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A292466 Triangle read by rows: T(n,k) = 4*T(n-1,k-1) + T(n,k-1) with T(2*m,0) = 0 and T(2*m+1,0) = 5^m.

%I A292466 #53 Sep 26 2017 09:26:26
%S A292466 0,1,1,0,4,8,5,5,21,53,0,20,40,124,336,25,25,105,265,761,2105,0,100,
%T A292466 200,620,1680,4724,13144,125,125,525,1325,3805,10525,29421,81997,0,
%U A292466 500,1000,3100,8400,23620,65720,183404,511392,625,625,2625,6625,19025,52625
%N A292466 Triangle read by rows: T(n,k) = 4*T(n-1,k-1) + T(n,k-1) with T(2*m,0) = 0 and T(2*m+1,0) = 5^m.
%H A292466 Seiichi Manyama, <a href="/A292466/b292466.txt">Rows n = 0..139, flattened</a>
%F A292466 T(n+1,n)^2 - 5*T(n,n)^2 = 11^n.
%e A292466 First few rows are:
%e A292466     0;
%e A292466     1,   1;
%e A292466     0,   4,   8;
%e A292466     5,   5,  21,   53;
%e A292466     0,  20,  40,  124,  336;
%e A292466    25,  25, 105,  265,  761,  2105;
%e A292466     0, 100, 200,  620, 1680,  4724, 13144;
%e A292466   125, 125, 525, 1325, 3805, 10525, 29421, 81997.
%e A292466 --------------------------------------------------------------
%e A292466 The diagonal is      {0, 1,  8,  53, 336, 2105, ...} and
%e A292466 the next diagonal is {1, 4, 21, 124, 761, 4724, ...}.
%e A292466 Two sequences have the following property:
%e A292466      1^2 - 5*   0^2 = 1      (= 11^0),
%e A292466      4^2 - 5*   1^2 = 11     (= 11^1),
%e A292466     21^2 - 5*   8^2 = 121    (= 11^2),
%e A292466    124^2 - 5*  53^2 = 1331   (= 11^3),
%e A292466    761^2 - 5* 336^2 = 14641  (= 11^4),
%e A292466   4724^2 - 5*2105^2 = 161051 (= 11^5),
%e A292466   ...
%Y A292466 The diagonal of the triangle is A091870.
%Y A292466 The next diagonal of the triangle is A108404.
%Y A292466 T(n,k) = b*T(n-1,k-1) + T(n,k-1): A292789 (b=-3), A292495 (b=-2), A117918 and A228405 (b=1), A227418 (b=2), this sequence (b=4).
%K A292466 nonn,tabl,look
%O A292466 0,5
%A A292466 _Seiichi Manyama_, Sep 22 2017