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.

1, 10, 100, 1, 1000, 20, 10000, 300, 1, 100000, 4000, 30, 1000000, 50000, 600, 1, 10000000, 600000, 10000, 40, 100000000, 7000000, 150000, 1000, 1, 1000000000, 80000000, 2100000, 20000, 50, 10000000000, 900000000, 28000000, 350000, 1500, 1, 100000000000, 10000000000, 360000000, 5600000, 35000, 60

Offset: 0

Author

Zagros Lalo, Jul 21 2018

Keywords

• tabf
• nonn
• easy

Comments

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).

The coefficients in the expansion of 1/(1-10*x-x^2) are given by the sequence generated by the row sums.

The row sums are Denominators of continued fraction convergents to sqrt(26), see A041041.

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))).

Examples

			Triangle begins:
  1;
  10;
  100, 1;
  1000, 20;
  10000, 300, 1;
  100000, 4000, 30;
  1000000, 50000, 600, 1;
  10000000, 600000, 10000, 40;
  100000000, 7000000, 150000, 1000, 1;
  1000000000, 80000000, 2100000, 20000, 50;
  10000000000, 900000000, 28000000, 350000, 1500, 1;
  100000000000, 10000000000, 360000000, 5600000, 35000, 60;
		

References

• Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 70, 102

Links

• Zagros Lalo, Left-justified triangle
• Zagros Lalo, Skew diagonals in center-justified triangle of coefficients in expansion of (1 + 10x)^n
• Zagros Lalo, Skew diagonals in center-justified triangle of coefficients in expansion of (10 + x)^n

Crossrefs

Row sums give A041041.

Cf. A013617, A038303, A176537.

Cf. A011557 (column 0), A053541 (column 1), A081140 (column 2).

Programs

Mathematica

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