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A069011 Triangle with T(n,k) = n^2 + k^2.

%I A069011 #31 Aug 07 2025 13:10:27
%S A069011 0,1,2,4,5,8,9,10,13,18,16,17,20,25,32,25,26,29,34,41,50,36,37,40,45,
%T A069011 52,61,72,49,50,53,58,65,74,85,98,64,65,68,73,80,89,100,113,128,81,82,
%U A069011 85,90,97,106,117,130,145,162,100,101,104,109,116,125,136,149,164,181,200
%N A069011 Triangle with T(n,k) = n^2 + k^2.
%C A069011 For any i,j >=0 a(i)*a(j) is a member of this sequence, since (a^2 + b^2)*(c^2 + d^2) = (a*c + b*d)^2 + (a*d - b*c)^2. - _Boris Putievskiy_, May 05 2013
%C A069011 A227481(n) = number of squares in row n. - _Reinhard Zumkeller_, Oct 11 2013
%C A069011 Norm of the complex numbers n +- i*k and k +- i*n, where i denotes the imaginary unit. - _Stefano Spezia_, Aug 07 2025
%H A069011 Reinhard Zumkeller, <a href="/A069011/b069011.txt">Rows n = 0..120 of triangle, flattened</a>
%F A069011 T(n+1,k+1) = T(n,k) + 2*(n+k+1), k=0..n; T(n+1,0) = T(n,0) + 2*n + 1. - _Reinhard Zumkeller_, Oct 11 2013
%F A069011 G.f.: x*(1 + 2*y + 5*x^3*y^2 - x^2*y*(2 + 5*y) + x*(1 - 4*y + 2*y^2))/((1 - x)^3*(1 - x*y)^3). - _Stefano Spezia_, Aug 04 2025
%e A069011 Triangle T(n,k) begins:
%e A069011     0;
%e A069011     1,   2;
%e A069011     4,   5,   8;
%e A069011     9,  10,  13,  18;
%e A069011    16,  17,  20,  25,  32;
%e A069011    25,  26,  29,  34,  41,  50;
%e A069011    36,  37,  40,  45,  52,  61,  72;
%e A069011    49,  50,  53,  58,  65,  74,  85,  98;
%e A069011    64,  65,  68,  73,  80,  89, 100, 113, 128;
%e A069011    81,  82,  85,  90,  97, 106, 117, 130, 145, 162;
%e A069011   100, 101, 104, 109, 116, 125, 136, 149, 164, 181, 200;
%e A069011   ...
%t A069011 Table[n^2 + k^2, {n, 0, 12}, {k, 0, n}] (* _Paolo Xausa_, Aug 07 2025 *)
%o A069011 (Haskell)
%o A069011 a069011 n k = a069011_tabl !! n !! k
%o A069011 a069011_row n = a069011_tabl !! n
%o A069011 a069011_tabl = map snd $ iterate f (1, [0]) where
%o A069011    f (i, xs@(x:_)) = (i + 2, (x + i) : zipWith (+) xs [i + 1, i + 3 ..])
%o A069011 -- _Reinhard Zumkeller_, Oct 11 2013
%Y A069011 Cf. A001481 for terms in this sequence, A000161 for number of times each term appears, A048147 for square array.
%Y A069011 Column k=0 gives A000290.
%Y A069011 Main diagonal gives A001105.
%Y A069011 Row sums give A132124.
%Y A069011 T(2n,n) gives A033429.
%K A069011 easy,nonn,tabl
%O A069011 0,3
%A A069011 _Henry Bottomley_, Apr 02 2002