cp's OEIS Frontend

A209304 Table T(n,k)=n+4*k-4 n, k > 0, read by antidiagonals.

%I A209304 #23 Mar 23 2025 23:33:30
%S A209304 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 A209304 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 A209304 19,16,13,10,41,38,35,32,29,26,23,20,17,14,11,45,42,39,36,33,30,27
%N A209304 Table T(n,k)=n+4*k-4 n, k > 0, read by antidiagonals.
%C A209304 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 A209304 For m=0 the result is A002260,
%C A209304 for m=1 the result is A002024,
%C A209304 for m=2 the result is A128076,
%C A209304 for m=3 the result is A131914.
%C A209304 This   sequence is result for m=4
%H A209304 Boris Putievskiy, <a href="/A209304/b209304.txt">Rows n = 1..140 of triangle, flattened</a>
%H A209304 Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations Integer Sequences And Pairing Functions</a>, arXiv:1212.2732 [math.CO], 2012.
%F A209304 For the general case
%F A209304 a(n) = m*A003056 -(m-1)*A002260.
%F A209304 a(n) = m*(t+1) + (m-1)*(t*(t+1)/2-n),
%F A209304 where t=floor((-1+sqrt(8*n-7))/2).
%F A209304 For m = 4
%F A209304 a(n) = 4*A003056 -3*A002260.
%F A209304 a(n) = 4*(t+1)+3*(t*(t+1)/2-n),
%F A209304 where t=floor((-1+sqrt(8*n-7))/2).
%e A209304 The start of the sequence as table for general case:
%e A209304   1...m+1...2*m+1...3*m+1...4*m+1...5*m+1...6*m+1  ...
%e A209304   2...m+2...2*m+2...3*m+2...4*m+2...5*m+2...6*m+2  ...
%e A209304   3...m+3...2*m+3...3*m+3...4*m+3...5*m+3...6*m+3  ...
%e A209304   4...m+4...2*m+4...3*m+4...4*m+4...5*m+4...6*m+4  ...
%e A209304   5...m+5...2*m+5...3*m+5...4*m+5...5*m+5...6*m+5  ...
%e A209304   6...m+6...2*m+6...3*m+6...4*m+6...5*m+6...6*m+6  ...
%e A209304   7...m+7...2*m+7...3*m+7...4*m+7...5*m+7...6*m+7  ...
%e A209304   ...
%e A209304 The start of the sequence as triangle array read by rows for general case:
%e A209304      1;
%e A209304     m+1,    2;
%e A209304   2*m+1,   m+2,   3;
%e A209304   3*m+1, 2*m+2,   m+3,   4;
%e A209304   4*m+1, 3*m+2, 2*m+3,   m+4,   5;
%e A209304   5*m+1, 4*m+2, 3*m+3, 2*m+4,   m+5,  6;
%e A209304   6*m+1, 5*m+2, 4*m+3, 3*m+4, 2*m+5, m+6, 7;
%e A209304   ...
%e A209304 Row number r contains r numbers: (r-1)*m+1, (r-2)*m+2,...m+r-1, r.
%e A209304 The start of the sequence as triangle array read by rows for m=4:
%e A209304   1;
%e A209304   5,2;
%e A209304   9,6,3;
%e A209304   13,10,7,4;
%e A209304   17,14,11,8,5;
%e A209304   21,18,15,12,9,6;
%e A209304   25,22,19,16,13,10,7;
%e A209304   ...
%o A209304 (Python)
%o A209304 t=int((math.sqrt(8*n-7) - 1)/ 2)
%o A209304 result = +4*(t+1) + 3*(t*(t+1)/2-n)
%Y A209304 Cf. A002260, A002024, A128076, A131914, A003056.
%K A209304 nonn,tabl
%O A209304 1,2
%A A209304 _Boris Putievskiy_, Jan 18 2013