A120027 - OEIS

%I #24 Feb 14 2022 03:56:10

%S 1,3,5,9,15,25,27,45,75,125,81,135,225,375,625,243,405,675,1125,1875,

%T 3125,729,1215,2025,3375,5625,9375,15625,2187,3645,6075,10125,16875,

%U 28125,46875,78125,6561,10935,18225,30375,50625,84375,140625,234375

%N Triangle, generated from (3^(n-k) * 5^k) table.

%C Row 1 of the array (3, 15, 75, 375, ...) = A005053, (3 * 5^n), deleting the "1".

%H Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations [Of] Integer Sequences And Pairing Functions</a>, arXiv preprint arXiv:1212.2732 [math.CO], 2012.

%F Antidiagonals of the (3^i * 5^j) multiplication table, as an array.

%F From _Boris Putievskiy_, Jan 09 2013: (Start)

%F T(n,k) = 3^(k-1)*5^(n-1) n, k >0 read by antidiagonals.

%F a(n) = 3^(A004736(n)-1) * 5^(A002260(n)-1), n > 0, or

%F a(n) = 3^(j-1) * 5^(i-1), n > 0,

%F where i = n - t*(t+1)/2, j = (t*t + 3*t + 4)/2 - n, t = floor((-1+sqrt(8*n-7))/2). (End)

%F G.f.: 1/((1 - 3*x)(1 - 5*x*y)). - _Ilya Gutkovskiy_, Jun 03 2017

%e First few rows of the array:

%e   1,  5,  25,  125, ...

%e   3, 15,  75,  375, ...

%e   9, 45, 225, 1125, ...

%e First few rows of the triangle are:

%e    1;

%e    3,  5;

%e    9, 15, 25;

%e   27, 45, 75, 125;

%e   ...

%e Example: a(17) = 675 = (3,2) in the array, = 3^3 * 5^2.

%t Table[3^(n - k)*5^k, {n, 0, 8}, {k, 0, n}] // Flatten (* _Robert G. Wilson v_, Jun 06 2006 *)

%Y Cf. A005053, A036561, A036565, A036566.

%K nonn,tabl

%O 0,2

%A _Gary W. Adamson_, Jun 04 2006

%E More terms from _Robert G. Wilson v_, Jun 06 2006