A117384 -id:A117384 - cp's OEIS frontend

A001614 Connell sequence: 1 odd, 2 even, 3 odd, ...

1, 2, 4, 5, 7, 9, 10, 12, 14, 16, 17, 19, 21, 23, 25, 26, 28, 30, 32, 34, 36, 37, 39, 41, 43, 45, 47, 49, 50, 52, 54, 56, 58, 60, 62, 64, 65, 67, 69, 71, 73, 75, 77, 79, 81, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 122

Offset: 1

N. J. A. Sloane

Next (2n-1) odd numbers alternating with next 2n even numbers. Squares (A000290(n)) occur at the A000217(n)-th entry. - Lekraj Beedassy, Aug 06 2004. - Comment corrected by Daniel Forgues, Jul 18 2009
a(t_n) = a(n(n+1)/2) = n^2 relates squares to triangular numbers. - Daniel Forgues
The natural numbers not included are A118011(n) = 4n - a(n) as n=1,2,3,... - Paul D. Hanna, Apr 10 2006
As a triangle with row sums = A069778 (1, 6, 21, 52, 105, ...): /Q 1;/Q 2, 4;/Q 5, 7, 9;/Q 10, 12, 14, 16;/Q ... . - Gary W. Adamson, Sep 01 2008
The triangle sums, see A180662 for their definitions, link the Connell sequence A001614 as a triangle with six sequences, see the crossrefs. - Johannes W. Meijer, May 20 2011
a(n) = A122797(n) + n - 1. - Reinhard Zumkeller, Feb 12 2012

			From _Omar E. Pol_, Aug 13 2013: (Start)
Written as a triangle the sequence begins:
   1;
   2,  4;
   5,  7,  9;
  10, 12, 14, 16;
  17, 19, 21, 23, 25;
  26, 28, 30, 32, 34, 36;
  37, 39, 41, 43, 45, 47, 49;
  50, 52, 54, 56, 58, 60, 62, 64;
  65, 67, 69, 71, 73, 75, 77, 79, 81;
  82, 84, 86, 88, 90, 92, 94, 96, 98, 100;
  ...
Right border gives A000290, n >= 1.
(End)

C. Pickover, Computers and the Imagination, St. Martin's Press, NY, 1991, p. 276.
C. A. Pickover, The Mathematics of Oz, Chapter 39, Camb. Univ. Press UK 2002.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this 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..1000
Ian Connell and Andrew Korsak, Problem E1382, Amer. Math. Monthly, 67 (1960), 380.
Douglas E. Iannucci and Donna Mills-Taylor, On Generalizing the Connell Sequence, J. Integer Sequences, Vol. 2, 1999, #99.1.7.
H. P. Robinson and N. J. A. Sloane, Correspondence, 1971-1972
N. J. A. Sloane, Handwritten notes on Self-Generating Sequences, 1970 (note that A1148 has now become A005282)
Gary E. Stevens, A Connell-Like Sequence, J. Integer Sequences, Vol. 1, 1998, #98.1.4.
Eric Weisstein's World of Mathematics, Connell Sequence

Cf. A117384, A118011 (complement), A118012.
Cf. A069778. - Gary W. Adamson, Sep 01 2008
From Johannes W. Meijer, May 20 2011: (Start)
Triangle columns: A002522, A117950 (n>=1), A117951 (n>=2), A117619 (n>=3), A154533 (n>=5), A000290 (n>=1), A008865 (n>=2), A028347 (n>=3), A028878 (n>=1), A028884 (n>=2), A054569 [T(2*n,n)].
Triangle sums (see the comments): A069778 (Row1), A190716 (Row2), A058187 (Related to Kn11, Kn12, Kn13, Kn21, Kn22, Kn23, Fi1, Fi2, Ze1 and Ze2), A000292 (Related to Kn3, Kn4, Ca3, Ca4, Gi3 and Gi4), A190717 (Related to Ca1, Ca2, Ze3, Ze4), A190718 (Related to Gi1 and Gi2). (End)
Cf. A014132, A057211, A023531.

a001614 n = a001614_list !! (n-1)
a001614_list = f 0 0 a057211_list where
   f c z (x:xs) = z' : f x z' xs where z' = z + 1 + 0 ^ abs (x - c)
-- Reinhard Zumkeller, Dec 30 2011

