cp's OEIS Frontend

A209302 Table T(n,k) = max{n+k-1, n+k-1} n, k > 0, read by sides of squares from T(1,n) to T(n,n), then from T(n,n) to T(n,1).

%I A209302 #28 Jun 09 2025 00:52:50
%S A209302 1,2,3,2,3,4,5,4,3,4,5,6,7,6,5,4,5,6,7,8,9,8,7,6,5,6,7,8,9,10,11,10,9,
%T A209302 8,7,6,7,8,9,10,11,12,13,12,11,10,9,8,7,8,9,10,11,12,13,14,15,14,13,
%U A209302 12,11,10,9,8,9,10,11,12,13,14,15,16,17,16,15,14
%N A209302 Table T(n,k) = max{n+k-1, n+k-1} n, k > 0, read by sides of squares from T(1,n) to T(n,n), then from T(n,n) to T(n,1).
%H A209302 Boris Putievskiy, <a href="/A209302/b209302.txt">Rows n = 1..140 of triangle, flattened</a>
%H A209302 Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations Integer Sequences And Pairing Functions</a>, arXiv:1212.2732 [math.CO], 2012.
%F A209302 In general, let m be a natural number. Table T(n,k) = max{m*n+k-m, n+m*k-m}. For the general case,
%F A209302 a(n) = (m+1)*sqrt(n-1) + 1 - |n - floor(sqrt(n-1))^2 - floor(sqrt(n-1))|.
%F A209302 For m=1,
%F A209302 a(n) = 2*sqrt(n-1) + 1 - |n - floor(sqrt(n-1))^2 - floor(sqrt(n-1))|.
%F A209302 a(n) = t + |t^2 - n|, where t = floor(sqrt(n)+1/2). - _Ridouane Oudra_, May 07 2019
%e A209302 The start of the sequence as a table for the general case:
%e A209302       1   m+1 2*m+1 3*m+1 4*m+1 5*m+1 6*m+1 ...
%e A209302     m+1   m+2 2*m+2 3*m+2 4*m+2 5*m+2 6*m+2 ...
%e A209302   2*m+1 2*m+2 2*m+3 3*m+3 4*m+3 5*m+3 6*m+3 ...
%e A209302   3*m+1 3*m+2 3*m+3 3*m+4 4*m+4 5*m+4 6*m+4 ...
%e A209302   4*m+1 4*m+2 4*m+3 4*m+4 4*m+5 5*m+5 6*m+5 ...
%e A209302   5*m+1 5*m+2 5*m+3 5*m+4 5*m+5 5*m+6 6*m+6 ...
%e A209302   6*m+1 6*m+2 6*m+3 6*m+4 6*m+5 6*m+6 6*m+7 ...
%e A209302   ...
%e A209302 The start of the sequence as a triangular array read by rows for general case:
%e A209302       1;
%e A209302     m+1,   m+2,   m+1;
%e A209302   2*m+1, 2*m+2, 2*m+3, 2*m+2, 2*m+1;
%e A209302   3*m+1, 3*m+2, 3*m+3, 3*m+4, 3*m+3, 3*m+2, 3*m+1;
%e A209302   4*m+1, 4*m+2, 4*m+3, 4*m+4, 4*m+5, 4*m+4, 4*m+3, 4*m+2, 4*m+1;
%e A209302   ...
%e A209302 Row r contains 2*r-1 terms: r*m+1, r*m+2, ... r*m+r, r*m+r+1, r*m+r, ..., r*m+2, r*m+1.
%e A209302 The start of the sequence as triangle array read by rows for m=1:
%e A209302   1;
%e A209302   2,  3,  2;
%e A209302   3,  4,  5,  4,  3;
%e A209302   4,  5,  6,  7,  6,  5,  4;
%e A209302   5,  6,  7,  8,  9,  8,  7,  6,  5;
%e A209302   6,  7,  8,  9, 10, 11, 10,  9,  8,  7,  6;
%e A209302   7,  8,  9, 10, 11, 12, 13, 12, 11, 10,  9,  8,  7;
%e A209302   ...
%o A209302 (Python)
%o A209302 result = 2*int(math.sqrt(n-1)) - abs(n-int(math.sqrt(n-1))**2 - int(math.sqrt(n-1)) -1) +1
%o A209302 (Python)
%o A209302 from math import isqrt
%o A209302 def A209302(n): return (k:=(m:=isqrt(n))+(n-m*(m+1)>=1))+abs(k**2-n) # _Chai Wah Wu_, Jun 08 2025
%Y A209302 Cf.  A187760.
%K A209302 nonn,tabf
%O A209302 1,2
%A A209302 _Boris Putievskiy_, Jan 18 2013