A006338 -id:A006338 - cp's OEIS frontend

A378401 Rectangular array read by descending antidiagonals: (row 1) = u, and for n >= 2, (row n) = u-inverse runlength sequence of u, where u = A006338. See Comments.

2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 2, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 2, 2, 2, 2, 1, 1, 1, 2, 1, 2, 2, 2, 2, 1, 1, 2, 1, 1

Offset: 1

Clark Kimberling, Dec 21 2024

If u and v are sequences, both consisting of 1's and 2's, we call v an inverse runlength sequence of u if u is the runlength sequence of v. Each u has two inverse runlength sequences, one with first term 1 and the other with first term 2. Consequently, an inverse runlength array, in which each row after the first is an inverse runlength sequence of the preceding row, is determined by its first column. Generally, if the first column is periodic with fundamental period p, then the array has p distinct limiting sequences; otherwise, there is no limiting sequence; however, if a segment, of any length, occurs in a row, then it also occurs in a subsequent row. See A378282 for details and related sequences.

			The corner of the array begins:
     2  1  2  1  1  2  1  2  1  2  1  1  2  1  2  1  1  2  1
     1  1  2  1  1  2  1  2  2  1  2  2  1  2  2  1  2  1  1
     2  1  2  2  1  2  1  1  2  1  1  2  2  1  2  2  1  1  2
     1  1  2  1  1  2  2  1  2  2  1  2  1  1  2  1  2  2  1
     1  2  1  1  2  1  2  2  1  1  2  1  1  2  2  1  2  2  1
     2  1  1  2  1  2  2  1  2  2  1  1  2  1  2  2  1  2  1
     1  1  2  1  2  2  1  2  2  1  1  2  1  1  2  2  1  2  1
     2  1  2  2  1  2  2  1  1  2  1  1  2  2  1  2  1  1  2
     1  1  2  1  1  2  2  1  2  2  1  1  2  1  2  2  1  2  1
     2  1  2  2  1  2  1  1  2  2  1  2  2  1  1  2  1  2  2
     1  1  2  1  1  2  2  1  2  2  1  2  1  1  2  2  1  2  2
     1  2  1  1  2  1  2  2  1  1  2  1  1  2  2  1  2  2  1

Cf. A006338, A378282, A378396, A378397, A378398, A378399.

invRE[seq_, k_] := Flatten[Map[ConstantArray[#[[2]], #[[1]]] &,
    Partition[Riffle[seq, {k, 2 - Mod[k + 1, 2]}, {2, -1, 2}], 2]]];
row1 = Differences[Table[Floor[n*Sqrt[2]+1/2], {n, 1, 20}]];
rows = {row1}; col = Take[row1, 12];
Do[AppendTo[rows, Take[invRE[Last[rows], col[[n]]], Length[row1]]], {n, 2, Length[col]}]
rows // ColumnForm  (* array *)
w[n_, k_] := rows[[n]][[k]]; Table[w[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (* sequence *)
(* Peter J. C. Moses, Nov 20 2024 *)

A006337 An "eta-sequence": a(n) = floor( (n+1)sqrt(2) ) - floor( nsqrt(2) ).

Original entry on oeis.org

1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1

Offset: 1

D. R. Hofstadter, Jul 15 1977

Defined by: (i) a(1) = 1; (ii) sequence consists of single 2's separated by strings of 1's; (iii) the sequence of lengths of runs of 1's in the sequence is equal to the sequence.
Equals its own "derivative", which is formed by counting the strings of 1's that lie between 2's.
First differences of A001951 (with a different offset). - Philippe Deléham, May 29 2006
Or number of perfect squares in interval (2*n^2, 2*(n+1)^2). In view of the uniform distribution mod 1 of sequence {sqrt(2)*n}, the density of 1's is 2-sqrt(2). - Vladimir Shevelev, Aug 05 2011
a(n) = number of repeating n's in A049472. - Reinhard Zumkeller, Jul 03 2015
Fixed point of the morphism 1 -> 12; 2 -> 121. - Jeffrey Shallit, Jan 19 2017
Also, let S be the increasing sequence of elements of the union N U N*sqrt(2), where N = {1, 2, 3, ...}. Then a(n) = { 1 if S(n) is integer, 2 if S(n) is irrational }. See A245222 for the analog with sqrt(3). - M. F. Hasler, Feb 06 2025

