OFFSET
3,2
LINKS
Robert Israel, Table of n, a(n) for n = 3..2370
Index entries for linear recurrences with constant coefficients, signature (9,-34,71,-90,71,-34,9,-1).
FORMULA
G.f.: x^3*(1+34*x+78*x^2-6*x^3-11*x^4) / ((1-x)^6*(1-3*x+x^2)). - Colin Barker, Feb 18 2016
From Robert Israel, Feb 18 2016: (Start)
By definition, a(n) is the coefficient of x^(2*n-6) in the Maclaurin series of (1+2*x)/((1-x-x^2)*(1-x)^6). This can be written explicitly:
a(n) = ((29-13*sqrt(5))/2)*((3-sqrt(5))/2)^n + ((29+13*sqrt(5))/2)*((3+sqrt(5))/2)^n - (4/5)*n^5 + (4/3)*n^4 - (25/3)*n^3 - (31/3)*n^2 - (433/15)*n - 29.
This confirms Colin Barker's g.f. (End)
From G. C. Greubel, Jun 16 2025: (Start)
a(n) = A000032(2*n+7) - (1/15)*(435 + 433*n + 155*n^2 + 125*n^3 - 20*n^4 + 12*n^5).
E.g.f.: exp(3*x/2)*(29*cosh(sqrt(5)*x/2) + 13*sqrt(5)*sinh(sqrt(5)*x/2)) - (1/15)*(435 + 705*x + 570*x^2 + 305*x^3 + 100*x^4 + 12*x^5)*exp(x). (End)
MAPLE
gf:= x^3*(1+34*x+78*x^2-6*x^3-11*x^4) / ((1-x)^6*(1-3*x+x^2)):
S:= series(gf, x, 100):
seq(coeff(S, x, n), n=3..100); # Robert Israel, Feb 18 2016
MATHEMATICA
LinearRecurrence[{9, -34, 71, -90, 71, -34, 9, -1}, {1, 43, 431, 2482, 10636, 38138, 122069, 362853}, 30] (* Vincenzo Librandi, Feb 19 2016 *)
PROG
(PARI) Vec(x^3*(1+34*x+78*x^2-6*x^3-11*x^4)/((1-x)^6*(1-3*x+x^2)) + O(x^40)) \\ Colin Barker, Feb 19 2016
(Magma)
A027002 := func< n | Lucas(2*n+7) -(435+433*n+155*n^2+125*n^3-20*n^4+12*n^5)/15 >;
[A027002(n): n in [3..50]]; // G. C. Greubel, Jun 16 2025
(SageMath)
def A027002(n): return lucas_number2(2*n+7, 1, -1) -(435+433*n+155*n^2+125*n^3 -20*n^4+12*n^5)/15
print([A027002(n) for n in range(3, 51)]) # G. C. Greubel, Jun 16 2025
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved