A317028 - OEIS

A317028

Triangle read by rows: T(0,0) = 1; T(n,k) = 8 * 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, 8, 64, 1, 512, 16, 4096, 192, 1, 32768, 2048, 24, 262144, 20480, 384, 1, 2097152, 196608, 5120, 32, 16777216, 1835008, 61440, 640, 1, 134217728, 16777216, 688128, 10240, 40, 1073741824, 150994944, 7340032, 143360, 960, 1, 8589934592, 1342177280, 75497472, 1835008, 17920, 48

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OFFSET

0,2

COMMENTS

The numbers in rows of the triangle are along skew diagonals pointing top-left in center-justified triangle given in A013615 ((1+8*x)^n) and along skew diagonals pointing top-right in center-justified triangle given in A038279 ((8+x)^n).

The coefficients in the expansion of 1/(1-8x-x^2) are given by the sequence generated by the row sums.

The row sums are Denominators of continued fraction convergents to sqrt(17), see A041025.

If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 8.12310562561766054982... (a metallic mean), when n approaches infinity (see A176458: (4+sqrt(17))).

REFERENCES

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, Pages 70, 98

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 + 8x)^n

Zagros Lalo, Skew diagonals in center-justified triangle of coefficients in expansion of (8 + x)^n

EXAMPLE

Triangle begins:

64, 1;

512, 16;

4096, 192, 1;