A071797 - OEIS

%I #70 Sep 12 2022 09:52:30

%S 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,

%T 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,

%U 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

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

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

%C 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

%C a(n) mod 2 = A071028(n). - _Boris Putievskiy_, Jul 24 2013

%C The triangle in the example is the triangle used by Kircheri in 1664. See the link "Mundus Subterraneus". - _Charles Kusniec_, Sep 11 2022

%H Reinhard Zumkeller, <a href="/A071797/b071797.txt">Table of n, a(n) for n = 1..1000</a>

%H Glen Joyce C. Dulatre, Jamilah V. Alarcon, Vhenedict M. Florida, Daisy Ann A. Disu, <a href="http://www.dmmmsu-sluc.com/wp-content/uploads/2018/03/CAS-Monitor-2016-2017-1.pdf">On Fractal Sequences</a>, DMMMSU-CAS Science Monitor (2016-2017) Vol. 15 No. 2, 109-113.

%H C. Kimberling, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa73/aa7321.pdf">Numeration systems and fractal sequences</a>, Acta Arithmetica 73 (1995) 103-117.

%H Athanasii Kircheri, <a href="https://archive.org/details/athanasiikircher00kirc_4/page/n53/mode/2up">Mundus Subterraneus</a>, (1664), pg. 24.

%H F. Smarandache, <a href="http://www.gallup.unm.edu/~smarandache/OPNS.pdf">Only Problems, Not Solutions!</a>, Phoenix,AZ: Xiquan,1993.

%H Michael Somos, <a href="/A073189/a073189.txt">Sequences used for indexing triangular or square arrays</a>

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

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

%F a(n) = n - floor(sqrt(n-1))^2, distance between n and the next smaller square. - _Marc LeBrun_, Jan 14 2004

%e a(1)=1; a(9)=5; a(10)=1;

%e From _Omar E. Pol_, May 29 2012: (Start)

%e Written as a triangle the sequence begins:

%e   1;

%e   1, 2, 3;

%e   1, 2, 3, 4, 5;

%e   1, 2, 3, 4, 5, 6, 7;

%e   1, 2, 3, 4, 5, 6, 7, 8, 9;

%e   1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11;

%e   1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13;

%e   1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15;

%e Row n has length 2*n - 1 = A005408(n-1). (End)

%p A071797 := proc(n)

%p     n-A048760(n-1) ;

%p end proc: # _R. J. Mathar_, May 29 2016

%t Array[Range[2# - 1]&, 10] // Flatten (* _Jean-François Alcover_, Jan 30 2018 *)

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

%o (Haskell)

%o import Data.List (inits)

%o a071797 n = a071797_list !! (n-1)

%o a071797_list = f $ tail $ inits [1..] where

%o    f (xs:_:xss) = xs ++ f xss

%o -- _Reinhard Zumkeller_, Apr 14 2014

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

%Y Cf. A074294.

%Y Row sums give positive terms of A000384.

%K easy,nonn,tabf

%O 1,3

%A _Antonio Esposito_, Jun 06 2002