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

0,2

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/(1-10x-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))).

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

Table of n, a(n) for n=0..41.

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

Example

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;
1000000000000, 110000000000, 4500000000, 84000000, 700000, 2100, 1;
10000000000000, 1200000000000, 55000000000, 1200000000, 12600000, 56000, 70;
100000000000000, 13000000000000, 660000000000, 16500000000, 210000000, 1260000, 2800, 1;

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.

Crossrefs

Row sums give A041041.

Cf. A013617

Cf. A038303

Cf. A176537

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

Sequence in context: A187019 A292429 A145644 * A284200 A355893 A228410

Adjacent sequences: A317052 A317053 A317054 * A317056 A317057 A317058

Keyword

tabf,nonn,easy

Author

Zagros Lalo, Jul 21 2018

Status

approved