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A305833 Triangle read by rows: T(0,0)=1; T(n,k) = 4*T(n-1,k) + T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0.

%I A305833 #20 Sep 05 2018 02:26:50
%S A305833 1,4,16,1,64,8,256,48,1,1024,256,12,4096,1280,96,1,16384,6144,640,16,
%T A305833 65536,28672,3840,160,1,262144,131072,21504,1280,20,1048576,589824,
%U A305833 114688,8960,240,1,4194304,2621440,589824,57344,2240,24,16777216,11534336,2949120,344064,17920,336,1
%N A305833 Triangle read by rows: T(0,0)=1; T(n,k) = 4*T(n-1,k) + T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0.
%C A305833 The numbers in rows of the triangle are along skew diagonals pointing top-left in center-justified triangle given in A013611 ((1+4*x)^n).
%C A305833 The coefficients in the expansion of 1/(1-4x-x^2) are given by the sequence generated by the row sums.
%C A305833 The row sums are A001076 (Denominators of continued fraction convergent to sqrt(5)).
%C A305833 If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 4.236067977...; a metallic mean (see A098317), when n approaches infinity.
%D A305833 Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 70, 72, 90, 373.
%H A305833 Shara Lalo, <a href="/A305833/a305833.pdf">Left justified triangle</a>
%H A305833 Shara Lalo, <a href="/A305833/a305833_1.pdf">Skew diagonals in triangle A013611</a>
%F A305833 G.f.: 1 / (1 - 4*t*x - t^2).
%e A305833 Triangle begins:
%e A305833          1;
%e A305833          4;
%e A305833         16,        1;
%e A305833         64,        8;
%e A305833        256,       48,        1;
%e A305833       1024,      256,       12;
%e A305833       4096,     1280,       96,       1;
%e A305833      16384,     6144,      640,      16;
%e A305833      65536,    28672,     3840,     160,      1;
%e A305833     262144,   131072,    21504,    1280,     20;
%e A305833    1048576,   589824,   114688,    8960,    240,    1;
%e A305833    4194304,  2621440,   589824,   57344,   2240,   24;
%e A305833   16777216, 11534336,  2949120,  344064,  17920,  336,  1;
%e A305833   67108864, 50331648, 14417920, 1966080, 129024, 3584, 28;
%t A305833 t[0, 0] = 1; t[n_, k_] := If[n < 0 || k < 0, 0, 4 t[n - 1, k] + t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 12}, {k, 0, Floor[n/2]}] // Flatten
%Y A305833 Row sums give A001076.
%Y A305833 Cf. A000302 (column 0), A002697 (column 1), A038845 (column 2), A038846 (column 3), A040075 (column 4).
%Y A305833 Cf. A013611.
%Y A305833 Cf. A098317.
%K A305833 tabf,nonn,easy
%O A305833 0,2
%A A305833 _Shara Lalo_, Jun 11 2018