A054347 -id:A054347 - cp's OEIS frontend

A194054 Natural interspersion of A054347; a rectangular array, by antidiagonals.

1, 4, 2, 8, 5, 3, 14, 9, 6, 7, 22, 15, 10, 11, 12, 31, 23, 16, 17, 18, 13, 42, 32, 24, 25, 26, 19, 20, 54, 43, 33, 34, 35, 27, 28, 21, 68, 55, 44, 45, 46, 36, 37, 29, 30, 84, 69, 56, 57, 58, 47, 48, 38, 39, 40, 101, 85, 70, 71, 72, 59, 60, 49, 50, 51, 41, 120, 102

Offset: 1

Clark Kimberling, Aug 15 2011

See A194029 for definitions of natural fractal sequence and natural interspersion. Every positive integer occurs exactly once (and every pair of rows intersperse), so that as a sequence, A194054 is a permutation of the positive integers; its inverse is A194055.

			Northwest corner:
1...4...8...14...22...31
2...5...9...15...23...32
3...6...10..16...24...33
7...11..17..25...34...45

Cf. A194029, A054347, A194053, A194058.

z = 40; g = GoldenRatio
c[k_] := Sum[Floor[j*g], {j, 1, k}];
c = Table[c[k], {k, 1, z}]  (* A054347 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
f = Table[f[n], {n, 1, 800}]  (* A194053 *)
r[n_] := Flatten[Position[f, n]]
t[n_, k_] := r[n][[k]]
TableForm[Table[t[n, k], {n, 1, 8}, {k, 1, 7}]]
p = Flatten[Table[t[k, n - k + 1], {n, 1, 16}, {k, 1, n}]]  (* A194054 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 80}]]  (* A194058 *)

A194053 Natural fractal sequence of A054347.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Aug 15 2011

nonn

See A194029 for definitions of natural fractal sequence and natural interspersion.

Cf. A194029.

z = 40; g = GoldenRatio
c[k_] := Sum[Floor[j*g], {j, 1, k}];
c = Table[c[k], {k, 1, z}]  (* A054347 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
f = Table[f[n], {n, 1, 800}]  (* A194053 *)
r[n_] := Flatten[Position[f, n]]
t[n_, k_] := r[n][[k]]
TableForm[Table[t[n, k], {n, 1, 8}, {k, 1, 7}]]
p = Flatten[Table[t[k, n - k + 1], {n, 1, 16}, {k, 1, n}]]  (* A194054 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 80}]]  (* A194058 *)

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

Original entry on oeis.org

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

A255977 The number of numbers j+k*r <= n, where r = golden ratio and j and k are nonnegative integers.

Original entry on oeis.org

1, 2, 4, 6, 9, 13, 17, 22, 27, 33, 40, 47, 55, 64, 73, 83, 93, 104, 116, 128, 141, 154, 168, 183, 198, 214, 231, 248, 266, 284, 303, 323, 343, 364, 386, 408, 431, 454, 478, 503, 528, 554, 580, 607, 635, 663, 692, 722, 752, 783, 814, 846, 879, 912, 946, 980

Offset: 1

Clark Kimberling, Mar 14 2015

The difference sequence is A019446.
From Thomas Anton, Oct 22 2018: (Start)
It appears that this sequence (apart from the first term) can be obtained through the following sieving process. Start with the positive integers. Then, at each stage, circle the first remaining number that has not already been circled, and delete all terms in the subsequence of terms that were not circled in previous stages with circled indices that have not yet been deleted. E.g., the first few iterations are
  1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...
We circle 1, and take the subsequence of previously uncircled numbers, which is the entire sequence, and delete all terms with circled indices that have not been deleted, in this case, just the 1st, 1.
  2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ...
We circle 2, and take the previously uncircled subsequence, which is again the entire sequence, and delete all terms with circled indices that have not been deleted, just the 2nd, 3.
  (2), 4, 5, 6, 7, 8, 9, 10, 11, 12, ...
We circle 4, and take the previously uncircled subsequence, which is all terms of the sequence except 2, and delete all terms in that subsequence with circled indices (as terms of the subsequence) that have not been deleted, the 2nd and 4th, respectively the 3rd and 5th terms of the entire sequence, 5 and 7.
  (2), (4), 6, 8, 9, 10, 11, 12, 13, 14, ...
etc.
(End)

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

Cf. A022776, A022778, A022780, A054347.

