A255977 -id:A255977 - 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

A084532 Signature sequence of 1/phi = phi-1 = (sqrt(5)-1)/2 = 0.61803...

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 3, 2, 1, 3, 2, 4, 1, 3, 2, 4, 1, 3, 5, 2, 4, 1, 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, 8, 5, 2, 7, 4, 1, 6, 3, 8, 5, 2, 7, 4, 9, 1, 6, 3, 8, 5, 2, 7, 4, 9, 1, 6, 3, 8, 5, 10, 2, 7, 4, 9, 1, 6, 3, 8, 5, 10, 2, 7, 4, 9, 1, 6, 11, 3, 8, 5, 10, 2, 7, 4, 9

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.

T. D. Noe, Table of n, a(n) for n = 1..1000
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. A084531.

a(A054347(n) + A255977(m) + m*n) = n - 1. - Alan Michael Gómez Calderón, Dec 09 2024

A022780 Place where n-th 1 occurs in A023118.

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 13, 17, 21, 26, 31, 36, 42, 48, 55, 62, 70, 78, 87, 96, 105, 115, 125, 136, 147, 159, 171, 184, 197, 210, 224, 238, 253, 268, 284, 300, 317, 334, 351, 369, 387, 406, 425, 445, 465, 486, 507, 529, 551, 573, 596, 619, 643, 667, 692

Offset: 1

Clark Kimberling

nonn

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A022776, A022778, A255977.

Table[n + 1 + Sum[Floor[(n - k)/Sqrt[5]], {k, 0, n}], {n, 0, 200}] (* A022780 *)
(* Clark Kimberling, Mar 14 2015 *)

a(n) = n + 1 + Sum{floor[(n - k)/sqrt(5)], k = 0..n}.

A283734 Rank array, R, of the golden ratio, read by antidiagonals downwards.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 6, 8, 10, 12, 9, 11, 14, 16, 19, 13, 15, 18, 21, 24, 28, 17, 20, 23, 26, 30, 34, 38, 22, 25, 29, 32, 36, 41, 45, 50, 27, 31, 35, 39, 43, 48, 53, 58, 63, 33, 37, 42, 46, 51, 56, 61, 67, 72, 78, 40, 44, 49, 54, 59, 65, 70, 76, 82, 88, 95, 47

Offset: 1

Clark Kimberling, Mar 16 2017

Every row intersperses all other rows, and every column intersperses all other columns. The array is the dispersion of the complement of column 1; column 1 is given by r(n) = r(n-1) + 1 + L(n), where L = lower Wythoff sequence (A000201).

			The corner of R begins:
1    2    4    6    9    13    17    22
3    5    8    11   15   20    25    31
7    10   14   18   23   29    35    42
12   16   21   26   32   39    46    54
19   24   30   36   43   51    59    68
28   34   41   48   56   65    74    84
38   45   53   61   70   80    90    101
50   58   67   76   86   97    108   120
Let t = golden ratio = (1 + sqrt(5))/2; then R(i,j) = rank of (j,i) when all nonnegative integer pairs (a,b) are ranked by the relation << defined as follows: (a,b) << (c,d) if a + b*t < c + d*t, and also (a,b) << (c,d) if a + b*t = c + d*t and b < d.  Thus R(2,1) = 10 is the rank of (1,2) in the list (0,0) << (1,0) << (0,1) << (2,0) << (1,1) << (3,0) << (0,2) << (2,1) << (4,0) << (1,2).

Clark Kimberling, Antidiagonals n = 1..60, flattened
Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.

Cf. A001622, A255977 (row 1), A283733 (column 1), A000201, A087465.

r = 40; r1 = 12;(*r=# rows of T,r1=# rows to show*);
c = 40; c1 = 12;(*c=# cols of T,c1=# cols to show*);
s[0] = 1; s[n_] := s[n] = s[n - 1] + 1 + Floor[n*GoldenRatio];
u = Table[s[n], {n, 0, 400}] (* A283733 *)
v = Complement[Range[Max[u]], u];
f[n_] := v[[n]]; Table[f[n], {n, 1, 30}]
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1,
Length[Union[list]]]; rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
w[i_, j_] := rows[[i, j]];
TableForm[Table[w[i, j], {i, 1, r1}, {j, 1, c1}]]   (* A283734, array *)
Flatten[Table[w[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A283734, sequence *)
TableForm[Table[w[i, 1] + w[1, j] + (i - 1)*(j - 1) - 1, {i, 1, r1}, {j, 1, c1}]] (* A283734, array, by formula *)

R(i,j) = R(i,0) + R(0,j) + i*j - 1, for i>=1, j>=1.

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

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A084532 Signature sequence of 1/phi = phi-1 = (sqrt(5)-1)/2 = 0.61803...

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A022780 Place where n-th 1 occurs in A023118.

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A283734 Rank array, R, of the golden ratio, read by antidiagonals downwards.

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