A274773 a(n) = floor(sqrt(2*n-1) + 1/2) - abs(2*(n-1) - (floor(sqrt(2*n-1) + 1/2))^2) + 1.

1, 1, 3, 1, 3, 3, 1, 3, 5, 3, 1, 3, 5, 5, 3, 1, 3, 5, 7, 5, 3, 1, 3, 5, 7, 7, 5, 3, 1, 3, 5, 7, 9, 7, 5, 3, 1, 3, 5, 7, 9, 9, 7, 5, 3, 1, 3, 5, 7, 9, 11, 9, 7, 5, 3, 1, 3, 5, 7, 9, 11, 11, 9, 7, 5, 3, 1, 3, 5, 7, 9, 11, 13, 11, 9, 7, 5, 3, 1, 3, 5, 7, 9, 11, 13, 13, 11, 9, 7, 5, 3, 1, 3, 5, 7, 9, 11, 13, 15, 13, 11, 9, 7, 5, 3, 1, 3, 5, 7, 9, 11, 13, 15, 15, 13, 11, 9, 7, 5, 3

Offset: 1

Author

Ilya Gutkovskiy, Aug 11 2016

Keywords

• nonn
• easy
• tabl

Comments

First bisection of A004738.

All terms are odd.

Examples

			Triangle begins:
  1;
  1, 3;
  1, 3, 3;
  1, 3, 5, 3;
  1, 3, 5, 5, 3;
  1, 3, 5, 7, 5, 3;
  1, 3, 5, 7, 7, 5, 3;
  1, 3, 5, 7, 9, 7, 5, 3;
  1, 3, 5, 7, 9, 9, 7, 5, 3;
  ...
Read like the Ulam spiral, starting with 1:
    x  7  x  5  x  3  x  1
    7  x  5  x  3  x  1  x
    x  5  x  3  x  1  x  3
    5  x  3  x  1  x  3  x
    x  3  x  1  x  3  x  5
    3  x  1  x  3  x  5  x
    x  1  x  3  x  5  x  7
    1  x  3  x  5  x  7  x
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000
• Ilya Gutkovskiy, Illustrations
• Eric Weisstein's World of Mathematics, Smarandache Sequences

Crossrefs

Cf. A004738, A005408.

Programs

Mathematica

Table[Floor[Sqrt[2 n - 1] + 1/2] - Abs[2 (n - 1) - Floor[Sqrt[2 n - 1] + 1/2]^2] + 1, {n, 1, 120}]

Python

from gmpy2 import isqrt_rem
def A274773(n):
    i, j = isqrt_rem(2*n-1)
    return int(i+2 - abs(j-2*(i+1)) if 4*(i-j) + 1 <= 0 else i+1 - abs(j-1)) # Chai Wah Wu, Aug 15 2016