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A060736 Array of square numbers read by antidiagonals in up direction.

%I A060736 #25 Feb 16 2025 08:32:44
%S A060736 1,2,4,5,3,9,10,6,8,16,17,11,7,15,25,26,18,12,14,24,36,37,27,19,13,23,
%T A060736 35,49,50,38,28,20,22,34,48,64,65,51,39,29,21,33,47,63,81,82,66,52,40,
%U A060736 30,32,46,62,80,100
%N A060736 Array of square numbers read by antidiagonals in up direction.
%C A060736 A simple permutation of natural numbers.
%C A060736 a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers. - _Boris Putievskiy_, Jan 09 2013
%H A060736 Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations [of] Integer Sequences And Pairing Functions</a> arXiv:1212.2732 [math.CO], 2012.
%H A060736 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PairingFunction.html">Pairing functions</a>
%H A060736 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A060736 T(n+1, k)=n*n+k, T(k, n+1)=(n+1)*(n+1)+1-k, 1 <= k <= n+1.
%F A060736 a(n)=i^2-j+1 if i >= j, a(n)=(j-1)^2 + i if i <  j, where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2). - _Boris Putievskiy_, Jan 09 2013
%e A060736 1 4 9 16 .. => a(1)= 1
%e A060736 2 3 8 15 .. => a(2)= 2, a(3)=4
%e A060736 5 6 7 14 .. => a(4)= 5, a(5)=3, a(6)=9
%e A060736 10 11 12 13 .. => a(7)=10, a(8)=6, a(9)=8, a(10)=16
%t A060736 Table[ If[n < 2*k-1, k^2 + k - n, (n-k)^2 + k], {n, 1, 10}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jan 09 2013 *)
%o A060736 (Python)
%o A060736 t=int((math.sqrt(8*n-7) - 1)/ 2)
%o A060736 i=n-t*(t+1)/2
%o A060736 j=(t*t+3*t+4)/2-n
%o A060736 if i>=j:
%o A060736    result=i**2-j+1
%o A060736 else:
%o A060736    result=(j-1)**2+i
%o A060736 # _Boris Putievskiy_, Jan 09 2013
%Y A060736 Cf. A060734. Inverse permutation: A064788, the first inverse function (numbers of rows) A194258, the second inverse function (numbers of columns) A194195.
%K A060736 nonn,tabl
%O A060736 1,2
%A A060736 _Frank Ellermann_, Apr 23 2001