|
%I #19 Feb 16 2022 18:24:39
%S 1,5,2,9,6,3,13,10,7,4,17,14,11,8,5,21,18,15,12,9,6,25,22,19,16,13,10,
%T 7,29,26,23,20,17,14,11,8,33,30,27,24,21,18,15,12,9,37,34,31,28,25,22,
%U 19,16,13,10,41,38,35,32,29,26,23,20,17,14,11,45,42,39,36,33,30,27
%N Table T(n,k)=n+4*k-4 n, k > 0, read by antidiagonals.
%C In general, let m be natural number. The first column of the table T(n,1) is the sequence of the natural numbers A000027. Every next column is formed from previous 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 This sequence is result for m=4
%H Boris Putievskiy, <a href="/A209304/b209304.txt">Rows n = 1..140 of triangle, flattened</a>
%H Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations Integer Sequences And Pairing Functions</a>, arXiv:1212.2732 [math.CO], 2012.
%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),
%F where t=floor((-1+sqrt(8*n-7))/2).
%F For m = 4
%F a(n) = 4*A003056 -3*A002260.
%F a(n) = 4*(t+1)+3*(t*(t+1)/2-n),
%F where t=floor((-1+sqrt(8*n-7))/2).
%e The start of the sequence as table for general case:
%e 1...m+1...2*m+1...3*m+1...4*m+1...5*m+1...6*m+1 ...
%e 2...m+2...2*m+2...3*m+2...4*m+2...5*m+2...6*m+2 ...
%e 3...m+3...2*m+3...3*m+3...4*m+3...5*m+3...6*m+3 ...
%e 4...m+4...2*m+4...3*m+4...4*m+4...5*m+4...6*m+4 ...
%e 5...m+5...2*m+5...3*m+5...4*m+5...5*m+5...6*m+5 ...
%e 6...m+6...2*m+6...3*m+6...4*m+6...5*m+6...6*m+6 ...
%e 7...m+7...2*m+7...3*m+7...4*m+7...5*m+7...6*m+7 ...
%e ...
%e The start of the sequence as triangle array read by rows for general case:
%e 1;
%e m+1, 2;
%e 2*m+1, m+2, 3;
%e 3*m+1, 2*m+2, m+3, 4;
%e 4*m+1, 3*m+2, 2*m+3, m+4, 5;
%e 5*m+1, 4*m+2, 3*m+3, 2*m+4, m+5, 6;
%e 6*m+1, 5*m+2, 4*m+3, 3*m+4, 2*m+5, m+6, 7;
%e ...
%e Row number r contains r numbers: (r-1)*m+1, (r-2)*m+2,...m+r-1, r.
%e The start of the sequence as triangle array read by rows for m=4:
%e 1;
%e 5,2;
%e 9,6,3;
%e 13,10,7,4;
%e 17,14,11,8,5;
%e 21,18,15,12,9,6;
%e 25,22,19,16,13,10,7;
%e ...
%o (Python)
%o t=int((math.sqrt(8*n-7) - 1)/ 2)
%o result = +4*(t+1) + 3*(t*(t+1)/2-n)
%Y Cf. A002260, A002024, A128076, A131914, A003056.
%K nonn,tabl
%O 1,2
%A _Boris Putievskiy_, Jan 18 2013
|
