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%I #17 Mar 05 2022 03:55:51

%S 1,4,2,7,5,3,10,8,6,4,13,11,9,7,5,16,14,12,10,8,6,19,17,15,13,11,9,7,

%T 22,20,18,16,14,12,10,8,25,23,21,19,17,15,13,11,9,28,26,24,22,20,18,

%U 16,14,12,10

%N 3*A002024 - 2*A051340.

%C Row sums = the hexagonal numbers, A000384: (1, 6, 15, 28, 45, ...).

%C From _Boris Putievskiy_, Jan 24 2013: (Start)

%C Table T(n,k) = n + 3*k - 3, n, k > 0, read by antidiagonals. General case A209304. Let m be a positive integer. The first column of the table T(n,1) is the sequence of the positive integers A000027. Every subsequent column is formed from the previous column, shifted by m elements.

%C For m=0 the result is A002260,

%C for m=1 the result is A002024,

%C for m=2 the result is A128076,

%C for m=3 the result is A131914,

%C for m=4 the result is A209304. (End)

%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 3*A002024 - 2*A051340 as infinite lower triangular matrices.

%F From _Boris Putievskiy_, Jan 24 2013: (Start)

%F For the general case

%F a(n) = m*A003056 - (m-1)*A002260.

%F a(n) = m*(t+1) + (m-1)*(t*(t+1)/2-n), where t = floor((-1+sqrt(8*n-7))/2).

%F For m = 3,

%F a(n) = 3*A003056 - 2*A002260.

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

%e First few rows of the triangle:

%e 1;

%e 4, 2;

%e 7, 5, 3;

%e 10, 8, 6, 4;

%e 13, 11, 9, 7, 5;

%e 16, 14, 12, 10, 8, 6;

%e 19, 17, 15, 13, 11, 9, 7;

%e ...

%Y Cf. A002024, A051340, A000384, A003056, A002260, A002024, A128076, A209304.

%K nonn,tabl

%O 1,2

%A _Gary W. Adamson_, Jul 27 2007