|
|
A164593
|
|
a(n) = ((5 + sqrt(18))*(2 + sqrt(8))^n + (5 - sqrt(18))*(2 - sqrt(8))^n)/2.
|
|
2
|
|
|
5, 22, 108, 520, 2512, 12128, 58560, 282752, 1365248, 6592000, 31828992, 153683968, 742051840, 3582943232, 17299980288, 83531694080, 403326697472, 1947433566208, 9403041054720, 45401898483712, 219219758153728
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,1
|
|
COMMENTS
|
Binomial transform of A096980 without initial 1. Second binomial transform of A164737. Inverse binomial transform of A101386.
|
|
LINKS
|
|
|
FORMULA
|
a(n) = 4*a(n-1) + 4*a(n-2) for n > 1; a(0) = 5, a(1) = 22.
G.f.: (5 + 2*x)/(1-4*x-4*x^2).
E.g.f.: exp(2*x)*(5*cosh(2*sqrt(2)*x) + 3*sqrt(2)*sinh(2*sqrt(2)*x)). - G. C. Greubel, Aug 12 2017
|
|
MAPLE
|
seq(coeff(series( (5+2*x)/(1-4*x-4*x^2) , x, n+1), x, n), n = 0..25); # G. C. Greubel, Apr 16 2020
|
|
MATHEMATICA
|
LinearRecurrence[{4, 4}, {5, 22}, 25] (* G. C. Greubel, Aug 12 2017 *)
Table[2^n*(Fibonacci[n+2, 2] + 3*Fibonacci[n+1, 2]), {n, 0, 25}] (* G. C. Greubel, Apr 16 2020 *)
|
|
PROG
|
(Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((5+3*r)*(2+2*r)^n+(5-3*r)*(2-2*r)^n)/2: n in [0..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 24 2009
(PARI) my(x='x+O('x^25)); Vec((5+2*x)/(1-4*x-4*x^2)) \\ G. C. Greubel, Aug 12 2017
(Sage) [2^n*(lucas_number1(n+2, 2, -1) + 3*lucas_number1(n+1, 2, -1)) for n in range(25)] # G. C. Greubel, Apr 16 2020
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
Al Hakanson (hawkuu(AT)gmail.com), Aug 17 2009
|
|
STATUS
|
approved
|
|
|
|