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A321122
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a(n) = n-th row common denominator of A321121.
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2
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4, 2, 3, 8, 36, 96, 44, 360, 492, 448, 1836, 5016, 2284, 18720, 25572, 23288, 95436, 260736, 118724, 973080, 1329252, 1210528, 4960836, 13553256, 6171364, 50581440, 69095532, 62924168, 257868036, 704508576, 320792204, 2629261800, 3591638412, 3270846208
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OFFSET
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0,1
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REFERENCES
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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.
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LINKS
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Harold J. Ahlberg, Edwin N. Nilson and Joseph L. Walsh, Chapter II. The Cubic Spline, Mathematics in Science and Engineering Volume 38 (1967), pp. 9-74.
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FORMULA
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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.
a(n) = 52*a(n-6) - a(n-12) for n >= 15 (conjectured).
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).
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MATHEMATICA
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s = -2 + Sqrt[3];
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));
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]];
a[n_] := LCM @@ Table[Denominator[FullSimplify[w[n, k]]], {k, 0, n}];
Join[{4, 3, 2}, Table[a[n], {n, 3, 50}]]
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PROG
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(Maxima)
s : -2 + sqrt(3)$
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))$
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$
a(n) := lcm(makelist(denom(fullratsimp(w(n, k))), k, 0, n))$
append([4, 2, 3], makelist(a(n), n, 3, 50));
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CROSSREFS
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KEYWORD
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nonn,easy,frac
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AUTHOR
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STATUS
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approved
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