t:=(1+sqrt(5))/2: a:=n->n+1+add(floor((n-k)/t),k=0..n): seq(a(n),n=0..55); # Muniru A Asiru, Oct 24 2018

Table[n + 1 + Sum[Floor[(n - k)/GoldenRatio], {k, 0, n}], {n, 0, 200}]

a(n) = n + 1 + Sum{floor[(n - k)/tau], k = 0..n}, where tau = (1 + sqrt(5))/2.

a(n) = A054347(n-1) - (n^2 - 3*n)/2. - Alan Michael Gómez Calderón, Nov 21 2024

A202869 Symmetric matrix based on the lower Wythoff sequence, A000201, by antidiagonals.

Original entry on oeis.org

1, 3, 3, 4, 10, 4, 6, 15, 15, 6, 8, 22, 26, 22, 8, 9, 30, 39, 39, 30, 9, 11, 35, 54, 62, 54, 35, 11, 12, 42, 66, 87, 87, 66, 42, 12, 14, 47, 79, 108, 126, 108, 79, 47, 14, 16, 54, 90, 132, 159, 159, 132, 90, 54, 16, 17, 62, 103, 151, 196, 207, 196, 151, 103, 62

Offset: 1

Clark Kimberling, Dec 26 2011

Let s=(1,3,4,6,8,...)=A000201 and let T be the infinite square matrix whose n-th row is formed by putting n-1 zeros before the terms of s. Let T' be the transpose of T. Then A202869 represents the matrix product M=T'*T. M is the self-fusion matrix of s, as defined at A193722. See A202870 for characteristic polynomials of principal submatrices of M, with interlacing zeros.

			Northwest corner:
1...3....4....6....8....9
3...10...15...22...30...35
4...15...26...39...54...66
6...22...39...62...87...108
8...30...54...87...126..159

Cf. A202870.

s[k_] := Floor[k*GoldenRatio];
U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[s[k], {k, 1, 15}]];
L = Transpose[U]; M = L.U; TableForm[M]
m[i_, j_] := M[[i]][[j]];
Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]
f[n_] := Sum[m[i, n], {i, 1, n}] + Sum[m[n, j], {j, 1, n - 1}]
Table[f[n], {n, 1, 12}]
Table[Sqrt[f[n]], {n, 1, 12}]   (* A054347 *)
Table[m[1, j], {j, 1, 12}]        (* A000201 *)

A283733 a(n) = a(n-1) + 1 + floor(n*golden ratio), with a(0) = 1.

Original entry on oeis.org

1, 3, 7, 12, 19, 28, 38, 50, 63, 78, 95, 113, 133, 155, 178, 203, 229, 257, 287, 318, 351, 385, 421, 459, 498, 539, 582, 626, 672, 719, 768, 819, 871, 925, 981, 1038, 1097, 1157, 1219, 1283, 1348, 1415, 1483, 1553, 1625, 1698, 1773, 1850, 1928, 2008, 2089

Offset: 0

Clark Kimberling, Mar 16 2017

This is column 1 of the rank array, A283734, of the golden ratio.

Clark Kimberling, Table of n, a(n) for n = 0..10000

Cf. A000201, A001622, A026351 (first differences), A054347, A283734.

a[0] = 1; a[n_] := a[n] = a[n - 1] + 1 + Floor[n*GoldenRatio]; Table[a[n], {n, 0, 100}]

g = (1 + sqrt(5))/2;
a(n) = if(n<1, 1, a(n - 1) + 1 + floor(n*g));
for(n=0, 100, print1(a(n),", ")) \\ Indranil Ghosh, Mar 17 2017

a(n) = A054347(n) + n + 1. - Alan Michael Gómez Calderón, Dec 03 2024

A088207 a(n) = Sum_{k=0..n} floor(k*phi^2) where phi=(1+sqrt(5))/2.

