A194374 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - <k*r>) = 0, where r=sqrt(5) and < > denotes fractional part.

4, 8, 12, 16, 72, 76, 80, 84, 88, 144, 148, 152, 156, 160, 216, 220, 224, 228, 232, 288, 292, 296, 300, 304, 1292, 1296, 1300, 1304, 1308, 1364, 1368, 1372, 1376, 1380, 1436, 1440, 1444, 1448, 1452, 1508, 1512, 1516, 1520, 1524, 1580, 1584, 1588, 1592, 1596, 2584, 2588, 2592, 2596

Offset

1,1

Comments

See A194368.

Links

Table of n, a(n) for n=1..53.

Mathematica

r = Sqrt[5]; c = 1/2;
x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 100}];
Flatten[Position[t1, 1]]   (* empty *)
t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 800}];
Flatten[Position[t2, 1]]   (* A194374 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 100}];
Flatten[Position[t3, 1]]   (* A194375 *)

Prog

(PARI) isok(m) = my(r=sqrt(5)); sum(k=1, m, frac(1/2+k*r)-frac(k*r)) == 0; \\ Michel Marcus, Jan 31 2023

Crossrefs

Cf. A194368, A194375.

Sequence in context: A071072 A175670 A355031 * A061085 A007883 A023706

Adjacent sequences: A194371 A194372 A194373 * A194375 A194376 A194377

Keyword

nonn

Author

Clark Kimberling, Aug 23 2011

Extensions

More terms from Michel Marcus, Jan 31 2023

Status

approved