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 A321122 a(n) = n-th row common denominator of A321121. 2
 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 (list; graph; refs; listen; history; text; internal format)
 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 2.5.3. LINKS 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. FORMULA 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). MATHEMATICA 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}]] PROG (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)); CROSSREFS Cf. A321121 (Numerators). Cf. A002176, A093736, A100621, A100646, A100648, A321119. Sequence in context: A016512 A201651 A026246 * A143051 A297022 A282888 Adjacent sequences:  A321119 A321120 A321121 * A321123 A321124 A321125 KEYWORD nonn,easy,frac AUTHOR Franck Maminirina Ramaharo, Nov 21 2018 STATUS approved

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Last modified May 15 02:21 EDT 2021. Contains 343909 sequences. (Running on oeis4.)