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A292789 Triangle read by rows: T(n,k) = (-3)*T(n-1,k-1) + T(n,k-1) with T(2*m,0) = 0 and T(2*m+1,0) = (-2)^m.

%I A292789 #19 Sep 25 2017 19:14:56
%S A292789 0,1,1,0,-3,-6,-2,-2,7,25,0,6,12,-9,-84,4,4,-14,-50,-23,229,0,-12,-24,
%T A292789 18,168,237,-450,-8,-8,28,100,46,-458,-1169,181,0,24,48,-36,-336,-474,
%U A292789 900,4407,3864,16,16,-56,-200,-92,916,2338,-362,-13583,-25175,0,-48
%N A292789 Triangle read by rows: T(n,k) = (-3)*T(n-1,k-1) + T(n,k-1) with T(2*m,0) = 0 and T(2*m+1,0) = (-2)^m.
%H A292789 Seiichi Manyama, <a href="/A292789/b292789.txt">Rows n = 0..139, flattened</a>
%F A292789 T(n+1,n)^2 + 2*T(n,n)^2 = 11^n.
%e A292789 First few rows are:
%e A292789    0;
%e A292789    1,   1;
%e A292789    0,  -3,  -6;
%e A292789   -2,  -2,   7,  25;
%e A292789    0,   6,  12,  -9,  -84;
%e A292789    4,   4, -14, -50,  -23,  229;
%e A292789    0, -12, -24,  18,  168,  237,  -450;
%e A292789   -8,  -8,  28, 100,   46, -458, -1169,  181;
%e A292789    0,  24,  48, -36, -336, -474,   900, 4407, 3864.
%e A292789 --------------------------------------------------------------
%e A292789 The diagonal is      {0,  1, -6, 25, -84, ...} and
%e A292789 the next diagonal is {1, -3,  7, -9, -23, ...}.
%e A292789 Two sequences have the following property:
%e A292789       1^2 + 2*    0^2 = 1      (= 11^0),
%e A292789    (-3)^2 + 2*    1^2 = 11     (= 11^1),
%e A292789       7^2 + 2* (-6)^2 = 121    (= 11^2),
%e A292789    (-9)^2 + 2*   25^2 = 1331   (= 11^3),
%e A292789   (-23)^2 + 2*(-84)^2 = 14641  (= 11^4),
%e A292789   ...
%Y A292789 T(n,k) = b*T(n-1,k-1) + T(n,k-1): this sequence (b=-3), A292495 (b=-2), A117918 and A228405 (b=1), A227418 (b=2), A292466 (b=4).
%K A292789 sign,tabl
%O A292789 0,5
%A A292789 _Seiichi Manyama_, Sep 23 2017