A190248 -id:A190248 - cp's OEIS frontend

A190249 Positions of 0 in A190248.

2, 5, 10, 13, 18, 23, 26, 31, 34, 36, 39, 44, 47, 52, 57, 60, 65, 68, 73, 78, 81, 86, 89, 91, 94, 99, 102, 107, 112, 115, 120, 123, 128, 133, 136, 141, 146, 149, 154, 157, 162, 167, 170, 175, 178, 180, 183, 188, 191, 196, 201, 204, 209, 212, 217, 222, 225, 230, 233, 235, 238, 243, 246, 251, 256, 259, 264, 267, 272, 277, 280, 285, 290

Offset: 1

Clark Kimberling, May 06 2011

nonn

G. C. Greubel, Table of n, a(n) for n = 1..2700
Burghard Herrmann, How integer sequences find their way into areas outside pure mathematics, The Fibonacci Quarterly (2019) Vol. 57, No. 5, 67-71.

Cf. A190248, A190250, A190251.

u = GoldenRatio; v = u^2; w=u^3;
f[n_] := Floor[n*u + n*v + n*w] - Floor[n*u] - Floor[n*v] - Floor[n*w]
t = Table[f[n], {n, 1, 120}] (* A190248 *)
Flatten[Position[t, 0]]      (* A190249 *)
Flatten[Position[t, 1]]      (* A190250 *)
Flatten[Position[t, 2]]      (* A190251 *)

A190251 Positions of 2 in A190248.

Original entry on oeis.org

3, 8, 11, 16, 21, 24, 29, 32, 37, 42, 45, 50, 53, 55, 58, 63, 66, 71, 76, 79, 84, 87, 92, 97, 100, 105, 110, 113, 118, 121, 126, 131, 134, 139, 142, 144, 147, 152, 155, 160, 165, 168, 173, 176, 181, 186, 189, 194, 197, 199, 202, 207, 210, 215, 220, 223, 228, 231, 236, 241, 244, 249, 254, 257, 262, 265, 270, 275, 278, 283, 286, 288

Offset: 1

Clark Kimberling, May 06 2011

nonn

G. C. Greubel, Table of n, a(n) for n = 1..2700
Burghard Herrmann, How integer sequences find their way into areas outside pure mathematics, The Fibonacci Quarterly (2019) Vol. 57, No. 5, 67-71.

Cf. A190248, A190249, A190250.

u = GoldenRatio; v = u^2; w=u^3;
f[n_] := Floor[n*u + n*v + n*w] - Floor[n*u] - Floor[n*v] - Floor[n*w]
t = Table[f[n], {n, 1, 120}] (* A190248 *)
Flatten[Position[t, 0]]      (* A190249 *)
Flatten[Position[t, 1]]      (* A190250 *)
Flatten[Position[t, 2]]      (* A190251 *)

A190250 Positions of 1 in A190248.

Original entry on oeis.org

1, 4, 6, 7, 9, 12, 14, 15, 17, 19, 20, 22, 25, 27, 28, 30, 33, 35, 38, 40, 41, 43, 46, 48, 49, 51, 54, 56, 59, 61, 62, 64, 67, 69, 70, 72, 74, 75, 77, 80, 82, 83, 85, 88, 90, 93, 95, 96, 98, 101, 103, 104, 106, 108, 109, 111, 114, 116, 117, 119, 122, 124, 125, 127, 129, 130, 132, 135, 137, 138, 140, 143, 145, 148, 150, 151, 153, 156, 158, 159, 161

Offset: 1

Clark Kimberling, May 06 2011

nonn

Numbers n such that 1/4 < {n*phi} < 3/4, where phi is the golden ratio (1+sqrt(5))/2 and { } denotes fractional part. - Burghard Herrmann, Nov 14 2017

G. C. Greubel, Table of n, a(n) for n = 1..5500
Burghard Herrmann, Characterization of some golden ratio sequences
Burghard Herrmann, How integer sequences find their way into areas outside pure mathematics, The Fibonacci Quarterly (2019) Vol. 57, No. 5, 67-71.

Cf. A190248, A190249, A190251.

u = GoldenRatio; v = u^2; w=u^3;
f[n_] := Floor[n*u + n*v + n*w] - Floor[n*u] - Floor[n*v] - Floor[n*w]
t = Table[f[n], {n, 1, 120}] (* A190248 *)
Flatten[Position[t, 0]]      (* A190249 *)
Flatten[Position[t, 1]]      (* A190250 *)
Flatten[Position[t, 2]]      (* A190251 *)

isok(n) = my(u=(1+sqrt(5))/2); floor(2*n+4*n*u)-floor(n*u)-floor(n+n*u)-floor(n+2*n*u) == 1; \\ Michel Marcus, Nov 14 2017

A190440 [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(golden ratio,4,0) and []=floor.

Original entry on oeis.org

2, 0, 3, 1, 0, 2, 1, 3, 2, 0, 3, 1, 0, 2, 1, 3, 2, 0, 2, 1, 3, 2, 0, 3, 1, 0, 2, 1, 3, 2, 0, 3, 1, 0, 2, 0, 3, 1, 0, 2, 1, 3, 2, 0, 3, 1, 0, 2, 1, 3, 2, 0, 3, 1, 3, 2, 0, 3, 1, 0, 2, 1, 3, 2, 0, 3, 1, 0, 2, 1, 3, 1, 0, 2, 1, 3, 2, 0, 3, 1, 0, 2, 1, 3, 2, 0, 3, 1, 0, 2, 0, 3, 1, 0, 2, 1, 3, 2, 0, 3, 1, 0, 2, 1, 3, 2, 0, 2, 1, 3, 2, 0

Offset: 1

Clark Kimberling, May 10 2011

nonn

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

Cf. A190889, A190442, A190443, A190251.

r = GoldenRatio;
f[n_] := Floor[4*n*r] - 4*Floor[n*r];
t = Table[f[n], {n, 1, 320}] (* A190440 *)
Flatten[Position[t, 0]]  (* A190240 *)
Flatten[Position[t, 1]]  (* A190249 *)
Flatten[Position[t, 2]]  (* A190442 *)
Flatten[Position[t, 3]]  (* A190443 *)
Flatten[Position[t, 4]]  (* A190248 *)

a(n)=[4nr]-4[nr], where r=golden ratio.

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A190249 Positions of 0 in A190248.

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A190251 Positions of 2 in A190248.

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A190250 Positions of 1 in A190248.

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A190440 [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(golden ratio,4,0) and []=floor.

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