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Natural interspersion of A054347; a rectangular array, by antidiagonals.
+20
3
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
COMMENTS
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.
EXAMPLE
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
MATHEMATICA
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 *)
Natural fractal sequence of A054347.
+20
2
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
COMMENTS
See A194029 for definitions of natural fractal sequence and natural interspersion.
MATHEMATICA
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 *)
Signature sequence of phi = (1+sqrt(5))/2 = 1.61803...
+10
22
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
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
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.
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 *)
Symmetric matrix based on the lower Wythoff sequence, A000201, by antidiagonals.
+10
3
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
COMMENTS
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.
EXAMPLE
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
MATHEMATICA
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 *)
a(n) = Sum_{k=0..n} floor(k*phi^2) where phi=(1+sqrt(5))/2.
+10
1
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
COMMENTS
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... 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))/((n*(n+1))))/( A054347(n)/(n*(n+1)/2) [Ambiguous, unbalanced parens - Editors of OEIS] -> (phi/2 + 1/2)/(phi/2), which equals phi. (End)
FORMULA
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
EXAMPLE
A001950(1) = 2, then 5, 7, 10, 13, ...; partial sums are 2, 7, 14, 24, 37, ...
MATHEMATICA
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 *)
PROG
(Python)
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))
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