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A317055 Triangle read by rows: T(0,0) = 1; T(n,k) = 10*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 A317055 #17 Aug 08 2025 07:05:14
%S A317055 1,10,100,1,1000,20,10000,300,1,100000,4000,30,1000000,50000,600,1,
%T A317055 10000000,600000,10000,40,100000000,7000000,150000,1000,1,1000000000,
%U A317055 80000000,2100000,20000,50,10000000000,900000000,28000000,350000,1500,1,100000000000,10000000000,360000000,5600000,35000,60
%N A317055 Triangle read by rows: T(0,0) = 1; T(n,k) = 10*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 A317055 The numbers in rows of the triangle are along skew diagonals pointing top-left in center-justified triangle given in A013617 ((1+10*x)^n) and along skew diagonals pointing top-right in center-justified triangle given in A038303 ((10+x)^n).
%C A317055 The coefficients in the expansion of 1/(1-10*x-x^2) are given by the sequence generated by the row sums.
%C A317055 The row sums are Denominators of continued fraction convergents to sqrt(26), see A041041.
%C A317055 If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 10.09901951359278483002... (a metallic mean) when n approaches infinity (see A176537: (5+sqrt(26))).
%D A317055 Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 70, 102
%H A317055 Zagros Lalo, <a href="/A317055/a317055.pdf">Left-justified triangle</a>
%H A317055 Zagros Lalo, <a href="/A317055/a317055_1.pdf">Skew diagonals in center-justified triangle of coefficients in expansion of (1 + 10x)^n</a>
%H A317055 Zagros Lalo, <a href="/A317055/a317055_2.pdf">Skew diagonals in center-justified triangle of coefficients in expansion of (10 + x)^n</a>
%e A317055 Triangle begins:
%e A317055   1;
%e A317055   10;
%e A317055   100, 1;
%e A317055   1000, 20;
%e A317055   10000, 300, 1;
%e A317055   100000, 4000, 30;
%e A317055   1000000, 50000, 600, 1;
%e A317055   10000000, 600000, 10000, 40;
%e A317055   100000000, 7000000, 150000, 1000, 1;
%e A317055   1000000000, 80000000, 2100000, 20000, 50;
%e A317055   10000000000, 900000000, 28000000, 350000, 1500, 1;
%e A317055   100000000000, 10000000000, 360000000, 5600000, 35000, 60;
%t A317055 t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, 10 t[n - 1, k] + t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 11}, {k, 0, Floor[n/2]}] // Flatten
%Y A317055 Row sums give A041041.
%Y A317055 Cf. A013617, A038303, A176537.
%Y A317055 Cf. A011557 (column 0), A053541 (column 1), A081140 (column 2).
%K A317055 tabf,nonn,easy
%O A317055 0,2
%A A317055 _Zagros Lalo_, Jul 21 2018