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 A304255 Triangle read by rows: T(0,0) = 1; T(n,k) = 6*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, 6, 36, 1, 216, 12, 1296, 108, 1, 7776, 864, 18, 46656, 6480, 216, 1, 279936, 46656, 2160, 24, 1679616, 326592, 19440, 360, 1, 10077696, 2239488, 163296, 4320, 30, 60466176, 15116544, 1306368, 45360, 540, 1, 362797056, 100776960, 10077696, 435456, 7560, 36 (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 A013613 ((1+6*x)^n). The coefficients in the expansion of 1/(1-6x-x^2) are given by the sequence generated by the row sums. The row sums are Denominators of continued fraction convergent to sqrt(10), see A005668. If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 6.162277660..., a metallic mean (see A176398), 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, 94. LINKS Zagros Lalo, Left-justified triangle EXAMPLE Triangle begins: 1; 6; 36, 1; 216, 12; 1296, 108, 1; 7776, 864, 18; 46656, 6480, 216, 1; 279936, 46656, 2160, 24; 1679616, 326592, 19440, 360, 1; 10077696, 2239488, 163296, 4320, 30; 60466176, 15116544, 1306368, 45360, 540, 1; 362797056, 100776960, 10077696, 435456, 7560, 36; 2176782336, 665127936, 75582720, 3919104, 90720, 756, 1; 13060694016, 4353564672, 554273280, 33592320, 979776, 12096, 42; 78364164096, 28298170368, 3990767616, 277136640, 9797760, 163296, 1008, 1; 470184984576, 182849716224, 28298170368, 2217093120, 92378880, 1959552, 18144, 48; MATHEMATICA t[0, 0] = 1; t[n_, k_] := If[n < 0 || k < 0, 0, 6 t[n - 1, k] + t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 11}, {k, 0, Floor[n/2]}] // Flatten PROG (PARI) T(n, k) = if ((n<0) || (k<0), 0, if ((n==0) && (k==0), 1, 6*T(n-1, k) + T(n-2, k-1))); tabf(nn) = for (n=0, nn, for (k=0, n\2, print1(T(n, k), ", ")); print); \\ Michel Marcus, May 26 2018 CROSSREFS Row sums give A005668. Cf. A000400 (column 0), A053469 (column 1), A081136 (column 2), A081144 (column 3). Cf. A013613. Cf. A176398. Sequence in context: A193001 A128298 A059059 * A050112 A250202 A036125 Adjacent sequences:  A304252 A304253 A304254 * A304256 A304257 A304258 KEYWORD tabf,nonn,easy AUTHOR Zagros Lalo, May 09 2018 STATUS approved

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Last modified September 17 03:42 EDT 2021. Contains 347478 sequences. (Running on oeis4.)