OFFSET
0,2
COMMENTS
Binomial transform of A164298. Third binomial transform of A164587. Inverse binomial transform of A164300.
This sequence is part of a class of sequences defined by the recurrence a(n,m) = 2*(m+1)*a(n-1,m) - ((m+1)^2 - 2)*a(n-2,m) with a(0) = 1 and a(1) = m+9. The generating function is Sum_{n>=0} a(n,m)*x^n = (1 - (m-7)*x)/(1 - 2*(m+1)*x + ((m+1)^2 - 2)*x^2) and has a series expansion in terms of Pell-Lucas numbers defined by a(n, m) = (1/2)*Sum_{k=0..n} binomial(n,k)*m^(n-k)*(5*Q(k) + 4*Q(k-1)). - G. C. Greubel, Mar 12 2021
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-7).
FORMULA
a(n) = 6*a(n-1) - 7*a(n-2) for n > 1; a(0) = 1, a(1) = 11.
G.f.: (1+5*x)/(1-6*x+7*x^2).
E.g.f.: (cosh(sqrt(2)*x) + 4*sqrt(2)*sinh(sqrt(2)*x))*exp(3*x). - G. C. Greubel, Sep 12 2017
From G. C. Greubel, Mar 12 2021: (Start)
a(n) = (1/2)*Sum_{k=0..n} binomial(n,k)*2^(n-k)*(5*Q(k) + 4*Q(k-1)), where Q(n) = Pell-Lucas(n) = A002203(n). (End)
MATHEMATICA
LinearRecurrence[{6, -7}, {1, 11}, 50] (* or *) CoefficientList[Series[(1 + 5*x)/(1 - 6*x + 7*x^2), {x, 0, 50}], x] (* G. C. Greubel, Sep 12 2017 *)
PROG
(Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((1+4*r)*(3+r)^n+(1-4*r)*(3-r)^n)/2: n in [0..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 17 2009
(PARI) my(x='x+O('x^50)); Vec((1+5*x)/(1-6*x+7*x^2)) \\ G. C. Greubel, Sep 12 2017
(Sage) [( (1+5*x)/(1-6*x+7*x^2) ).series(x, n+1).list()[n] for n in (0..30)] # G. C. Greubel, Mar 12 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Al Hakanson (hawkuu(AT)gmail.com), Aug 12 2009
EXTENSIONS
Edited and extended beyond a(5) by Klaus Brockhaus, Aug 17 2009
STATUS
approved