[2*n-Round(Sqrt(2*n)): n in [1..80]]; // Vincenzo Librandi, Apr 17 2015

A001614:=proc(n): 2*n - floor((1+sqrt(8*n-7))/2) end: seq(A001614(n),n=1..67); # Johannes W. Meijer, May 20 2011

lst={};i=0;For[j=1, j<=4!, a=i+1;b=j;k=0;For[i=a, i<=9!, k++;AppendTo[lst, i];If[k>=b, Break[]];i=i+2];j++ ];lst (* Vladimir Joseph Stephan Orlovsky, Aug 29 2008 *)
row[n_] := 2*Range[n+1]+n^2-1; Table[row[n], {n, 0, 11}] // Flatten (* Jean-François Alcover, Oct 25 2013 *)

a(n)=2*n - round(sqrt(2*n)) \\ Charles R Greathouse IV, Apr 20 2015

from math import isqrt
def A001614(n): return (m:=n<<1)-(k:=isqrt(m))-int((m<<2)>(k<<2)*(k+1)+1) # Chai Wah Wu, Jul 26 2022

a(n) = 2*n - floor( (1+ sqrt(8*n-7))/2 ).
a(n) = A005843(n) - A002024(n). - Lekraj Beedassy, Aug 06 2004
a(n) = A118012(A118011(n)). A117384( a(n) ) = n; A117384( 4*n - a(n) ) = n. - Paul D. Hanna, Apr 10 2006
a(1) = 1; then a(n) = a(n-1)+1 if a(n-1) is a square, a(n) = a(n-1)+2 otherwise. For example, a(21)=36 is a square therefore a(22)=36+1=37 which is not a square so a(23)=37+2=39 ... - Benoit Cloitre, Feb 07 2007
T(n,k) = (n-1)^2 + 2*k - 1. - Omar E. Pol, Aug 13 2013
a(n)^2 = a(n*(n+1)/2). - Ivan N. Ianakiev, Aug 15 2013
Let the sequence be written in the form of the triangle in the EXAMPLE section below and let a(n) and a(n+1) belong to the same row of the triangle. Then a(n)*a(n+1) + 1 = a(A000217(A118011(n))) = A000290(A118011(n)). - Ivan N. Ianakiev, Aug 16 2013
a(n) = 2*n-round(sqrt(2*n)). - Gerald Hillier, Apr 15 2015
From Robert Israel, Apr 20 2015 (Start):
G.f.: 2*x/(1-x)^2 - (x/(1-x))*Sum_{n>=0} x^(n*(n+1)/2) = 2*x/(1-x)^2 - (Theta2(0,x^(1/2)))*x^(7/8)/(2*(1-x)) where Theta2 is a Jacobi theta function.
a(n) = 2*n-1 - Sum_{i=0..n-2} A023531(i).  (End)
a(n) = 3*n-A014132(n). - Chai Wah Wu, Oct 19 2024

More terms from Larry Reeves (larryr(AT)acm.org), Mar 16 2001

A118011 Complement of the Connell sequence (A001614); a(n) = 4*n - A001614(n).

Original entry on oeis.org

3, 6, 8, 11, 13, 15, 18, 20, 22, 24, 27, 29, 31, 33, 35, 38, 40, 42, 44, 46, 48, 51, 53, 55, 57, 59, 61, 63, 66, 68, 70, 72, 74, 76, 78, 80, 83, 85, 87, 89, 91, 93, 95, 97, 99, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 123, 125, 127, 129, 131, 133, 135, 137, 139

Offset: 1

Paul D. Hanna, Apr 10 2006

nonn

a(n) is the position of the second appearance of n in A117384, where A117384(m) = A117384(k) and k = 4*A117384(m) - m. The Connell sequence (A001614) is generated as: 1 odd, 2 even, 3 odd, ...

Cf. A001614, A014132, A023531, A117384, A118012.

A171152 gives partial sums.

[2*n+Round(Sqrt(2*n)): n in [1..70]]; // Vincenzo Librandi, Apr 16 2015

Table[2 n + Round[Sqrt[2 n]], {n, 70}] (* Vincenzo Librandi, Apr 16 2015 *)

from math import isqrt
def A118011(n): return (m:=n<<1)+(k:=isqrt(m))+int((m<<2)>(k<<2)*(k+1)+1) # Chai Wah Wu, Jul 26 2022

A001614(n) = A118012(a(n)).
a(n) = 2n+[(1+sqrt(8n-7))/2]. - Juri-Stepan Gerasimov Aug 25 2009
a(n) = 2*n+round(sqrt(2*n)). - Gerald Hillier, Apr 16 2015
From Robert Israel, Apr 20 2015 (Start):
a(n) = 2*n + 1 + Sum_{j=0..n-2} A023531(j).
G.f.: 2*x/(1-x)^2 + x/(1-x) * Sum_{j=0..oo} x^(j*(j+1)/2) = 2*x/(1-x)^2 + x^(7/8)/(2-2*x) * Theta2(0,sqrt(x)), where Theta2 is a Jacobi theta function. (End)
a(n) = n+A014132(n). - Chai Wah Wu, Oct 19 2024

A117385 Positive integers, each occurring twice in the sequence, such that a(n) = a(k) when n+k = 5*a(n), starting with a(1)=1 and filling the next vacant position with the smallest unused number.

Original entry on oeis.org

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

Offset: 1

Paul D. Hanna, Mar 11 2006

nonn

Cf. A117384 (a(4*a(n)-n)=a(n)), A117386 (a(6*a(n)-n)=a(n)).

{a(n)=local(A=vector(n),m=1); for(k=1,n,if(A[k]==0,A[k]=m;if(5*m-k<=#A,A[5*m-k]=m);m+=1));A[n]}

a(5*a(n)-n) = a(n). Conjecture: Lim_inf a(n)/n = (5-sqrt(5))/10; Lim_sup a(n)/n = (5+sqrt(5))/10.

