A244840 - OEIS

%I #27 Jul 28 2014 16:27:14

%S 1,1,2,1,2,1,1,1,1,2,1,1,3,2,1,1,2,1,4,1,2,1,2,1,4,5,2,1,1,1,3,1,5,3,

%T 1,2,1,1,1,1,5,1,7,2,1,1,2,1,4,1,2,7,8,1,2,1,2,3,4,1,6,7,8,9,2,1,1,1,

%U 1,2,5,1,7,4,3,5,1,2

%N Denominators of the triangle T(n,k) = (n*(n+1)/2+k+1)/(k+1) for n >= k >= 0.

%C Numerators: A244734(n,k).

%C See A244734 for the first entries of the rational triangle T(n,k).

%F a(n,k) = denominator((n*(n+1)/2 + k + 1)/(k+1)) for n >= k >= 0.

%e T(0,0) = 1/1, T(1,0) = 2/1, T(1,1) = 3/2,... .

%e The triangle a(n,k) begins:

%e n/k  0 1 2 3 4 5 6 7 8  9 10 11 12 13 14 15 16 17 18 19 20 ...

%e 0:   1

%e 1:   1 2

%e 2:   1 2 1

%e 3:   1 1 1 2

%e 4:   1 1 3 2 1

%e 5:   1 2 1 4 1 2

%e 6:   1 2 1 4 5 2 1

%e 7:   1 1 3 1 5 3 1 2

%e 8:   1 1 1 1 5 1 7 2 1

%e 9:   1 2 1 4 1 2 7 8 1  2

%e 10:  1 2 3 4 1 6 7 8 9  2  1

%e 11:  1 1 1 2 5 1 7 4 3  5  1  2

%e 12:  1 1 1 2 5 1 7 4 3  5 11  2 1

%e 13:  1 2 3 4 5 6 1 8 9 10 11 12 1   2

%e 14:  1 2 1 4 1 2 1 8 3  2 11  4 13  2  1

%e 15:  1 1 1 1 1 1 7 1 3  1 11  1 13  7  1  2

%e 16:  1 1 3 1 5 3 7 1 9  5 11  3 13  7 15  2  1

%e 17:  1 2 1 4 5 2 7 8 1 10 11  4 13 14  5 16  1  2

%e 18:  1 2 1 4 5 2 7 8 1 10 11  4 13 14  5 16 17  2  1

%e 19:  1 1 3 2 1 3 7 4 9  1 11  6 13  7  3  8 17  9  1  2

%e 20:  1 1 1 2 1 1 1 4 3  1 11  2 13  1  1  8 17  3 19  2  1

%e n/k  0 1 2 3 4 5 6 7 8  9 10 11 12 13 14 15 16 17 18 19 20 ...

%e .. reformatted - _Wolfdieter Lang_, Jul 28 2014 .

%e The second column is of period 4: repeat 2, 2, 1, 1. From A014695 or A130658.

%e The third column is of period 3: repeat 1, 1, 3. From A109007.

%e The fourth column is of period 8: repeat 2, 2, 4, 4, 1, 1, 4, 4.

%e The fifth column is of period 5: repeat 1, 1, 5, 5, 5.

%e The sixth column is of period 12: repeat 2, 2, 3, 1, 2, 6, 1, 1, 6, 2, 1, 3 .

%e The seventh column is of period 7: repeat 1, 1, 7, 7, 7, 7, 7.

%e Hence the positive terms of A022998.

%e Main diagonal: A000034(n).

%e Alternate main and second diagonal: A130658(n).

%e Common denominator by row: 1, 2, 2, 2, 6, 4, 20, 30, 70, ... .

%t Table[(n*(n+1)/2+k+1)/(k+1) // Denominator, {n, 0, 11}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 08 2014 *)

%Y Cf. A014695, A130658, A109007, A002260, A022998, A244734, A000034.

%K nonn,frac,tabl,easy

%O 0,3

%A _Paul Curtz_, Jul 07 2014

%E Editse: Name reformulated, comment with T(n,k) reference added. - _Wolfdieter Lang_, Jul 28 2014