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

0,1

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

Table of n, a(n) for n=0..33.

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).

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.

Example

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

Mathematica

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

Prog

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

Crossrefs

Cf. A002176 (common denominators of Cotesian numbers).

Cf. A321118, A321120.

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

Sequence in context: A296477 A292964 A050128 * A246202 A134042 A051239

Adjacent sequences: A321116 A321117 A321118 * A321120 A321121 A321122

Keyword

nonn,easy,frac

Author

Franck Maminirina Ramaharo, Nov 01 2018

Status

approved