login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

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

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

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