login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A219021
Sum of cubes of first n terms of Lucas sequence U(4,1) (A001353) divided by sum of their first powers.
2
1, 13, 172, 2356, 32661, 454329, 6325816, 88099144, 1227032521, 17090245381, 238035989412, 3315412063548, 46177727142301, 643172746439665, 8958240642814960, 124772195953666576, 1737852501591502353, 24205162822158610557, 337134426993071036956, 4695676815022772628676, 65402340983109050660389
OFFSET
1,2
COMMENTS
For a Lucas sequence U(k,1), the sum of the cubes of the first n terms is divisible by the sum of the first n terms. This sequence corresponds to the case of k=4.
FORMULA
a(n) = Sum_{k=1..n} A001353(k)^3 / Sum_{k=1..n} A001353(k).
a(n) = Sum_{k=1..n} A001353(k)^3 / A061278(n).
From Colin Barker, Dec 08 2015: (Start)
a(n) = 19*a(n-1)-76*a(n-2)+76*a(n-3)-19*a(n-4)+a(n-5) for n>5.
G.f.: x*(1-6*x+x^2) / ((1-x)*(1-14*x+x^2)*(1-4*x+x^2)).
(End)
MATHEMATICA
CoefficientList[Series[(1 - 6 x + x^2)/((1 - x) (1 - 14 x + x^2) (1 - 4 x + x^2)), {x, 0, 33}], x] (* Vincenzo Librandi, Dec 09 2015 *)
PROG
(PARI) Vec(x*(1-6*x+x^2)/((1-x)*(1-14*x+x^2)*(1-4*x+x^2)) + O(x^30)) \\ Colin Barker, Dec 08 2015
(Magma) I:=[1, 13, 172, 2356, 32661]; [n le 5 select I[n] else 19*Self(n-1)-76*Self(n-2)+76*Self(n-3)-19*Self(n-4)+Self(n-5): n in [1..30]]; // Vincenzo Librandi, Dec 09 2015
CROSSREFS
Sequence in context: A041314 A275293 A296585 * A065544 A362859 A096719
KEYWORD
nonn,easy
AUTHOR
Max Alekseyev, Nov 09 2012
STATUS
approved