqrt(2)) and v(n) = floor((1+sqrt(2))/2)*n), so that r = sqrt(2), s = (1+sqrt(2))/2, r' = (2+sqrt(2))/2, s' = 1 + 1/sqrt(2).
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LINKS
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Table of n, a(n) for n=1..49.
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EXAMPLE
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(1) u ^ v = (2, 4, 7, 9, 12, 14, 16, 19, 21, 24, 26, 28, 31, 33, ...) = A003151
(2) u ^ v' = (1, 5, 8, 11, 15, 18, 22, 25, 29, 32, 35, 39, 42, ...) = A001954
(3) u' ^ v = (284, 287, 289, 292, 294, 296, 299, 301, 304, 306, ...) = A356219
(4) u' ^ v' = (3, 6, 10, 13, 17, 20, 23, 27, 30, 34, 37, 40, 44, ...) = A003152
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MATHEMATICA
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z = 200;
r = Sqrt[2]; u = Table[Floor[n*r], {n, 1, z}] (* A001951 *)
u1 = Take[Complement[Range[1000], u], z] (* A001952 *)
r1 = 1 + Sqrt[2]; v = Table[Floor[n*r1], {n, 1, z}] (* A003151 *)
v1 = Take[Complement[Range[1000], v], z] (* A003152 *)
t1 = Intersection[u, v] (* A003151 *)
t2 = Intersection[u, v1] (* A001954 *)
t3 = Intersection[u1, v] (* A356219 *)
t4 = Intersection[u1, v1] (* A001952 *)
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CROSSREFS
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Cf. A001951, A001952, A003151, A003152, A001954, A184922 (results of compositions instead of intersections), A341239 (reversed compositions).
Sequence in context: A263670 A108826 A285890 * A061310 A333930 A259996
Adjacent sequences: A356216 A356217 A356218 * A356220 A356221 A356222
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KEYWORD
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nonn,easy
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AUTHOR
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Clark Kimberling, Nov 13 2022
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STATUS
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approved
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