Douglas Hofstadter, "Fluid Concepts and Creative Analogies", Chapter 1: "To seek whence cometh a sequence".
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
F. M. Dekking, On the structure of self-generating sequences, Seminar on Number Theory, 1980-1981 (Talence, 1980-1981), Exp. No. 31, 6 pp., Univ. Bordeaux I, Talence, 1981. Math. Rev. 83e:10075.
D. R. Hofstadter, Eta-Lore [Cached copy, with permission]
D. R. Hofstadter, Pi-Mu Sequences [Cached copy, with permission]
D. R. Hofstadter and N. J. A. Sloane, Correspondence, 1977 and 1991
Clark Kimberling, Problem 6281, Amer. Math. Monthly 86 (1979), no. 9, 793.
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)

Cf. A006338. Exchanging 1's and 2's gives A080763. Essentially same as A004641 + 1.
Cf. A049472.
The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A003151 as the parent: A003151, A001951, A001952, A003152, A006337, A080763, A082844 (conjectured), A097509, A159684, A188037, A245219 (conjectured), A276862. - N. J. A. Sloane, Mar 09 2021
Cf. A245222 (an analog with sqrt(3) instead of sqrt(2)).

a006337 n = a006337_list !! (n-1)
a006337_list = f [1] where
   f xs = ys ++ f ys where
          ys = concatMap (\z -> if z == 1 then [1,2] else [1,1,2]) xs
-- Reinhard Zumkeller, May 06 2012

Digits := 100; sq2 := sqrt(2.); A006337 := n->floor((n+1)*sq2)-floor(n*sq2);

Flatten[ Table[ Nest[ Flatten[ # /. {1 -> {1, 2}, 2 -> {1, 1, 2}}] &, {1}, n], {n, 5}]] (* Robert G. Wilson v, May 06 2005 *)
Differences[ Table[ Floor[ n*Sqrt[2]], {n, 1, 106}]] (* Jean-François Alcover, Apr 06 2012 *)

a(n)=sqrt(2)*(n+1)\1-sqrt(2)*n\1 \\ Charles R Greathouse IV, Apr 06 2012

a(n)=sqrtint(2*n^2+4*n+2)-sqrtint(2*n^2) \\ Charles R Greathouse IV, Apr 06 2012

from math import isqrt
def A006337(n): return -isqrt(m:=n*n<<1)+isqrt(m+(n<<2)+2) # Chai Wah Wu, Aug 03 2022

Let S(0) = 1; obtain S(k) from S(k-1) by applying 1 -> 12, 2 -> 112; sequence is S(0), S(1), S(2), ... - Matthew Vandermast, Mar 25 2003
a(A003152(n)) = 1 and a(A003151(n)) = 2. - Philippe Deléham, May 29 2006
a(n) = A159684(n-1) + 1. - Filip Zaludek, Oct 28 2016

A022846 Nearest integer to n*sqrt(2).

Original entry on oeis.org

0, 1, 3, 4, 6, 7, 8, 10, 11, 13, 14, 16, 17, 18, 20, 21, 23, 24, 25, 27, 28, 30, 31, 33, 34, 35, 37, 38, 40, 41, 42, 44, 45, 47, 48, 49, 51, 52, 54, 55, 57, 58, 59, 61, 62, 64, 65, 66, 68, 69, 71, 72, 74, 75, 76, 78, 79, 81, 82, 83, 85, 86, 88, 89, 91, 92, 93, 95, 96

Offset: 0

Clark Kimberling

Let R(i,j) be the rectangle with antidiagonals 1; 2,3; 4,5,6; ...; n^2 is in antidiagonal number a(n). Proof: n^2 is in antidiagonal m iff A000217(m-1)< n^2 <=A000217(m), where A000217(m)=m*(m+1)/2. So m = A002024(n^2) = round(n*sqrt(2)) = a(n). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Mar 07 2003
In the rectangle R(i,j), n^2 is the number in row i=A057049(n) and column j=A057050(n), so that for n >= 1, a(n) = -1 + A057049(n) + A057050(n). - Clark Kimberling, Jan 31 2011
Number of triangular numbers less than n^2. - Philippe Deléham, Mar 08 2013

			n = 4, n^2 = 16; 0, 1, 3, 6, 10, 15 are triangular numbers in interval [0, 16); a(4) = 6. - _Philippe Deléham_, Mar 08 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Clark Kimberling, Beatty sequences and trigonometric functions, Integers 16 (2016), #A15.

Cf. A063957 (complement of this set).

Cf. A214848 (first differences), also A006338.

a022846 = round . (* sqrt 2) . fromIntegral
-- Reinhard Zumkeller, Mar 03 2014

[Round(n*Sqrt(2)): n in [0..60]]; // Vincenzo Librandi, Oct 22 2011

Round[Sqrt[2]Range[0,70]] (* Harvey P. Dale, Jun 18 2013 *)

PARI
```
a(n)=round(n*sqrt(2))
```

from math import isqrt
def A022846(n): return isqrt(n**2<<3)+1>>1 # Chai Wah Wu, Feb 10 2023

a(n) = A002024(n^2).

a(n+1) - a(n) = 1 or 2. - Philippe Deléham, Mar 08 2013

A214848 First difference of A022846.

