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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 Table of n, a(n) for n=0..48. Shara Lalo, Left justified triangle Shara Lalo, Skew diagonals in triangle A013611 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 Adjacent sequences: A305830 A305831 A305832 * A305834 A305835 A305836 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.)