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A281385 Triangular array T(n, k) = n^2 + n*k - k^2.

%I A281385 #19 Jan 05 2025 19:51:41
%S A281385 0,1,1,4,5,4,9,11,11,9,16,19,20,19,16,25,29,31,31,29,25,36,41,44,45,
%T A281385 44,41,36,49,55,59,61,61,59,55,49,64,71,76,79,80,79,76,71,64,81,89,95,
%U A281385 99,101,101,99,95,89,81,100,109,116,121,124,125,124,121,116,109,100
%N A281385 Triangular array T(n, k) = n^2 + n*k - k^2.
%C A281385 Let {y0, y1, ...} a sequence satisfying y(m) = y(m-1) + y(m-2), then y(m)^2 - y(m-1)*y(m+1) = T(y0, y1)*(-1)^m. See the Fib. Quart. link.
%H A281385 Robert Israel, <a href="/A281385/b281385.txt">Table of n, a(n) for n = 0..10010</a> (rows 0 to 140, flattened)
%H A281385 F. D. Parker, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/13-1/advanced13-1.pdf">The Very Existence: Problem H-248 and solution</a>, The Fibonacci Quarterly, Vol 15, Nr 1, February 1977.
%F A281385 From _Robert Israel_, Jan 23 2017: (Start)
%F A281385 G.f. as triangle: (1 + x + y - x*y - 4*x^2*y + x*y^2 - 4*x^2*y^2 + 5*x^3*y^2)*x/((1-x*y)^3*(1-x)^3).
%F A281385 G.f. as sequence: (1-4*x+x^2 + (3-4*x+x^2)*Sum_{k>=0} k*x^(k*(k+1)/2) + (-1+3*x-2*x^2)*Sum_{k>=0} x^(k*(k+1)/2))/(1-x)^3.
%F A281385 -(5*k-1)*T(n,k-1) + (5*k-2)*T(n,k) + (5*k-3)*T(n-1,k-1) - (5*k-4)*T(n-1,k) = 0 for 1 <= k <= n-1.
%F A281385 (End)
%e A281385 Triangle begins:
%e A281385    0;
%e A281385    1,  1;
%e A281385    4,  5,  4;
%e A281385    9, 11, 11,  9;
%e A281385   16, 19, 20, 19, 16;
%e A281385   25, 29, 31, 31, 29, 25;
%e A281385   36, 41, 44, 45, 44, 41, 36;
%e A281385   ...
%e A281385 A000032 begins {2, 1 ...} and satisfies y(m)^2-y(m-1)*y(m+1) = 5*(-1)^m.
%p A281385 seq(seq(n^2+n*k-k^2, k=0..n),n=0..10); # _Robert Israel_, Jan 23 2017
%t A281385 Table[n^2+n*k-k^2,{n,0,10},{k,0,n}]//Flatten (* _Harvey P. Dale_, May 25 2024 *)
%o A281385 (PARI) T(n, k) = n^2 + n*k - k^2;
%o A281385 lista(nn) = for (n=0, nn, for (k=0,n, print1(T(n, k), ", ")); print());
%Y A281385 Cf. A000032, A000045, A000290.
%K A281385 nonn,tabl
%O A281385 0,4
%A A281385 _Michel Marcus_, Jan 23 2017