A321119 a(n) = ((1 - sqrt(3))^n + (1 + sqrt(3))^n)/2^floor((n - 1)/2); n-th row common denominator of A321118.

4, 2, 8, 10, 28, 38, 104, 142, 388, 530, 1448, 1978, 5404, 7382, 20168, 27550, 75268, 102818, 280904, 383722, 1048348, 1432070, 3912488, 5344558, 14601604, 19946162, 54493928, 74440090, 203374108, 277814198, 759002504, 1036816702, 2832635908, 3869452610

Offset: 0

Author

Franck Maminirina Ramaharo, Nov 01 2018

Keywords

• nonn
• easy
• frac

Examples

			a(0) = ((1 - sqrt(3))^0 + (1 + sqrt(3))^0)/2^floor((0 - 1)/2) = 2*(1 + 1) = 4.
		

References

• Harold J. Ahlberg, Edwin N. Nilson and Joseph L. Walsh, The Theory of Splines and Their Applications, Academic Press, 1967. See p. 47, Table 2.5.2.

Links

• Encyclopedia of Mathematics, Quadrature formula
• John C. Holladay, A smoothest curve approximation, Math. Comp. Vol. 11 (1957), 233-243.
• Peter Köhler, On the weights of Sard's quadrature formulas, CALCOLO Vol. 25 (1988), 169-186.
• Leroy F. Meyers and Arthur Sard, Best approximate integration formulas, J. Math. Phys. Vol. 29 (1950), 118-123.
• Arthur Sard, Best approximate integration formulas; best approximation formulas, American Journal of Mathematics Vol. 71 (1949), 80-91.
• Isaac J. Schoenberg, Spline interpolation and best quadrature formulae, Bull. Amer. Math. Soc. Vol. 70 (1964), 143-148.
• Frans Schurer, On natural cubic splines, with an application to numerical integration formulae, EUT report. WSK, Dept. of Mathematics and Computing Science Vol. 70-WSK-04 (1970), 1-32.
• Index entries for linear recurrences with constant coefficients, signature (0,4,0,-1).

Crossrefs

Cf. A002176 (common denominators of Cotesian numbers).

Cf. A321118, A321120.

Cf. A001834, A002530, A002531, A003500, A005246, A048788, A083336, A125905.

Programs

Mathematica

LinearRecurrence[{0, 4, 0, -1}, {4, 2, 8, 10}, 50]

Maxima

a(n) := ((1 - sqrt(3))^n + (1 + sqrt(3))^n)/2^floor((n - 1)/2)$
makelist(ratsimp(a(n)), n, 0, 50);

Formula

a(n) = (((sqrt(2) - sqrt(6))/2)^n + ((sqrt(6) + sqrt(2))/2)^n)*((2 - sqrt(2))*(-1)^n + 2 + sqrt(2))/2.

a(-n) = (-1)^n*a(n).

a(n) = 2*A000034(n+1)*A002531(n).

a(2*n) = 2*A001834(n).

a(2*n+1) = 2*A003500(n).

a(n) = 4*a(n-2) - a(n-4) with a(0) = 4, a(1) = 2, a(2) = 8, a(3) = 10.

a(2*n+3) = a(2*n+1) + a(2*n+2).

a(2*n+2) = a(2*n) + 2*a(2*n+1).

G.f.: 2*(1 - x)*(2 + 3*x - x^2)/(1 - 4*x^2 + x^4).

E.g.f.: (1 + exp(-sqrt(6)*x))*((2 - sqrt(2))*exp(sqrt(2 - sqrt(3))*x) + (2 + sqrt(2))*exp(sqrt(2 + sqrt(3))*x))/2.

Lim_{n->infinity} a(2*n+1)/a(2*n) = (1 + sqrt(3))/2.