A001614 Connell sequence: 1 odd, 2 even, 3 odd, ... (Formerly M0962 N0359)

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

1,2

Comments

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

References

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).

Links

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

Formula

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)

Example

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)

Maple

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

Mathematica

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 *)

Prog

(Haskell)
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
(Magma) [2*n-Round(Sqrt(2*n)): n in [1..80]]; // Vincenzo Librandi, Apr 17 2015
(PARI) a(n)=2*n - round(sqrt(2*n)) \\ Charles R Greathouse IV, Apr 20 2015
(Python)
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

Crossrefs

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. A057211, A023531.

Sequence in context: A190228 A286667 A083120 * A284535 A244222 A360531

Adjacent sequences: A001611 A001612 A001613 * A001615 A001616 A001617

Keyword

nonn,easy,nice,tabl

Author

N. J. A. Sloane

Extensions

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

Status

approved