A084532 -id:A084532 - cp's OEIS frontend

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

Henry Bottomley, May 28 2003

nonn

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

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

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

Cf. A084532, A167267, A255977.

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 *)

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

A118276 Signature sequence of Phi^2 = 2.618033989... (A104457), where Phi is the golden ratio A001622.

Original entry on oeis.org

1, 2, 3, 1, 4, 2, 5, 3, 6, 1, 4, 7, 2, 5, 8, 3, 6, 1, 9, 4, 7, 2, 10, 5, 8, 3, 11, 6, 1, 9, 4, 12, 7, 2, 10, 5, 13, 8, 3, 11, 6, 14, 1, 9, 4, 12, 7, 15, 2, 10, 5, 13, 8, 16, 3, 11, 6, 14, 1, 9, 17, 4, 12, 7, 15, 2, 10, 18, 5, 13, 8, 16, 3, 11, 19, 6, 14, 1, 9, 17, 4, 12, 20, 7, 15, 2, 10, 18, 5, 13

Offset: 1

Casey Mongoven, Apr 21 2006

nonn

Equals A023119 in the first 98 terms, then the sequences differ. [From R. J. Mathar, Aug 08 2008]

C. Kimberling, "Fractal Sequences and Interspersions", Ars Combinatoria, vol. 45 p 157 1997.

T. D. Noe, Table of n, a(n) for n=1..1000
Eric Weisstein's World of Mathematics, Signature Sequence
Index entries for sequences related to signature sequences

Cf. A084531, A084532, A167970.

terms = 90; m = Ceiling[Sqrt[terms]]; s0 = {}; While[s = (Table[i + j*GoldenRatio^2, {i, 1, m}, {j, 1, m}] // Flatten // SortBy[#, N] &)[[1 ;; terms]] /. GoldenRatio -> 0; s != s0, s0 = s; m = 2 m]; s (* Jean-François Alcover, Jan 08 2017 *)

A167968 Signature sequence of phi^4 = 0.14589803375032..., where phi is the golden ratio minus 1 (0.61803398874989...).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 4, 3, 2, 1, 4, 3, 2, 1, 4, 3, 2, 1, 4, 3, 2, 1, 4, 3, 2, 1, 4, 3, 2, 1, 4, 3, 2, 1, 5, 4, 3, 2, 1, 5, 4, 3, 2, 1, 5, 4, 3, 2, 1, 5, 4, 3, 2, 1, 5, 4, 3, 2, 1, 5, 4, 3, 2, 1, 5, 4, 3, 2, 1

Offset: 1

Casey Mongoven, Nov 15 2009

nonn

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

Index entries for sequences related to signature sequences

Cf. A084531, A084532, A167972.

terms = 105; m = Ceiling[Sqrt[terms]]; s0 = {}; While[s = (Table[i + j*(GoldenRatio-1)^4, {i, 1, m}, {j, 1, m}] // Flatten // SortBy[#, N]&)[[1 ;; terms]] /. GoldenRatio -> 1; s != s0, s0 = s; m = 2m]; s (* Jean-François Alcover, Jan 08 2017 *)

A167969 Signature sequence of phi^3 = 0.23606797749979..., where phi is the golden ratio 0.61803398874989... .

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 6, 2, 3, 4, 5, 1, 6, 2, 3, 4, 5, 1, 6, 2, 3, 4, 5, 1, 6, 2, 3, 4, 5, 1, 6, 2, 7, 3, 4, 5, 1, 6, 2, 7, 3, 4, 5, 1, 6

Offset: 1

Casey Mongoven, Nov 15 2009

nonn

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

Index entries for sequences related to signature sequences

Cf. A084531, A084532, A167971.

terms = 105; m = Ceiling[Sqrt[terms]]; s0 = {}; While[s = (Table[i + j/GoldenRatio^3, {i, 1, m}, {j, 1, m}] // Flatten // SortBy[#, N] &)[[1 ;; terms]] /. GoldenRatio -> \[Infinity]; s != s0, s0 = s; m = 2 m]; s (* Jean-François Alcover, Jan 08 2017 *)

A167970 Signature sequence of phi^2 = 0.38196601125011..., where phi is the golden ratio 0.61803398874989... .

