A007401 Add n-1 to n-th term of 'n appears n times' sequence (A002024).

1, 3, 4, 6, 7, 8, 10, 11, 12, 13, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76

Offset: 1

Author

N. J. A. Sloane, Mira Bernstein

Keywords

• nonn
• easy

Comments

Complement of A000096 = increasing sequence of positive integers excluding n*(n+3)/2. - Jonathan Vos Post, Aug 13 2005

As a triangle: (1; 3,4; 6,7,8; 10,11,12,13; ...), Row sums = A127736: (1, 7, 21, 46, 85, 141, 217, ...). - Gary W. Adamson, Oct 25 2007

Odd primes are a subsequence except 5, cf. A004139. - Reinhard Zumkeller, Jul 18 2011

A023532(a(n)) = 1. - Reinhard Zumkeller, Dec 04 2012

T(n,k) = ((n+k)^2+n-k)/2 - 1, n,k > 0, read by antidiagonals. - Boris Putievskiy, Jan 14 2013

A023531(a(n)) = 0. - Reinhard Zumkeller, Feb 14 2015

Examples

			From _Boris Putievskiy_, Jan 14 2013: (Start)
The start of the sequence as table:
   1,  3,  6, 10, 15, 21, 28, ...
   4,  7, 11, 16, 22, 29, 37, ...
   8, 12, 17, 23, 30, 38, 47, ...
  13, 18, 24, 31, 39, 48, 58, ...
  19, 25, 32, 40, 49, 59, 70, ...
  26, 33, 41, 50, 60, 71, 83, ...
  34, 42, 51, 61, 72, 84, 97, ...
  ...
The start of the sequence as triangle array read by rows:
   1;
   3,  4;
   6,  7,  8;
  10, 11, 12, 13;
  15, 16, 17, 18, 19;
  21, 22, 23, 24, 25, 26;
  28, 29, 30, 31, 32, 33, 34;
  ...
Row number r contains r numbers r*(r+1)/2, r*(r+1)/2+1, ..., r*(r+1)/2+r-1. (End)
		

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
• J. Riordan, The enumeration of trees by height and diameter, IBM Journal 4 (1960), 473-478. (Annotated scanned copy)
• T. R. S. Walsh & N. J. A. Sloane, Correspondence, 1991
• T. R. S. Walsh, Data (Preprint 1, Part 1)
• T. R. S. Walsh, Data (Preprint 1, Part 2)
• T. R. S. Walsh, Data (Preprint 1, Part 3)
• T. R. S. Walsh, Notes
• T. R. S. Walsh, Number of sensed planar maps with n edges and m vertices
• N. C. Wormald, On the number of planar maps, Can. J. Math. 33.1 (1981), 1-11. (Annotated scanned copy)

Crossrefs

Cf. A002024, A000096, A127736, A014132, A002260, A004736, A003057, A114327, A023531.

Programs

Haskell

a007401 n = a007401_list !! n
a007701_list = [x | x <- [0..], a023531 x == 0]
-- Reinhard Zumkeller, Feb 14 2015, Dec 04 2012

Mathematica

f[n_] := n + Floor[ Sqrt[2n] - 1/2]; Array[f, 66]; (* Robert G. Wilson v, Feb 13 2011 *)

PARI

a(n)=n+floor(sqrt(n+n)-1/2) \\ Charles R Greathouse IV, Feb 13 2011

PARI

for(m=1,9, for(n=m*(m+1)/2,(m^2+3*m-2)/2, print1(n", "))) \\ Charles R Greathouse IV, Feb 13 2011

Python

from math import isqrt
def A007401(n): return n-1+(isqrt(n<<3)+1>>1) # Chai Wah Wu, Oct 18 2022

Formula

From Boris Putievskiy, Jan 14 2013: (Start)

a(n) = A014132(n) - 1.

a(n) = A003057(n)^2 - A114327(n) - 1.

a(n) = ((t+2)^2 + i - j)/2-1, where

i = n-t*(t+1)/2,

j = (t*t+3*t+4)/2-n,

t = floor((-1+sqrt(8*n-7))/2). (End)