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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
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
KEYWORD
tabf,nonn,easy
AUTHOR
Zagros Lalo, May 09 2018
STATUS
approved

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Last modified March 29 06:44 EDT 2024. Contains 371265 sequences. (Running on oeis4.)