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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. 3

%I

%S 0,1,1,0,4,8,5,5,21,53,0,20,40,124,336,25,25,105,265,761,2105,0,100,

%T 200,620,1680,4724,13144,125,125,525,1325,3805,10525,29421,81997,0,

%U 500,1000,3100,8400,23620,65720,183404,511392,625,625,2625,6625,19025,52625

%N 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 Seiichi Manyama, <a href="/A292466/b292466.txt">Rows n = 0..139, flattened</a>

%F T(n+1,n)^2 - 5*T(n,n)^2 = 11^n.

%e First few rows are:

%e 0;

%e 1, 1;

%e 0, 4, 8;

%e 5, 5, 21, 53;

%e 0, 20, 40, 124, 336;

%e 25, 25, 105, 265, 761, 2105;

%e 0, 100, 200, 620, 1680, 4724, 13144;

%e 125, 125, 525, 1325, 3805, 10525, 29421, 81997.

%e --------------------------------------------------------------

%e The diagonal is {0, 1, 8, 53, 336, 2105, ...} and

%e the next diagonal is {1, 4, 21, 124, 761, 4724, ...}.

%e Two sequences have the following property:

%e 1^2 - 5* 0^2 = 1 (= 11^0),

%e 4^2 - 5* 1^2 = 11 (= 11^1),

%e 21^2 - 5* 8^2 = 121 (= 11^2),

%e 124^2 - 5* 53^2 = 1331 (= 11^3),

%e 761^2 - 5* 336^2 = 14641 (= 11^4),

%e 4724^2 - 5*2105^2 = 161051 (= 11^5),

%e ...

%Y The diagonal of the triangle is A091870.

%Y The next diagonal of the triangle is A108404.

%Y 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 nonn,tabl,look

%O 0,5

%A _Seiichi Manyama_, Sep 22 2017

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Last modified August 20 22:45 EDT 2019. Contains 326155 sequences. (Running on oeis4.)