A084531 Signature sequence of phi = (1+sqrt(5))/2 = 1.61803...

1, 2, 1, 3, 2, 4, 1, 3, 5, 2, 4, 1, 6, 3, 5, 2, 7, 4, 1, 6, 3, 8, 5, 2, 7, 4, 9, 1, 6, 3, 8, 5, 10, 2, 7, 4, 9, 1, 6, 11, 3, 8, 5, 10, 2, 7, 12, 4, 9, 1, 6, 11, 3, 8, 13, 5, 10, 2, 7, 12, 4, 9, 1, 14, 6, 11, 3, 8, 13, 5, 10, 2, 15, 7, 12, 4, 9, 1, 14, 6, 11, 3, 16, 8, 13, 5, 10, 2, 15, 7, 12, 4, 17, 9, 1

Offset: 1

Author

Henry Bottomley, May 28 2003

Keywords

• nonn

Comments

Arrange the numbers i+j*x (i,j >= 1) in increasing order; the sequence of i's is the signature of x; the sequence of j's is the signature of 1/x.

As a fractal sequence, if the first occurrence of each n is deleted, the remaining sequence is the original. That is, the upper trim of A084531 is A084531. Also, the lower trim of A084531 is A084531, meaning that if 1 is subtracted from every term and then all 0's are deleted, the result is the original sequence. Every fractal sequence begets an interspersion; the interspersion of A084531 is A167267. - Clark Kimberling, Oct 31 2009

The positions of the first occurrence of i in this sequence, i>=1, form sequence A255977. That is, 1 occurs for the first time at position 1, 2 at position 2, 3 at position 4, 4 at position 6, and 1,2,4,6, ... is A255977. - Jeffrey Shallit, Jun 28 2024

References

• Clark Kimberling, "Fractal Sequences and Interspersions," Ars Combinatoria 45 (1997) 157-168.

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Glen Joyce C. Dulatre, Jamilah V. Alarcon, Vhenedict M. Florida, and Daisy Ann A. Disu, On Fractal Sequences, DMMMSU-CAS Science Monitor (2016-2017) Vol. 15 No. 2, 109-113.
• Casey Mongoven, Sonification of multiple Fibonacci-related sequences, Annales Mathematicae et Informaticae, 41 (2013) pp. 175-192.
• Index entries for sequences related to signature sequences

Crossrefs

Cf. A084532, A167267, A255977.

Programs

Mathematica

x = GoldenRatio; Take[Transpose[Sort[Flatten[Table[{i + j*x, i}, {i, 30}, {j, 20}], 1], #1[[1]] < #2[[1]] &]][[2]], 100] (* Clark Kimberling, Nov 10 2012 *)

Formula

a(A054347(n) + A255977(m) + m*n) = m. - Alan Michael Gómez Calderón, Nov 21 2024