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A131914 3*A002024 - 2*A051340.

%I A131914 #17 Mar 05 2022 03:55:51
%S A131914 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 A131914 22,20,18,16,14,12,10,8,25,23,21,19,17,15,13,11,9,28,26,24,22,20,18,
%U A131914 16,14,12,10
%N A131914 3*A002024 - 2*A051340.
%C A131914 Row sums = the hexagonal numbers, A000384: (1, 6, 15, 28, 45, ...).
%C A131914 From _Boris Putievskiy_, Jan 24 2013: (Start)
%C A131914 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 A131914 For m=0 the result is A002260,
%C A131914 for m=1 the result is A002024,
%C A131914 for m=2 the result is A128076,
%C A131914 for m=3 the result is A131914,
%C A131914 for m=4 the result is A209304. (End)
%H A131914 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 A131914 3*A002024 - 2*A051340 as infinite lower triangular matrices.
%F A131914 From _Boris Putievskiy_, Jan 24 2013: (Start)
%F A131914 For the general case
%F A131914 a(n) = m*A003056 - (m-1)*A002260.
%F A131914 a(n) = m*(t+1) + (m-1)*(t*(t+1)/2-n), where t = floor((-1+sqrt(8*n-7))/2).
%F A131914 For m = 3,
%F A131914 a(n) = 3*A003056 - 2*A002260.
%F A131914 a(n) = 3*(t+1) + 2*(t*(t+1)/2-n), where t = floor((-1+sqrt(8*n-7))/2). (End)
%e A131914 First few rows of the triangle:
%e A131914    1;
%e A131914    4,  2;
%e A131914    7,  5,  3;
%e A131914   10,  8,  6,  4;
%e A131914   13, 11,  9,  7,  5;
%e A131914   16, 14, 12, 10,  8,  6;
%e A131914   19, 17, 15, 13, 11,  9,  7;
%e A131914   ...
%Y A131914 Cf. A002024, A051340, A000384, A003056, A002260, A002024, A128076, A209304.
%K A131914 nonn,tabl
%O A131914 1,2
%A A131914 _Gary W. Adamson_, Jul 27 2007