Original entry on oeis.org

1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1

Offset: 0

Philippe Deléham, Mar 08 2013

Number of triangular numbers in interval [n^2, (n+1)^2).
From Michel Dekking, Sep 20 2022: (Start)
(a(n)) is an inhomogeneous Sturmian sequence s(alpha, rho) with slope alpha = sqrt(2) and intercept 1/2, since A022846(n) = floor(n*sqrt(2) + 1/2).
(a(n)) is the fixed point of the morphism 1->12121, 2->1212121.
This is proved by writing the 0-1 version psi: 0->01010, 1->0101010 of this morphism as a composition
     psi = psi_1 psi_3 psi_1 psi_4,
where the psi_i are the three elementary Sturmian morphisms
     psi_1: 0->01, 1->0,   psi_3: 0->0, 1->01,   psi_4: 0->0, 1->10.
By Lemma 2.2.18 in Lothaire it then follows that the 0-1 word (a(n)-1) = A214848 is fixed by the morphism psi (note that in Lothaire psi_1 is phi, psi_3 is G, and psi_4 is G^~). (End)

			28 is in [25, 36), a(5) = 1.
36 and 45 are in [36, 49), a(6) = 2.

S.-I. Yasutomi, On Sturmian sequences which are invariant under some substitutions, in Number theory and its applications (Kyoto, 1997), pp. 347-373, Kluwer Acad. Publ., Dordrecht, 1999.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Svetlana Jitomirskaya, Small denominators and multiplicative Jensen's formula, ICM 2022. See the initial slides "Playing with numbers".
M. Lothaire, Algebraic combinatorics on words, Cambridge University Press. Online publication date: April 2013; Print publication year: 2002.
S.-I. Yasutomi, On Sturmian sequences which are invariant under some substitutions, on ResearchGate.

Cf. A022846, A006337, A006338.

a214848 n = a214848_list !! n
a214848_list = zipWith (-) (tail a022846_list) a022846_list
-- Reinhard Zumkeller, Mar 03 2014

Differences[Round[Sqrt[2]Range[0,100]]] (* Harvey P. Dale, Jun 14 2020 *)

For n > 0: a(n) = A006338(n). - Reinhard Zumkeller, Mar 03 2014

A106802 Trajectory of 2 under the morphism 1->{2, 1, 2, 1, 1, 2, 2, 1}, 2->{1, 1, 1, 2, 2, 1, 2}.

Original entry on oeis.org

2, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2

Offset: 0

Roger L. Bagula, May 17 2005

nonn

T. S. Blyth and E. F. Robertson, Essential Student Algebra: volume 5: Groups: Chapman and Hall, 1986, page 9.

Index entries for sequences that are fixed points of mappings

Cf. A001030, A006338.

s[1, 1] = {1}; s[2, 1] = {2};; s[1, 2] = {2}; s[2, 2] = {1};; s[1, 3] = {1, 2}; s[2, 3] = {1};; s[1, 4] = {1}; s[2, 4] = {1, 2};; s[1, 5] = {1, 2}; s[2, 5] = {2};; s[1, 6] = {2}; s[2, 6] = {1};; w[i_] = s[1, 1 + Mod[i, 6]] v[i_] = s[2, 1 + Mod[i, 6]] S[1] = Flatten[Table[w[i], {i, 1, 6}]] S[2] = Flatten[Table[v[i], {i, 1, 6}]] t[a_] := Flatten[S /@ a]; p[0] = {1}; p[1] = t[p[0]]; p[n_] := t[p[n - 1]] aa = p[3]
Nest[Flatten[#]/.{1->{2,1,2,1,1,2,2,1},2->{1,1,1,2,2,1,2}}&,2,4]//Flatten (* Harvey P. Dale, Apr 09 2019 *)

Edited by N. J. A. Sloane, Nov 12 2006

cp's OEIS Frontend

A378401 Rectangular array read by descending antidiagonals: (row 1) = u, and for n >= 2, (row n) = u-inverse runlength sequence of u, where u = A006338. See Comments.

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A006337 An "eta-sequence": a(n) = floor( (n+1)*sqrt(2) ) - floor( n*sqrt(2) ).

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A022846 Nearest integer to n*sqrt(2).

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A214848 First difference of A022846.

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A106802 Trajectory of 2 under the morphism 1->{2, 1, 2, 1, 1, 2, 2, 1}, 2->{1, 1, 1, 2, 2, 1, 2}.

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A006337 An "eta-sequence": a(n) = floor( (n+1)sqrt(2) ) - floor( nsqrt(2) ).