A302253 Positions of 3 in A190436.

8, 21, 29, 42, 55, 63, 76, 97, 110, 118, 131, 144, 152, 165, 186, 199, 207, 220, 241, 254, 262, 275, 288, 296, 309, 330, 343, 351, 364, 377, 385, 398, 406, 419, 432, 440, 453, 474, 487, 495, 508, 521, 529, 542, 563, 576, 584, 597, 618, 631, 639, 652, 665, 673, 686, 707, 720, 728

Offset: 1

Author

G. C. Greubel, Apr 04 2018

Keywords

• nonn

Comments

Write a(n) = [(bn+c)r] - b[nr] - [cr]. If r>0 and b and c are integers satisfying b >= 2 and 0 <= c <= b-1, then 0 <= a(n) <= b. The positions of 0 in the sequence a are of interest, as are the position sequences for 1,2,...,b. These b+1 position sequences comprise a partition of the positive integers.

Examples:

(golden ratio,2,0): A078588, A005653, A005652

(golden ratio,2,1): A190427-A190430

(golden ratio,3,0): A140397-A190400

(golden ratio,3,1): A140431-A190435

(golden ratio,3,2): A140436-A190439

(golden ratio,4,c): A140440-A190461

Links

• G. C. Greubel, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A190436, A190437, A190438, A190439.

Programs

Mathematica

r = GoldenRatio; b = 3; c = 2;
f[n_] := Floor[(b*n + c)*r] - b*Floor[n*r] - Floor[c*r];
t = Table[f[n], {n, 1, 500}] (* A190436 *)
Flatten[Position[t, 0]] (* A190437 *)
Flatten[Position[t, 1]] (* A190438 *)
Flatten[Position[t, 2]] (* A190439 *)
Flatten[Position[t, 3]] (* A302253 *)