A164607 a(n) = 4*a(n-1) + 4*a(n-2) for n > 1; a(0) = 1, a(1) = 10.

1, 10, 44, 216, 1040, 5024, 24256, 117120, 565504, 2730496, 13184000, 63657984, 307367936, 1484103680, 7165886464, 34599960576, 167063388160, 806653394944, 3894867132416, 18806082109440, 90803796967424, 438439516307456

Offset

0,2

Comments

Binomial transform of A083100. Second binomial transform of A164683. Inverse binomial transform of A054490.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..164 from Vincenzo Librandi)

Index entries for linear recurrences with constant coefficients, signature (4, 4).

Formula

a(n) = 4*a(n-1) + 4*a(n-2) for n > 1; a(0) = 1, a(1) = 10.

a(n) = ((2+4*sqrt(2))*(2+2*sqrt(2))^n + (2-4*sqrt(2))*(2-2*sqrt(2))^n)/4.

G.f.: (1+6*x)/(1-4*x-4*x^2).

G.f.: G(0)/(2*x) - 1/x, where G(k)= 1 + 1/(1 - x*(8*k-1)/(x*(8*k+7) - (1-x)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013

E.g.f.: exp(2*x)*(cosh(2*sqrt(2)*x) + 2*sqrt(2)*sinh(2*sqrt(2)*x)). - G. C. Greubel, Aug 10 2017

Mathematica

LinearRecurrence[{4, 4}, {1, 10}, 40] (* Harvey P. Dale, Jun 28 2011 *)

Prog

(Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((2+4*r)*(2+2*r)^n+(2-4*r)*(2-2*r)^n)/4: n in [0..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 22 2009
(PARI) x='x+O('x^50); Vec((1+6*x)/(1-4*x-4*x^2)) \\ G. C. Greubel, Aug 10 2017

Crossrefs

Cf. A083100, A164683, A054490.

Sequence in context: A097215 A281993 A126397 * A200189 A240383 A160970

Adjacent sequences: A164604 A164605 A164606 * A164608 A164609 A164610

Keyword

nonn,easy

Author

Al Hakanson (hawkuu(AT)gmail.com), Aug 17 2009

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 22 2009

Status

approved