|
|
A094706
|
|
Convolution of Pell(n) and 2^n.
|
|
8
|
|
|
0, 1, 4, 13, 38, 105, 280, 729, 1866, 4717, 11812, 29365, 72590, 178641, 438064, 1071153, 2613138, 6362965, 15470140, 37565389, 91125206, 220864377, 534951112, 1294960905, 3133261530, 7578261181, 18323338324, 44292046693, 107041649438
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
LINKS
|
|
|
FORMULA
|
G.f.: x/((1-2*x-x^2)*(1-2*x)).
a(n) = Sum_{k=0..n} ((1+sqrt(2))^n - (1-sqrt(2))^n)/(2*sqrt(2))*2^(n-k).
a(n) = (1 + 3*sqrt(2)/4)*(1 + sqrt(2))^n + (1 - 3*sqrt(2)/4)*(1-sqrt(2))^n - 2^(n+1).
a(n) = 4*a(n-1) - 3*a(n-2) - 2*a(n-3).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k+1)*2^(n-2k-1);
a(n) = Sum_{k=0..n} binomial(k, n-k+1)*2^k*(1/2)^(n-k+1). - Paul Barry, Oct 07 2004
|
|
MATHEMATICA
|
|
|
PROG
|
(Magma) I:=[0, 1, 4]; [n le 3 select I[n] else 4*Self(n-1)-3*Self(n-2)-2*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 24 2012
(Sage) [lucas_number1(n+2, 2, -1) - 2^(n+1) for n in (0..30)] # G. C. Greubel, Sep 16 2021
|
|
CROSSREFS
|
|
|
KEYWORD
|
easy,nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|