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a(n) = n-th row common denominator of A321121.
2

%I #11 Dec 31 2018 08:17:18

%S 4,2,3,8,36,96,44,360,492,448,1836,5016,2284,18720,25572,23288,95436,

%T 260736,118724,973080,1329252,1210528,4960836,13553256,6171364,

%U 50581440,69095532,62924168,257868036,704508576,320792204,2629261800,3591638412,3270846208

%N a(n) = n-th row common denominator of A321121.

%D Harold J. Ahlberg, Edwin N. Nilson and Joseph L. Walsh, The Theory of Splines and Their Applications, Academic Press, 1967. See p. 47, Table 2.5.3.

%H Harold J. Ahlberg, Edwin N. Nilson and Joseph L. Walsh, <a href="https://doi.org/10.1016/S0076-5392(08)61988-8">Chapter II. The Cubic Spline</a>, Mathematics in Science and Engineering Volume 38 (1967), pp. 9-74.

%F Let s = -2 + sqrt(3), and define e(n) = s*(2 + s)*(-1 + s^n)/(2*(1 - s)*(-s + s^n)), f(n,k) = 6*s^(1 - k)*(s^(2*k) + s^n)/((1 - s)*(-s + s^n)), and w(n,0) = 1/4 + e(n)/6, w(n,1) = 2 - (1 + 1/6)*e(n), w(n,k) = 1 + f(n,k)/4 for 2 <= k <= n - 2. Then a(n) = LCM of denominators of {w(n,k), 0 <= k <= n} for n >= 3.

%F a(n) = 52*a(n-6) - a(n-12) for n >= 15 (conjectured).

%F G.f.: (4 + 2*x + 3*x^2 + 8*x^3 + 36*x^4 + 96*x^5 - 164*x^6 + 256*x^7 + 336*x^8 + 32*x^9 - 36*x^10 + 24*x^11 + 2*x^13 - 9*x^14)/(1 - 52*x^6 + x^12) (conjectured).

%t s = -2 + Sqrt[3];

%t e[n_] := s*(2 + s)*(-1 + s^n)/(2*(1 - s)*(-s + s^n));

%t f[n_, k_] := 6*s^(1 - k)*(s^(2*k) + s^n)/((1 - s)*(-s + s^n));

%t w[n_, k_] := If[k == 0 || k == n, 1/4 + e[n]/6, If[k == 1 || k == n - 1, 2 - (1 + 1/6)*e[n], 1 + f[n, k]/4]];

%t a[n_] := LCM @@ Table[Denominator[FullSimplify[w[n, k]]], {k, 0, n}];

%t Join[{4, 3, 2}, Table[a[n], {n, 3, 50}]]

%o (Maxima)

%o s : -2 + sqrt(3)$

%o e(n) := s*(2 + s)*(-1 + s^n)/(2*(1 - s)*(-s + s^n))$

%o f(n, k) := 6*s^(1 - k)*(s^(2*k) + s^n)/((1 - s)*(-s + s^n))$

%o w(n, k) := if k = 0 or k = n then 1/4 + e(n)/6 else if k = 1 or k = n - 1 then 2 - (1 + 1/6)*e(n) else 1 + f(n, k)/4$

%o a(n) := lcm(makelist(denom(fullratsimp(w(n, k))), k, 0, n))$

%o append([4, 2, 3], makelist(a(n), n, 3, 50));

%Y Cf. A321121 (Numerators).

%Y Cf. A002176, A093736, A100621, A100646, A100648, A321119.

%K nonn,easy,frac

%O 0,1

%A _Franck Maminirina Ramaharo_, Nov 21 2018