A083479 The natural numbers with all terms of A033638 inserted.

0, 1, 1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 7, 7, 8, 9, 10, 10, 11, 12, 13, 13, 14, 15, 16, 17, 17, 18, 19, 20, 21, 21, 22, 23, 24, 25, 26, 26, 27, 28, 29, 30, 31, 31, 32, 33, 34, 35, 36, 37, 37, 38, 39, 40, 41, 42, 43, 43, 44, 45, 46, 47, 48, 49, 50, 50, 51, 52, 53, 54, 55, 56, 57, 57

Offset: 0

Author

Alford Arnold, Jun 08 2003

Keywords

• easy
• nonn
• tabf

Comments

Row n of A049597 has a(n+1) nonzero values.

When considering the set of nested parabolas defined by -(x^2) + p*x for integer values of p, a(n) tells us how many parabolas are intersected by the line from (1,n) to (n,n). - Gregory R. Bryant, Apr 01 2013

Number of distinct perimeters for polyominoes with n square cells. - Wesley Prosser, Sep 06 2017

Examples

			There are three 1's, one from the natural numbers and two from A033638.
When viewed as an array the sequence begins:
   0
   1
   1  1
   2  2
   3  3  4
   5  5  6
   7  7  8  9
  10 10 11 12
  13 13 14 15 16
  17 17 18 19 20
  21 21 22 23 24 25
  26 26 27 28 29 30
  ...
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A002620, A027434, A027709, A033638, A049068, A049597.

Cf. A054243, A060510, A080037, A083480, A083906.

Programs

Haskell

a083479 n = a083479_list !! n
a083479_list = m [0..] a033638_list where
   m xs'@(x:xs) ys'@(y:ys) | x <= y    = x : m xs ys'
                           | otherwise = y : m xs' ys
-- Reinhard Zumkeller, Apr 06 2012

Magma

[n eq 0 select 0 else (n+2)-Ceiling(Sqrt(4*n)): n in [0..100]]; // G. C. Greubel, Feb 17 2024

Mathematica

Table[(n + 2) - Ceiling@ Sqrt[4 n] - 2 Boole[n == 0], {n, 0, 73}] (* Michael De Vlieger, Sep 05 2017 *)

Maxima

a(n):=((n+2)-ceiling(sqrt(4*n))); /* Gregory R. Bryant, Apr 01 2013 */

Python

from math import isqrt
def A083479(n): return n+1-isqrt((n<<2)-1) if n else 0 # Chai Wah Wu, Jul 28 2022

SageMath

[(n+2)-ceil(sqrt(4*n)) -2*int(n==0) for n in range(101)] # G. C. Greubel, Feb 17 2024

Formula

a(n) = (n+2) - ceiling(sqrt(4*n)), for n > 0. - Gregory R. Bryant, Apr 01 2013

From Wesley Prosser, Sep 06 2017: (Start)

a(n) = (n+2) - A027709(n)/2.

a(n) = (n+2) - A027434(n).

a(n) = (2n+2) - A049068(n).

a(n) = (2n+3) - A080037(n).

(End)

Extensions

Edited and extended by David Wasserman, Nov 16 2004