A097337 -id:A097337 - cp's OEIS frontend

A194981 Interspersion fractally induced by A194979, a rectangular array, by antidiagonals.

1, 2, 3, 4, 6, 5, 7, 10, 8, 9, 11, 15, 12, 14, 13, 16, 21, 17, 20, 18, 19, 22, 28, 23, 27, 24, 25, 26, 29, 36, 30, 35, 31, 32, 34, 33, 37, 45, 38, 44, 39, 40, 43, 41, 42, 46, 55, 47, 54, 48, 49, 53, 50, 52, 51, 56, 66, 57, 65, 58, 59, 64, 60, 63, 61, 62, 67, 78, 68

Offset: 1

Clark Kimberling, Sep 07 2011

See A194959 for a discussion of fractalization and the interspersion fractally induced by a sequence. Every pair of rows eventually intersperse. As a sequence, A194981 is a permutation of the positive integers, with inverse A194982.

			Northwest corner:
1...2...4...7...11..16..22
3...6...10..15..21..28..36
5...8...12..17..23..30..38
9...14..20..27..35..44..54
13..18..24..31..39..48..58

Cf. A194959, A019480, A194982.

r = Sqrt[3]; p[n_] := 1 + Floor[n/r]
Table[p[n], {n, 1, 90}]  (* A194979 = 1+ A097337 *)
g[1] = {1}; g[n_] := Insert[g[n - 1], n, p[n]]
f[1] = g[1]; f[n_] := Join[f[n - 1], g[n]]
f[20] (* A194980 *)
row[n_] := Position[f[30], n];
u = TableForm[Table[row[n], {n, 1, 5}]]
v[n_, k_] := Part[row[n], k];
w = Flatten[Table[v[k, n - k + 1], {n, 1, 13},
{k, 1, n}]]  (* A194981 *)
q[n_] := Position[w, n]; Flatten[Table[q[n],
{n, 1, 80}]]  (* A194982 *)

A194979 a(n) = 1 + floor(n/sqrt(3)).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 07 2011

The fractalization of this sequence is A194980.
Least number k such that k*tan(1/k) - 1 < 1/n^2. - Clark Kimberling, Dec 02 2014
The integers k such that a(k) = a(k+1) give A054406. - Michel Marcus, Nov 01 2021

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Cf. A054406, A097337, A194980, A194981.

[1+Floor(n/Sqrt(3)): n in [1..80] ];// Vincenzo Librandi, Sep 10 2011

p[n_]:=1+Floor[n/Sqrt[3]]
Table[p[n],{n,1,90}] (* A194979 *)

a(n)=1+sqrtint(n^2\3) \\ Charles R Greathouse IV, Sep 02 2015

a(n) = 1 + A097337(n).

A194980 Fractalization of (1+[n/sqrt(3)]), where [ ]=floor.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 07 2011

nonn

See A194959 for a discussion of fractalization and the interspersion fractally induced by a sequence. The sequence (1+[n/sqrt(3)]) is A194979.

Cf. A194959, A194879, A194981, A194982.

r = Sqrt[3]; p[n_] := 1 + Floor[n/r]
Table[p[n], {n, 1, 90}]  (* A194979 = 1+ A097337 *)
g[1] = {1}; g[n_] := Insert[g[n - 1], n, p[n]]
f[1] = g[1]; f[n_] := Join[f[n - 1], g[n]]
f[20] (* A194980 *)
row[n_] := Position[f[30], n];
u = TableForm[Table[row[n], {n, 1, 5}]]
v[n_, k_] := Part[row[n], k];
w = Flatten[Table[v[k, n - k + 1], {n, 1, 13},
{k, 1, n}]]  (* A194981 *)
q[n_] := Position[w, n]; Flatten[Table[q[n],
{n, 1, 80}]]  (* A194982 *)

A183142 Beatty sequence for 2/(3+sqrt(3)).

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 2, 3, 3, 4, 4, 5, 5, 5, 6, 6, 7, 7, 8, 8, 8, 9, 9, 10, 10, 10, 11, 11, 12, 12, 13, 13, 13, 14, 14, 15, 15, 16, 16, 16, 17, 17, 18, 18, 19, 19, 19, 20, 20, 21, 21, 21, 22, 22, 23, 23, 24, 24, 24, 25, 25, 26, 26, 27, 27, 27, 28, 28, 29, 29, 30

Offset: 1

Clark Kimberling, Dec 26 2010

nonn

A097337 (B.s. for 1/sqrt(3)) + A183142 = A001477 (the nonnegative integers).

Cf. A097337.

With[{s=(3+Sqrt[3])/2},Floor[Range[100]/s]] (* Harvey P. Dale, Feb 15 2016 *)

default(realprecision,100); s=(3+sqrt(3))/2; for(n=1,99,print1(floor(n/s),", "))

a(n) = floor(2*n/(3+sqrt(3))).

