A071797 Restart counting after each new odd integer (a fractal sequence).

1, 1, 2, 3, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11

Offset: 1

Author

Antonio Esposito, Jun 06 2002

Keywords

• easy
• nonn
• tabf

Comments

The following sequences all have the same parity: A004737, A006590, A027052, A071028, A071797, A078358, A078446.

This is also a triangle read by rows in which row n lists the first 2*n-1 positive integers, n >= 1 (see example). - Omar E. Pol, May 29 2012

a(n) mod 2 = A071028(n). - Boris Putievskiy, Jul 24 2013

The triangle in the example is the triangle used by Kircheri in 1664. See the link "Mundus Subterraneus". - Charles Kusniec, Sep 11 2022

Examples

			a(1)=1; a(9)=5; a(10)=1;
From _Omar E. Pol_, May 29 2012: (Start)
Written as a triangle the sequence begins:
  1;
  1, 2, 3;
  1, 2, 3, 4, 5;
  1, 2, 3, 4, 5, 6, 7;
  1, 2, 3, 4, 5, 6, 7, 8, 9;
  1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11;
  1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13;
  1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15;
Row n has length 2*n - 1 = A005408(n-1). (End)
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
• Glen Joyce C. Dulatre, Jamilah V. Alarcon, Vhenedict M. Florida, Daisy Ann A. Disu, On Fractal Sequences, DMMMSU-CAS Science Monitor (2016-2017) Vol. 15 No. 2, 109-113.
• C. Kimberling, Numeration systems and fractal sequences, Acta Arithmetica 73 (1995) 103-117.
• Athanasii Kircheri, Mundus Subterraneus, (1664), pg. 24.
• F. Smarandache, Only Problems, Not Solutions!, Phoenix,AZ: Xiquan,1993.
• Michael Somos, Sequences used for indexing triangular or square arrays

Crossrefs

Cf. A002260, A004737, A006590, A027052, A071028, A078358, A078446.

Cf. A074294.

Row sums give positive terms of A000384.

Programs

Haskell

import Data.List (inits)
a071797 n = a071797_list !! (n-1)
a071797_list = f $ tail $ inits [1..] where
   f (xs:_:xss) = xs ++ f xss
-- Reinhard Zumkeller, Apr 14 2014

Maple

A071797 := proc(n)
    n-A048760(n-1) ;
end proc: # R. J. Mathar, May 29 2016

Mathematica

Array[Range[2# - 1]&, 10] // Flatten (* Jean-François Alcover, Jan 30 2018 *)

PARI

a(n)=if(n<1,0,n-sqrtint(n-1)^2)

Formula

a(n) = 1 + A053186(n-1).

a(n) = n - 1 - ceiling(sqrt(n))*(ceiling(sqrt(n))-2); n > 0.

a(n) = n - floor(sqrt(n-1))^2, distance between n and the next smaller square. - Marc LeBrun, Jan 14 2004