login
A161947
a(n) = ((4+sqrt(2))*(5+sqrt(2))^n + (4-sqrt(2))*(5-sqrt(2))^n)/4.
2
2, 11, 64, 387, 2398, 15079, 95636, 609543, 3895802, 24938531, 159781864, 1024232427, 6567341398, 42116068159, 270111829436, 1732448726703, 11111915190002, 71272831185851, 457154262488464, 2932267507610067, 18808127038865998, 120639117713628439, 773804255242366436
OFFSET
0,1
COMMENTS
Fifth binomial transform of A135530.
FORMULA
a(n) = 10*a(n-1) - 23*a(n-2) for n>1; a(0) = 2; a(1) = 11.
G.f.: (2-9*x)/(1-10*x+23*x^2).
E.g.f.: exp(5*x)*(4*cosh(sqrt(2)*x) + sqrt(2)*sinh(sqrt(2)*x))/2. - Stefano Spezia, Oct 24 2023
MAPLE
seq(simplify(((4+sqrt(2))*(5+sqrt(2))^n+(4-sqrt(2))*(5-sqrt(2))^n)*1/4), n = 0 .. 20); # Emeric Deutsch, Jun 28 2009
MATHEMATICA
LinearRecurrence[{10, -23}, {2, 11}, 50] (* G. C. Greubel, Aug 17 2018 *)
Table[(((4+Sqrt[2])(5+Sqrt[2])^n)+((4-Sqrt[2])(5-Sqrt[2])^n))/4, {n, 0, 20}]//Simplify (* Harvey P. Dale, Mar 07 2020 *)
PROG
(Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((4+r)*(5+r)^n+(4-r)*(5-r)^n)/4: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 01 2009
(GAP) a := [2, 11];; for n in [3..10^2] do a[n] := 10*a[n-1] - 23*a[n-2]; od; a; # Muniru A Asiru, Feb 02 2018
(PARI) x='x+O('x^30); Vec((2-9*x)/(1-10*x+23*x^2)) \\ G. C. Greubel, Aug 17 2018
CROSSREFS
Cf. A135530.
Sequence in context: A179120 A373890 A038725 * A349023 A001565 A357845
KEYWORD
nonn,easy
AUTHOR
Al Hakanson (hawkuu(AT)gmail.com), Jun 22 2009
EXTENSIONS
Edited and extended beyond a(4) by Klaus Brockhaus, Jul 01 2009
Extended by Emeric Deutsch, Jun 28 2009
STATUS
approved