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 4, 1, 3, 2, 4, 1, 3, 2, 4, 1, 3, 5, 2, 4, 1, 3, 5, 2, 4, 1, 3, 5, 2, 4, 1, 6, 3, 5, 2, 4, 1, 6, 3, 5, 2, 4, 1, 6, 3, 5, 2, 7, 4, 1, 6, 3, 5, 2, 7, 4, 1, 6, 3, 5, 2, 7, 4, 1, 6, 3, 8, 5, 2, 7, 4, 1, 6, 3, 8, 5, 2, 7, 4, 1, 6, 3, 8, 5, 2, 7, 4, 9, 1, 6, 3, 8, 5, 2

Offset: 1

Casey Mongoven, Nov 15 2009

nonn

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

Index entries for sequences related to signature sequences

Cf. A084531, A084532, A118276.

terms = 105; m = Ceiling[Sqrt[terms]]; s0 = {}; While[s = (Table[i + j/GoldenRatio^2, {i, 1, m}, {j, 1, m}] // Flatten // SortBy[#, N] &)[[1 ;; terms]] /. GoldenRatio -> \[Infinity]; s != s0, s0 = s; m = 2 m]; s (* Jean-François Alcover, Jan 08 2017 *)

A167971 Signature sequence of Phi^3 = 4.2360679774998..., where Phi is the golden ratio 1.6180339887499... .

Original entry on oeis.org

1, 2, 3, 4, 5, 1, 6, 2, 7, 3, 8, 4, 9, 5, 1, 10, 6, 2, 11, 7, 3, 12, 8, 4, 13, 9, 5, 1, 14, 10, 6, 2, 15, 11, 7, 3, 16, 12, 8, 4, 17, 13, 9, 5, 1, 18, 14, 10, 6, 2, 19, 15, 11, 7, 3, 20, 16, 12, 8, 4, 21, 17, 13, 9, 5, 22, 1, 18, 14, 10, 6, 23, 2, 19, 15, 11, 7, 24, 3, 20, 16, 12, 8, 25, 4, 21

Offset: 1

Casey Mongoven, Nov 15 2009

nonn

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

Index entries for sequences related to signature sequences

Cf. A084531, A084532, A167969.

terms = 86; m = Ceiling[Sqrt[terms]]; s0 = {}; While[s = (Table[i + j*GoldenRatio^3, {i, 1, m}, {j, 1, m}] // Flatten // SortBy[#, N] &)[[1 ;; terms]] /. GoldenRatio -> 0; s != s0, s0 = s; m = 2 m]; s (* Jean-François Alcover, Jan 08 2017 *)

A167972 Signature sequence of Phi^4 = 6.8541019662497..., where Phi is the golden ratio 1.6180339887499... .

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 1, 8, 2, 9, 3, 10, 4, 11, 5, 12, 6, 13, 7, 14, 1, 8, 15, 2, 9, 16, 3, 10, 17, 4, 11, 18, 5, 12, 19, 6, 13, 20, 7, 14, 21, 1, 8, 15, 22, 2, 9, 16, 23, 3, 10, 17, 24, 4, 11, 18, 25, 5, 12, 19, 26, 6, 13, 20, 27, 7, 14, 21, 28, 1, 8, 15, 22, 29, 2, 9, 16, 23, 30, 3, 10, 17

Offset: 1

Casey Mongoven, Nov 15 2009

nonn

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

Index entries for sequences related to signature sequences

Cf. A084531, A084532, A167968.

terms = 83; m = Ceiling[Sqrt[terms]]; s0 = {}; While[s = (Table[i + j*GoldenRatio^4, {i, 1, m}, {j, 1, m}] // Flatten // SortBy[#, N] &)[[1 ;; terms]] /. GoldenRatio -> 0; s != s0, s0 = s; m = 2 m]; s (* Jean-François Alcover, Jan 08 2017 *)

A167973 Signature sequence of Phi^5 = 11.090169943749..., where Phi is the golden ratio 1.6180339887499... .

