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A122196 Fractal sequence: count down by 2's from successive integers.

%I A122196 #44 Jun 05 2021 06:18:49
%S A122196 1,2,3,1,4,2,5,3,1,6,4,2,7,5,3,1,8,6,4,2,9,7,5,3,1,10,8,6,4,2,11,9,7,
%T A122196 5,3,1,12,10,8,6,4,2,13,11,9,7,5,3,1,14,12,10,8,6,4,2,15,13,11,9,7,5,
%U A122196 3,1,16,14,12,10,8,6,4,2,17,15,13,11,9,7,5,3,1,18,16,14,12,10,8,6,4,2,19,17
%N A122196 Fractal sequence: count down by 2's from successive integers.
%C A122196 First differences of A076644. Fractal - deleting the first occurrence of each integer leaves the original sequence. Also, original sequence plus 1. 1's occur at square indices. New values occur at indices m^2+1 and m^2+m+1.
%C A122196 Ordinal transform of A122197.
%C A122196 Row sums give A002620. - _Gary W. Adamson_, Nov 29 2008
%C A122196 From _Gary W. Adamson_, Dec 05 2009: (Start)
%C A122196 A122196 considered as an infinite lower triangular matrix * [1,2,3,...] =
%C A122196 A006918 starting (1, 2, 5, 8, 14, 20, 30, 40, ...).
%C A122196 Let A122196 = an infinite lower triangular matrix M; then lim_{n->infinity} M^n = A171238, a left-shifted vector considered as a matrix. (End)
%C A122196 A122196 is the fractal sequence associated with the dispersion A082156; that is, A122196(n) is the number of the row of A082156 that contains n. - _Clark Kimberling_, Aug 12 2011
%C A122196 From _Johannes W. Meijer_, Sep 09 2013: (Start)
%C A122196 The alternating row sums lead to A004524(n+2).
%C A122196 The antidiagonal sums equal A001840(n). (End)
%H A122196 Reinhard Zumkeller, <a href="/A122196/b122196.txt">Table of n, a(n) for n = 1..10000</a>
%F A122196 From _Boris Putievskiy_, Sep 09 2013: (Start)
%F A122196 a(n) = 2*(1-A122197(n)) + A000267(n-1).
%F A122196 a(n) = floor(sqrt(4*n-1)) - 2*((n-1) mod (t+1)), where t = floor((sqrt(4*n-3)-1)/2). (End)
%F A122196 From _Johannes W. Meijer_, Sep 09 2013: (Start)
%F A122196 T(n, k) = n - 2*k + 2, for n >= 1 and 1 <= k <= floor((n+1)/2).
%F A122196 T(n, k) = A002260(n, n-2*k+2). (End)
%e A122196 The first few rows of the sequence a(n) as a triangle T(n, k):
%e A122196 n/k  1   2   3
%e A122196 1    1
%e A122196 2    2
%e A122196 3    3,  1
%e A122196 4    4,  2
%e A122196 5    5,  3,  1
%e A122196 6    6,  4,  2
%p A122196 From _Johannes W. Meijer_, Sep 09 2013: (Start)
%p A122196 a := proc(n) local t: t:=floor((sqrt(4*n-3)-1)/2): floor(sqrt(4*n-1))-2*((n-1) mod (t+1)) end: seq(a(n), n=1..92); # End first program.
%p A122196 T := (n, k) -> n-2*k+2: seq(seq(T(n, k), k=1..floor((n+1)/2)), n=1..18); # End second program. (End)
%t A122196 Flatten@Range[Range[10], 1, -2] (* _Birkas Gyorgy_, Apr 07 2011 *)
%o A122196 (Haskell)
%o A122196 a122196 n = a122196_list !! (n-1)
%o A122196 a122196_list = concatMap (\x -> enumFromThenTo x (x - 2) 1) [1..]
%o A122196 -- _Reinhard Zumkeller_, Jul 19 2012
%Y A122196 Cf. A076644, A122197, A000290, A033638, A002620, A006918, A171238, A082156, A000267.
%K A122196 easy,nonn,tabf
%O A122196 1,2
%A A122196 _Franklin T. Adams-Watters_, Aug 25 2006