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A329442 Irregular triangular array, read by rows: row n shows the coefficients of the polynomial p(n,x) defined in Comments. 2
1, 2, 3, 14, 36, 27, 590, 3024, 6156, 5832, 2187, 1044302, 10704960, 49225968, 132339744, 227246796, 255091680, 182815704, 76527504, 14348907, 3271700001614, 67075266827520, 652229166810816, 3990988066439808, 17193623473530864, 55281675697126272 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Let f(x) = 3 x^2 + 2, u(0,x) = 1, u(n,x) = f(u(n-1,x)), and p(n,x) = u(n,sqrt(x)). Then the sequence (p(n,0)) = (1, 2, 14, 590, 1044302, 3271700001614, ...) is a strong divisibility sequence, as implied by Dickson's record of a statement by J. J. Sylvester proved by W. S. Foster in 1889.
REFERENCES
L. E. Dickson, History of the Theory of Numbers, vol. 1, Chelsea, New York, 1952, p. 403.
LINKS
EXAMPLE
Rows 0..4:
1;
2, 3;
14, 36, 27;
590, 3024, 6156, 5832, 2187;
1044302, 10704960, 49225968, 132339744, 227246796, 255091680, 182815704, 76527504, 14348907.
Rows 0..4, the polynomials u(n,x):
1;
2 + 3 x^2;
14 + 36 x^2 + 27 x^4;
590 + 3024 x^2 + 6156 x^4 + 5832 x^6 + 2187 x^8;
1044302 + 10704960 x^2 + 49225968 x^4 + 132339744 x^6 + 227246796 x^8 + 255091680 x^10 + 182815704 x^12 + 76527504
x^14 + 14348907 x^16.
MATHEMATICA
f[x_] := 3 x^2 + 2; u[0, x_] := 1;
u[1, x_] := f[x]; u[n_, x_] := f[u[n - 1, x]]
Column[Table [Expand[u[n, x]], {n, 0, 5}]] (* A329442 polynomials u(n, x) *)
Table[CoefficientList[u[n, Sqrt[x], x], {n, 0, 5}] (* A329442 array *)
CROSSREFS
Sequence in context: A059188 A080768 A203578 * A281486 A185895 A358651
KEYWORD
nonn,tabf
AUTHOR
Clark Kimberling, Dec 07 2019
STATUS
approved

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Last modified March 28 17:25 EDT 2024. Contains 371254 sequences. (Running on oeis4.)