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 13, 2, 14, 3, 15, 4, 16, 5, 17, 6, 18, 7, 19, 8, 20, 9, 21, 10, 22, 11, 23, 12, 1, 24, 13, 2, 25, 14, 3, 26, 15, 4, 27, 16, 5, 28, 17, 6, 29, 18, 7, 30, 19, 8, 31, 20, 9, 32, 21, 10, 33, 22, 11, 34, 23, 12, 1, 35, 24, 13, 2, 36, 25, 14, 3, 37, 26

Offset: 1

Casey Mongoven, Nov 15 2009

nonn

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

Index entries for sequences related to signature sequences

Cf. A084531, A084532.

terms = 80; m = Ceiling[Sqrt[terms]]; s0 = {}; While[s = (Table[i + j*GoldenRatio^5, {i, 1, m}, {j, 1, m}] // Flatten // SortBy[#, N] &)[[1 ;; terms]] /. GoldenRatio -> 0; s != s0, s0 = s; m = 2 m]; s (* Jean-François Alcover, Jan 08 2017 *)

A167974 Signature sequence of Phi^6 = 17.944271909999..., where Phi is the golden ratio 1.6180339887499... .

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 1, 19, 2, 20, 3, 21, 4, 22, 5, 23, 6, 24, 7, 25, 8, 26, 9, 27, 10, 28, 11, 29, 12, 30, 13, 31, 14, 32, 15, 33, 16, 34, 17, 35, 18, 36, 1, 19, 37, 2, 20, 38, 3, 21, 39, 4, 22, 40, 5, 23, 41, 6, 24, 42, 7, 25, 43, 8, 26, 44, 9

Offset: 1

Casey Mongoven, Nov 15 2009

nonn

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

Index entries for sequences related to signature sequences

Cf. A084531, A084532.

terms = 79; m = Ceiling[Sqrt[terms]]; s0 = {}; While[s = (Table[i + j*GoldenRatio^6, {i, 1, m}, {j, 1, m}] // Flatten // SortBy[#, N] &)[[1 ;; terms]] /. GoldenRatio -> 0; s != s0, s0 = s; m = 2 m]; s (* Jean-François Alcover, Jan 08 2017 *)

A167966 Signature sequence of phi^6 = 0.055728090000841..., where phi is the inverse golden ratio A094214.

Original entry on oeis.org

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

Offset: 1

Casey Mongoven, Nov 15 2009

nonn

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

Index entries for sequences related to signature sequences

Cf. A084531, A084532, A167974.

terms = 105; m = Ceiling[Sqrt[terms]]; s0 = {}; While[s = (Table[i + j/GoldenRatio^6, {i, 1, m}, {j, 1, m}] // Flatten // SortBy[#, N] &)[[1 ;; terms]] /. GoldenRatio -> \[Infinity]; s != s0, s0 = s; m = 2 m]; s (* Jean-François Alcover, Jan 08 2017 *)

cp's OEIS Frontend

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

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A118276 Signature sequence of Phi^2 = 2.618033989... (A104457), where Phi is the golden ratio A001622.

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A167968 Signature sequence of phi^4 = 0.14589803375032..., where phi is the golden ratio minus 1 (0.61803398874989...).

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A167969 Signature sequence of phi^3 = 0.23606797749979..., where phi is the golden ratio 0.61803398874989... .

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A167970 Signature sequence of phi^2 = 0.38196601125011..., where phi is the golden ratio 0.61803398874989... .

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A167971 Signature sequence of Phi^3 = 4.2360679774998..., where Phi is the golden ratio 1.6180339887499... .

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A167972 Signature sequence of Phi^4 = 6.8541019662497..., where Phi is the golden ratio 1.6180339887499... .

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A167973 Signature sequence of Phi^5 = 11.090169943749..., where Phi is the golden ratio 1.6180339887499... .

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References

Links

Crossrefs

Programs

Mathematica

A167974 Signature sequence of Phi^6 = 17.944271909999..., where Phi is the golden ratio 1.6180339887499... .

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