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

%I A321119 #12 Dec 01 2018 08:59:18
%S A321119 4,2,8,10,28,38,104,142,388,530,1448,1978,5404,7382,20168,27550,75268,
%T A321119 102818,280904,383722,1048348,1432070,3912488,5344558,14601604,
%U A321119 19946162,54493928,74440090,203374108,277814198,759002504,1036816702,2832635908,3869452610
%N A321119 a(n) = ((1 - sqrt(3))^n + (1 + sqrt(3))^n)/2^floor((n - 1)/2); n-th row common denominator of A321118.
%D A321119 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.
%H A321119 Encyclopedia of Mathematics, <a href="https://www.encyclopediaofmath.org/index.php/Quadrature_formula">Quadrature formula</a>
%H A321119 John C. Holladay, <a href="https://doi.org/10.1090/S0025-5718-1957-0093894-6">A smoothest curve approximation</a>, Math. Comp. Vol. 11 (1957), 233-243.
%H A321119 Peter Köhler, <a href="https://doi.org/10.1007/BF02575942">On the weights of Sard's quadrature formulas</a>, CALCOLO Vol. 25 (1988), 169-186.
%H A321119 Leroy F. Meyers and Arthur Sard, <a href="https://doi.org/10.1002/sapm1950291118">Best approximate integration formulas</a>, J. Math. Phys. Vol. 29 (1950), 118-123.
%H A321119 Arthur Sard, <a href="https://doi.org/10.2307/2372095">Best approximate integration formulas; best approximation formulas</a>, American Journal of Mathematics Vol. 71 (1949), 80-91.
%H A321119 Isaac J. Schoenberg, <a href="https://projecteuclid.org/euclid.bams/1183525790">Spline interpolation and best quadrature formulae</a>, Bull. Amer. Math. Soc. Vol. 70 (1964), 143-148.
%H A321119 Frans Schurer, <a href="https://research.tue.nl/en/publications/on-natural-cubic-splines-with-an-application-to-numerical-integra">On natural cubic splines, with an application to numerical integration formulae</a>, EUT report. WSK, Dept. of Mathematics and Computing Science Vol. 70-WSK-04 (1970), 1-32.
%H A321119 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,4,0,-1).
%F A321119 a(n) = (((sqrt(2) - sqrt(6))/2)^n + ((sqrt(6) + sqrt(2))/2)^n)*((2 - sqrt(2))*(-1)^n + 2 + sqrt(2))/2.
%F A321119 a(-n) = (-1)^n*a(n).
%F A321119 a(n) = 2*A000034(n+1)*A002531(n).
%F A321119 a(2*n) = 2*A001834(n).
%F A321119 a(2*n+1) = 2*A003500(n).
%F A321119 a(n) = 4*a(n-2) - a(n-4) with a(0) = 4, a(1) = 2, a(2) = 8, a(3) = 10.
%F A321119 a(2*n+3) = a(2*n+1) + a(2*n+2).
%F A321119 a(2*n+2) = a(2*n) + 2*a(2*n+1).
%F A321119 G.f.: 2*(1 - x)*(2 + 3*x - x^2)/(1 - 4*x^2 + x^4).
%F A321119 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.
%F A321119 Lim_{n->infinity} a(2*n+1)/a(2*n) = (1 + sqrt(3))/2.
%e A321119 a(0) = ((1 - sqrt(3))^0 + (1 + sqrt(3))^0)/2^floor((0 - 1)/2) = 2*(1 + 1) = 4.
%t A321119 LinearRecurrence[{0, 4, 0, -1}, {4, 2, 8, 10}, 50]
%o A321119 (Maxima) a(n) := ((1 - sqrt(3))^n + (1 + sqrt(3))^n)/2^floor((n - 1)/2)$
%o A321119 makelist(ratsimp(a(n)), n, 0, 50);
%Y A321119 Cf. A002176 (common denominators of Cotesian numbers).
%Y A321119 Cf. A321118, A321120.
%Y A321119 Cf. A001834, A002530, A002531, A003500, A005246, A048788, A083336, A125905.
%K A321119 nonn,easy,frac
%O A321119 0,1
%A A321119 _Franck Maminirina Ramaharo_, Nov 01 2018