A007401 Add n-1 to n-th term of 'n appears n times' sequence (A002024). (Formerly M2316)

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

1,2

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

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)

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)

Example

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)

Mathematica

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

Prog

(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
(Haskell)
a007401 n = a007401_list !! n
a007701_list = [x | x <- [0..], a023531 x == 0]
-- Reinhard Zumkeller, Feb 14 2015, Dec 04 2012
(Python)
from math import isqrt
def A007401(n): return n-1+(isqrt(n<<3)+1>>1) # Chai Wah Wu, Oct 18 2022

Crossrefs

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

Sequence in context: A298644 A075747 A143344 * A275481 A234349 A042954

Adjacent sequences: A007398 A007399 A007400 * A007402 A007403 A007404

Keyword

nonn,easy

Author

N. J. A. Sloane, Mira Bernstein

Status

approved