Definition modified and offset changed to 1 by Georg Fischer, Jul 09 2021

A187319 Rank transform of the sequence floor(n/sqrt(3)); complement of A187410.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 10, 11, 12, 13, 15, 16, 18, 19, 20, 22, 23, 24, 25, 27, 29, 30, 32, 33, 35, 36, 37, 39, 40, 42, 43, 44, 46, 47, 49, 50, 52, 53, 54, 56, 57, 59, 60, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 76, 77, 78, 79, 81, 83, 84, 85, 86, 88, 89, 91, 93, 94, 95, 96, 98, 99, 101, 102, 103, 105, 106, 108, 110, 111, 112, 113, 115, 116, 118, 119, 120, 122, 123, 125, 126, 127, 129

Offset: 1

Clark Kimberling, Mar 08 2011

nonn

See A187224.

Cf. A187224, A187410.

m = 3^(-1/2);
seqA = Table[Floor[m*n], {n, 1, 180}]  (* A097337 *)
seqB = Table[n, {n, 1, 80}];                   (* A000027 *)
jointRank[{seqA_, seqB_}] := {Flatten@Position[#1, {_, 1}],
Flatten@Position[#1, {_, 2}]} &[Sort@Flatten[{{#1, 1} & /@ seqA,
{#1, 2} & /@ seqB}, 1]];
limseqU = FixedPoint[jointRank[{seqA, #1[[1]]}] &, jointRank
[{seqA, seqB}]][[1]]                                      (* A187319 *)
Complement[Range[Length[seqA]], limseqU]  (* A187410 *)
(* by Peter J. C. Moses, Mar 09 2011 *)

A263574 Beatty sequence for 1/sqrt(3) - log(phi)/3575 where phi is the golden ratio, A001622.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 3, 4, 4, 5, 5, 6, 6, 7, 8, 8, 9, 9, 10, 10, 11, 12, 12, 13, 13, 14, 15, 15, 16, 16, 17, 17, 18, 19, 19, 20, 20, 21, 21, 22, 23, 23, 24, 24, 25, 25, 26, 27, 27, 28, 28, 29, 30, 30, 31, 31, 32, 32, 33, 34, 34, 35, 35, 36, 36, 37, 38, 38, 39, 39, 40, 40, 41, 42, 42, 43

Offset: 0

Karl V. Keller, Jr., Oct 21 2015

nonn

The number 1/sqrt(3) - log(phi)/3575 (=0.577215664483...) is an approximation to Euler's constant (A001620) (=0.577215664901...).

M. Hudson found a similar Euler-Mascheroni constant approximation (see link), 1/sqrt(3)-1/7429 (=0.57721566157...).

			For n=9, floor(9*(0.577215664483)) = floor(5.194940980347) = 5.

Karl V. Keller, Jr., Table of n, a(n) for n = 0..100000
Xavier Gourdon and Pascal Sebah, Collection of formulas for Euler's constant,Euler's constant.
Eric Weisstein's World of Mathematics, Beatty Sequence.
Eric Weisstein's World of Mathematics, Euler-Mascheroni Constant.
Eric Weisstein's World of Mathematics, Euler-Mascheroni Constant Approximations.

Cf. A001620, A020760 (1/sqrt(3)), A038128 (Beatty sequence for Euler's constant), A097337 (Beatty sequence for 1/sqrt(3)).

phi:= (1+Sqrt(5))/2; [Floor(n*(1/Sqrt(3) - Log(phi)/3575)): n in [0..100]]; // G. C. Greubel, Sep 05 2018

Table[Floor[n (1/Sqrt@ 3 - Log[GoldenRatio]/3575)], {n, 0, 75}] (* Michael De Vlieger, Nov 12 2015 *)

{phi = (1+sqrt(5))/2}; vector(100, n, n--; floor(n*(1/sqrt(3) - log(phi)/3575))) \\ G. C. Greubel, Sep 05 2018

from sympy import floor, log, sqrt
for n in range(0,101):print(floor(n*(1/sqrt(3)-log(1/2+sqrt(5)/2)/3575)),end=', ')

a(n) = floor(n*(1/sqrt(3) - log(phi)/3575)).

a(n) = A038128(n) for n < 58628.

cp's OEIS Frontend

A194981 Interspersion fractally induced by A194979, a rectangular array, by antidiagonals.

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A194979 a(n) = 1 + floor(n/sqrt(3)).

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A194980 Fractalization of (1+[n/sqrt(3)]), where [ ]=floor.

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A183142 Beatty sequence for 2/(3+sqrt(3)).

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A187319 Rank transform of the sequence floor(n/sqrt(3)); complement of A187410.

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A263574 Beatty sequence for 1/sqrt(3) - log(phi)/3575 where phi is the golden ratio, A001622.

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