OFFSET
0,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-4).
FORMULA
a(n) = B(2*n, 2)/B(2*n), where B(n, p) = Sum_{i=0..n} p^i * Sum_{j=0..i} binomial(n,j)*B(j))) with B(k) = k-th Bernoulli number.
a(n) = 3*4^n - 2.
a(n) = 5*a(n-1) - 4*a(n-2).
a(n) = 4*a(n-1) + 6. First differences give A002063. - Paul Curtz, Jul 07 2008
From G. C. Greubel, Jan 22 2023: (Start)
a(n) = 3*A000302(n) - 2.
G.f.: (1+5*x)/(1-x)*(1-4*x)).
E.g.f.: 3*exp(4*x) - 2*exp(x). (End)
MATHEMATICA
a[n_]:= Sum[2^k*Sum[Binomial[2*n, j]*BernoulliB[j], {j, 0, k}], {k, 0, 2*n}]/BernoulliB[2*n]; Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Jan 14 2015 *)
NestList[4#+6&, 1, 30] (* Harvey P. Dale, Dec 27 2016 *)
PROG
(PARI) a(n)=sum(i=0, 2*n, 2^i*sum(j=0, i, binomial(2*n, j)*bernfrac(j)))/bernfrac(2*n)
(Magma) [3*4^n-2: n in [0..30]]; // Vincenzo Librandi, Aug 13 2011
(SageMath) [3*4^n-2 for n in range(41)] # G. C. Greubel, Jan 22 2023
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Benoit Cloitre, Jun 17 2004
STATUS
approved