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A101037 Triangle read by rows: T(n,1) = T(n,n) = n and for 1 T(n,k) = floor((T(n-1,k-1)+T(n-1,k))/2).

%I A101037 #15 Jan 16 2025 23:30:49
%S A101037 1,2,2,3,2,3,4,2,2,4,5,3,2,3,5,6,4,2,2,4,6,7,5,3,2,3,5,7,8,6,4,2,2,4,
%T A101037 6,8,9,7,5,3,2,3,5,7,9,10,8,6,4,2,2,4,6,8,10,11,9,7,5,3,2,3,5,7,9,11,
%U A101037 12,10,8,6,4,2,2,4,6,8,10,12,13,11,9,7,5,3,2,3,5,7,9,11,13,14,12,10,8,6,4,2,2,4,6,8,10,12,14
%N A101037 Triangle read by rows: T(n,1) = T(n,n) = n and for 1<k<n: T(n,k) = floor((T(n-1,k-1)+T(n-1,k))/2).
%C A101037 For n>1: sum of n-th row = A007590(n+1).
%H A101037 Robert Israel, <a href="/A101037/b101037.txt">Table of n, a(n) for n = 1..10011</a> (rows 1 to 141, flattened)
%F A101037 From _Robert Israel_, Jan 30 2018: (Start)
%F A101037 T(n,k) = n - 2*k + 2 if k < (n+1)/2.
%F A101037 T(n,(n+1)/2) = 2 if n>1 is odd.
%F A101037 T(n,k) = 2*k - n if k > (n+1)/2.
%F A101037 G.f. as triangle: x*y*(x^6*y^3-2*x^5*y^3-2*x^5*y^2+x^4*y^3+3*x^4*y^2+x^4*y-3*x^2*y+1)/((1-x^2*y)*(1-x)^2*(1-x*y)^2).
%F A101037 (End)
%e A101037 Triangle begins:
%e A101037   1;
%e A101037   2, 2;
%e A101037   3, 2, 3;
%e A101037   4, 2, 2, 4;
%e A101037   5, 3, 2, 3, 5;
%e A101037   6, 4, 2, 2, 4, 6;
%e A101037   7, 5, 3, 2, 3, 5, 7;
%e A101037   ...
%p A101037 T:= proc(n,k) if k < (n+1)/2 then n-2*k+2 elif k=(n+1)/2 then 2 else 2*k-n fi end proc:
%p A101037 T(1,1):= 1:
%p A101037 seq(seq(T(n,k),k=1..n),n=1..20); # _Robert Israel_, Jan 30 2018
%t A101037 T[n_, 1] := n; T[n_, n_] := n; T[n_, k_] := T[n, k] = Which[k < (n + 1)/2, n - 2*k + 2, k == (n + 1)/2, 2, True, 2*k - n];
%t A101037 Table[T[n, k], {n, 1, 13}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Mar 04 2019 *)
%K A101037 nonn,tabl
%O A101037 1,2
%A A101037 _Reinhard Zumkeller_, Nov 27 2004