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A138099 Irregular triangle read by rows: T(n,k) = k + floor(n/2), 1 <= k <= ceiling(n/2).

%I A138099 #26 Sep 08 2022 08:45:33
%S A138099 1,2,2,3,3,4,3,4,5,4,5,6,4,5,6,7,5,6,7,8,5,6,7,8,9,6,7,8,9,10,6,7,8,9,
%T A138099 10,11,7,8,9,10,11,12,7,8,9,10,11,12,13,8,9,10,11,12,13,14,8,9,10,11,
%U A138099 12,13,14,15,9,10,11,12,13,14,15,16,9,10,11,12,13
%N A138099 Irregular triangle read by rows: T(n,k) = k + floor(n/2), 1 <= k <= ceiling(n/2).
%C A138099 Previous name was: Base for 8 X 32 Janet periodic table of elements. Ordered energy levels of atoms.
%F A138099 a(n) = n - floor((1/4)*(floor(sqrt(4*n-3))-1)^2). - _Mircea Merca_, Feb 05 2012
%F A138099 From _Benedict W. J. Irwin_, May 14 2016: (Start)
%F A138099 Conjecture: a(n) = HookSum(T(n))-HookSum(T(n-1)), where HookSum(A) is the sum of all hooks across a tableau A, and T(n) are the spiraling sequence of tableaux:
%F A138099                                     o   oo  ooo
%F A138099 Empty, o, oo, oo, oo, oo  , ooo , ooo, ooo, ooo, ...
%F A138099               o   oo  ooo   ooo   ooo  ooo  ooo
%F A138099 (End)
%e A138099 Triangle begins:
%e A138099 1;
%e A138099 2;
%e A138099 2, 3;
%e A138099 3, 4;
%e A138099 3, 4, 5;
%e A138099 4, 5, 6;
%e A138099 4, 5, 6, 7;
%e A138099 5, 6, 7, 8;
%e A138099 ...
%t A138099 Table[k+Floor[n/2],{n,20},{k,Ceiling[n/2]}]//Flatten (* _Harvey P. Dale_, Aug 15 2017 *)
%o A138099 (Magma) lst:=[]; for n in [1..16] do for k in [1..Ceiling(n/2)] do Append(~lst, k+Floor(n/2)); end for; end for; lst; // _Arkadiusz Wesolowski_, Jun 15 2014
%K A138099 nonn,easy,tabf
%O A138099 1,2
%A A138099 _Paul Curtz_, May 03 2008
%E A138099 Name changed and more terms added by _Arkadiusz Wesolowski_, Jun 15 2014