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A305833 Triangle read by rows: T(0,0)=1; T(n,k) = 4*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
1, 4, 16, 1, 64, 8, 256, 48, 1, 1024, 256, 12, 4096, 1280, 96, 1, 16384, 6144, 640, 16, 65536, 28672, 3840, 160, 1, 262144, 131072, 21504, 1280, 20, 1048576, 589824, 114688, 8960, 240, 1, 4194304, 2621440, 589824, 57344, 2240, 24, 16777216, 11534336, 2949120, 344064, 17920, 336, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
The numbers in rows of the triangle are along skew diagonals pointing top-left in center-justified triangle given in A013611 ((1+4*x)^n).
The coefficients in the expansion of 1/(1-4x-x^2) are given by the sequence generated by the row sums.
The row sums are A001076 (Denominators of continued fraction convergent to sqrt(5)).
If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 4.236067977...; a metallic mean (see A098317), when n approaches infinity.
REFERENCES
Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 70, 72, 90, 373.
LINKS
FORMULA
G.f.: 1 / (1 - 4*t*x - t^2).
EXAMPLE
Triangle begins:
1;
4;
16, 1;
64, 8;
256, 48, 1;
1024, 256, 12;
4096, 1280, 96, 1;
16384, 6144, 640, 16;
65536, 28672, 3840, 160, 1;
262144, 131072, 21504, 1280, 20;
1048576, 589824, 114688, 8960, 240, 1;
4194304, 2621440, 589824, 57344, 2240, 24;
16777216, 11534336, 2949120, 344064, 17920, 336, 1;
67108864, 50331648, 14417920, 1966080, 129024, 3584, 28;
MATHEMATICA
t[0, 0] = 1; t[n_, k_] := If[n < 0 || k < 0, 0, 4 t[n - 1, k] + t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 12}, {k, 0, Floor[n/2]}] // Flatten
CROSSREFS
Row sums give A001076.
Cf. A000302 (column 0), A002697 (column 1), A038845 (column 2), A038846 (column 3), A040075 (column 4).
Cf. A013611.
Cf. A098317.
Sequence in context: A165410 A016486 A065659 * A110650 A334840 A232515
KEYWORD
tabf,nonn,easy
AUTHOR
Shara Lalo, Jun 11 2018
STATUS
approved

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Last modified April 18 21:51 EDT 2024. Contains 371781 sequences. (Running on oeis4.)