This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%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