A117386 Positive integers, each occurring twice in the sequence, such that a(n) = a(k) when n+k = 6*a(n), starting with a(1)=1 and filling the next vacant position with the smallest unused number.

Original entry on oeis.org

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

Offset: 1

Paul D. Hanna, Mar 11 2006

nonn

Cf. A117384 (a(4*a(n)-n)=a(n)), A117385 (a(5*a(n)-n)=a(n)).

{a(n)=local(A=vector(n),m=1); for(k=1,n,if(A[k]==0,A[k]=m;if(6*m-k<=#A,A[6*m-k]=m);m+=1));A[n]}

a(6*a(n)-n) = a(n). Conjecture: Lim_inf a(n)/n = (3-sqrt(3))/6; Lim_sup a(n)/n = (3+sqrt(3))/6.

A344482 Primes, each occurring twice, such that a(C(n)) = a(4*n-C(n)) = prime(n), where C is the Connell sequence (A001614).

Original entry on oeis.org

2, 3, 2, 5, 7, 3, 11, 5, 13, 17, 7, 19, 11, 23, 13, 29, 31, 17, 37, 19, 41, 23, 43, 29, 47, 53, 31, 59, 37, 61, 41, 67, 43, 71, 47, 73, 79, 53, 83, 59, 89, 61, 97, 67, 101, 71, 103, 73, 107, 109, 79, 113, 83, 127, 89, 131, 97, 137, 101, 139, 103, 149, 107, 151

Offset: 1

Paolo Xausa, Aug 16 2021

Terms can be arranged in an irregular triangle read by rows in which row r is a permutation P of the primes in the interval [prime(s), prime(s+rlen-1)], where s = 1+(r-1)*(r-2)/2, rlen = 2*r-1 = A005408(r-1) and r >= 1 (see example).
P is the alternating (first term > second term < third term > fourth term < ...) permutation m -> 1, 1 -> 2, m+1 -> 3, 2 -> 4, m+2 -> 5, 3 -> 6, ..., rlen -> rlen where m = ceiling(rlen/2).
The triangle has the following properties.
Row lengths are the positive odd numbers (A005408).
First column is A078721.
Column 3 is A078722 (for n >= 1).
Column 5 is A078724 (for n >= 2).
Column 7 is A078725 (for n >= 3).
Each even column is equal to the column preceding it.
Row records (A011756) are in the right border.
Indices of row records are the positive terms of A000290.
Each row r contains r terms that are duplicated in the next row.
In each row, the sum of terms which are not already listed in the sequence give A007468.
For rows r >= 2, row sum is A007468(r)+A007468(r-1) and row product is A007467(r)*A007467(r-1).

			Written as an irregular triangle the sequence begins:
   2;
   3,   2,   5;
   7,   3,  11,   5,  13;
  17,   7,  19,  11,  23,  13,  29;
  31,  17,  37,  19,  41,  23,  43,  29,  47;
  53,  31,  59,  37,  61,  41,  67,  43,  71,  47,  73;
  79,  53,  83,  59,  89,  61,  97,  67, 101,  71, 103,  73, 107;
  ...
The triangle can be arranged as shown below so that, in every row, each odd position term is equal to the term immediately below it.
                2
             3  2  5
          7  3 11  5 13
      17  7 19 11 23 13 29
   31 17 37 19 41 23 43 29 47
              ...

Cf. A000040, A117384, A000290, A001614, A005408, A007467, A007468, A011756.

Cf. A078721, A078722, A078724, A078725.

nterms=64;a=ConstantArray[0,nterms];For[n=1;p=1,n<=nterms,n++,If[a[[n]]==0,a[[n]]=Prime[p];If[(d=4p-n)<=nterms,a[[d]]=a[[n]]];p++]]; a
(* Second program, triangle rows *)
nrows=8;Table[rlen=2r-1;Permute[Prime[Range[s=1+(r-1)(r-2)/2,s+rlen-1]],Join[Range[2,rlen,2],Range[1,rlen,2]]],{r,nrows}]

a(A001614(n)) = a(4*n-A001614(n)) = prime(n).

cp's OEIS Frontend

A118012 a(n) = 4*A117384(n) - n; a self-inverse permutation of the natural numbers.

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A001614 Connell sequence: 1 odd, 2 even, 3 odd, ...

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A118011 Complement of the Connell sequence (A001614); a(n) = 4*n - A001614(n).

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A117385 Positive integers, each occurring twice in the sequence, such that a(n) = a(k) when n+k = 5*a(n), starting with a(1)=1 and filling the next vacant position with the smallest unused number.

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A117386 Positive integers, each occurring twice in the sequence, such that a(n) = a(k) when n+k = 6*a(n), starting with a(1)=1 and filling the next vacant position with the smallest unused number.

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A344482 Primes, each occurring twice, such that a(C(n)) = a(4*n-C(n)) = prime(n), where C is the Connell sequence (A001614).

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