A138099 - OEIS

%I #26 Sep 08 2022 08:45:33

%S 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 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 12,13,14,15,9,10,11,12,13,14,15,16,9,10,11,12,13

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

%C Previous name was: Base for 8 X 32 Janet periodic table of elements. Ordered energy levels of atoms.

%F a(n) = n - floor((1/4)*(floor(sqrt(4*n-3))-1)^2). - _Mircea Merca_, Feb 05 2012

%F From _Benedict W. J. Irwin_, May 14 2016: (Start)

%F 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                                     o   oo  ooo

%F Empty, o, oo, oo, oo, oo  , ooo , ooo, ooo, ooo, ...

%F               o   oo  ooo   ooo   ooo  ooo  ooo

%F (End)

%e Triangle begins:

%e 1;

%e 2;

%e 2, 3;

%e 3, 4;

%e 3, 4, 5;

%e 4, 5, 6;

%e 4, 5, 6, 7;

%e 5, 6, 7, 8;

%e ...

%t Table[k+Floor[n/2],{n,20},{k,Ceiling[n/2]}]//Flatten (* _Harvey P. Dale_, Aug 15 2017 *)

%o (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 nonn,easy,tabf

%O 1,2

%A _Paul Curtz_, May 03 2008

%E Name changed and more terms added by _Arkadiusz Wesolowski_, Jun 15 2014