A317026 Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1,k) + 8 * T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0.

1, 1, 1, 8, 1, 16, 1, 24, 64, 1, 32, 192, 1, 40, 384, 512, 1, 48, 640, 2048, 1, 56, 960, 5120, 4096, 1, 64, 1344, 10240, 20480, 1, 72, 1792, 17920, 61440, 32768, 1, 80, 2304, 28672, 143360, 196608, 1, 88, 2880, 43008, 286720, 688128, 262144, 1, 96, 3520, 61440, 516096, 1835008, 1835008

Offset

0,4

Comments

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

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

The row sums are Generalized Fibonacci numbers (see A015443).

If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 3.3722813232690143..., when n approaches infinity; see A235162 (Decimal expansion of (sqrt(33)+1)/2).

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..55.

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

Formula

T(n,k) = 8^k*binomial(n-k,k), n >= 0, 0 <= k <= floor(n/2).

Example

Triangle begins:
1;
1;
1, 8;
1, 16;
1, 24, 64;
1, 32, 192;
1, 40, 384, 512;
1, 48, 640, 2048;
1, 56, 960, 5120, 4096;
1, 64, 1344, 10240, 20480;
1, 72, 1792, 17920, 61440, 32768;
1, 80, 2304, 28672, 143360, 196608;
1, 88, 2880, 43008, 286720, 688128, 262144;
1, 96, 3520, 61440, 516096, 1835008, 1835008;
1, 104, 4224, 84480, 860160, 4128768, 7340032, 2097152;
1, 112, 4992, 112640, 1351680, 8257536, 22020096, 16777216;
1, 120, 5824, 146432, 2027520, 15138816, 55050240, 75497472, 16777216;

Mathematica

t[0, 0] = 1; t[n_, k_] := If[n < 0 || k < 0, 0,  t[n - 1, k] + 8 t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 11}, {k, 0, Floor[n/2]}] // Flatten.
Table[8^k Binomial[n - k, k], {n, 0, 15}, {k, 0, Floor[n/2]}].

Prog

(GAP) Flat(List([0..13], n->List([0..Int(n/2)], k->8^k*Binomial(n-k, k)))); # Muniru A Asiru, Jul 19 2018

Crossrefs

Row sums give A015443.

Cf. A013615, A038279, A235162.

Sequence in context: A332941 A107929 A040071 * A126000 A326992 A333509

Adjacent sequences: A317023 A317024 A317025 * A317027 A317028 A317029

Keyword

tabf,nonn,easy

Author

Zagros Lalo, Jul 19 2018

Status

approved