Original entry on oeis.org

0, 2, 7, 14, 24, 37, 52, 70, 90, 113, 139, 167, 198, 232, 268, 307, 348, 392, 439, 488, 540, 594, 651, 711, 773, 838, 906, 976, 1049, 1124, 1202, 1283, 1366, 1452, 1541, 1632, 1726, 1822, 1921, 2023, 2127, 2234, 2343, 2455, 2570, 2687, 2807, 2930, 3055, 3183

Offset: 0

Gary W. Adamson, Sep 23 2003

Partial sums of A001950.
A001950 is the upper Beatty sequence for the constant phi^2, where phi = (1 + sqrt(5))/2 and the sequence is generated by floor(n*phi). A054347 = partial sums of the lower Beatty sequence (A000201).
Conjecture: a(n)/A054347(n) tends to phi. Example: a(28)/A054347(28) = 1049/643 = 1.6314...
From Michel Dekking, Aug 19 2019: (Start)
Proof of Adamson's conjecture. We know that lim_{n->oo} A054347(n)/(n*(n+1)) = phi/2 (see A054347).
Using that floor(k*phi^2) = floor(k*phi)+k, for k=1,...,n, we obtain a(n)/A054347(n) = (A054347(n)+n*(n+1)/2)/A054347(n) = 1+(n*(n+1)/2)/A054347(n) -> 1+(2/phi)*(1/2), which equals phi. [parentheses corrected by Jason Yuen, Jan 12 2025]
(End)

			A001950(1) = 2, then 5, 7, 10, 13, ...; partial sums are 2, 7, 14, 24, 37, ...

Cf. A001622 (phi), A001950, A054347, A000201, A000217 (triangular numbers).

a[0] = 0; a[n_] := a[n] = (a[n - 1] + Floor[n*(1 + Sqrt[5])^2/4]); Table[ a[n], {n, 1, 50}] (* Robert G. Wilson v, Sep 27 2003 *)
Accumulate[Floor[GoldenRatio^2 Range[0,50]]] (* Harvey P. Dale, Aug 11 2021 *)

from math import isqrt
from itertools import islice, count, accumulate
def A088207_gen(): # generator of terms
    return accumulate((n+isqrt(5*n**2)>>1)+n for n in count(0))
A088207_list = list(islice(A088207_gen(),10)) # Chai Wah Wu, Aug 29 2022

a(n) = Sum_{k=1..n} floor(k*phi^2).
a(n) = floor((n*(n+1)/2)*phi^2 - n/2) + (0 or 1). - Benoit Cloitre, Sep 27 2003
a(n) = Sum_{k=1..n} floor(k*phi)+k = A054347(n)+n*(n+1)/2. - Jason Yuen, Jan 12 2025

More terms from Robert G. Wilson v and Benoit Cloitre, Sep 27 2003

A151889 a(1)=2, a(2)=3; a(2k-1)=2a(2k-2)+a(2k-3), a(2k)=3a(2k-2)+2a(2k-3), k >= 2.

Original entry on oeis.org

2, 3, 8, 13, 34, 55, 144, 233, 610, 987, 2584, 4181, 10946, 17711, 46368, 75025, 196418, 317811, 832040, 1346269, 3524578, 5702887, 14930352, 24157817, 63245986, 102334155, 267914296, 433494437, 1134903170, 1836311903

Offset: 1

Antonia Redondo Buitrago (aredondo(AT)sabuco.com), Jul 23 2009

Because of G.f. in Formula section, numbers k + 1 such that A054347(k)/k is an integer. - Ctibor O. Zizka, Apr 02 2025

Index entries for linear recurrences with constant coefficients, signature (0,4,0,1).

Cf. A000045, A000201, A054347.

LinearRecurrence[{0,4,0,1},{2,3,8,13},30] (* Harvey P. Dale, Dec 31 2021 *)

a(n) = if (n==1, 2, if (n==2, 3, if (n%2, 2*a(n-1)+a(n-2), 3*a(n-2)+2*a(n-3)))) \\ Michel Marcus, Jun 20 2013

a(2k-1) = Fib(3k-1), a(2k) = Fib(3k).

G.f.: -(x^3+3*x+2)*x/(x^4+4*x^2-1). - Alois P. Heinz, Jun 20 2013

cp's OEIS Frontend

A194054 Natural interspersion of A054347; a rectangular array, by antidiagonals.

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A194053 Natural fractal sequence of A054347.

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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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A255977 The number of numbers j+k*r <= n, where r = golden ratio and j and k are nonnegative integers.

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A202869 Symmetric matrix based on the lower Wythoff sequence, A000201, by antidiagonals.

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A283733 a(n) = a(n-1) + 1 + floor(n*golden ratio), with a(0) = 1.

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A088207 a(n) = Sum_{k=0..n} floor(k*phi^2) where phi=(1+sqrt(5))/2.

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