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

Author

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

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A083100, A164683, A054490.

Programs

Magma

Z:=PolynomialRing(Integers()); N:=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

Mathematica

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

PARI

x='x+O('x^50); Vec((1+6*x)/(1-4*x-4*x^2)) \\ G. C. Greubel, Aug 10 2017

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

Extensions

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