A164546 a(n) = 8*a(n-1) - 8*a(n-2) for n > 1; a(0) = 1, a(1) = 10.

1, 10, 72, 496, 3392, 23168, 158208, 1080320, 7376896, 50372608, 343965696, 2348744704, 16038232064, 109515898880, 747821334528, 5106443485184, 34868977205248, 238100269760512, 1625850340442112, 11102000565452800

Offset: 0

Author

Al Hakanson (hawkuu(AT)gmail.com), Aug 15 2009

Keywords

• nonn
• easy

Comments

Binomial transform of A038761. Fourth binomial transform of A164640. Inverse binomial transform of A164547.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..149
• Index entries for linear recurrences with constant coefficients, signature (8,-8).

Crossrefs

Cf. A038761, A164640, A164547.

Programs

Magma

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((2+3*r)*(4+2*r)^n+(2-3*r)*(4-2*r)^n)/4: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 19 2009

Mathematica

LinearRecurrence[{8,-8}, {1,10}, 30] (* G. C. Greubel, Jul 17 2021 *)

Sage

[2*(2*sqrt(2))^(n-1)*(sqrt(2)*chebyshev_U(n, sqrt(2)) + chebyshev_U(n-1, sqrt(2))) for n in (0..30)] # G. C. Greubel, Jul 17 2021

Formula

a(n) = 8*a(n-1) - 8*a(n-2) for n > 1; a(0) = 1, a(1) = 10.

a(n) = ((2+3*sqrt(2))*(4+2*sqrt(2))^n + (2-3*sqrt(2))*(4-2*sqrt(2))^n)/4.

G.f.: (1 + 2*x)/(1 - 8*x + 8*x^2).

a(n) = 2*(2*sqrt(2))^(n-1)*(sqrt(2)*chebyshev_U(n, sqrt(2)) + chebyshev_U(n-1, sqrt(2))). - G. C. Greubel, Jul 17 2021

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 19 2009