A147959 a(n) = ((8 + sqrt(2))^n + (8 - sqrt(2))^n)/2.

1, 8, 66, 560, 4868, 43168, 388872, 3545536, 32618512, 302072960, 2810819616, 26244590336, 245642629184, 2303117466112, 21620036448384, 203127300275200, 1909594544603392, 17959620096591872, 168959059780059648

Offset: 0

Author

Al Hakanson (hawkuu(AT)gmail.com), Nov 17 2008

Keywords

• nonn
• easy

Comments

Binomial transform of A147958. Inverse binomial transform of A147960. 8th binomial transform of A077957. - Philippe Deléham, Nov 30 2008

Examples

			a(3) = ((8 + sqrt(2))^3 + (8 - sqrt(2))^3)/2
     = (8^3 + 3*8^2*sqrt(2) + 3*8*2 + 2*sqrt(2)
      + 8^3 - 3*8^2*sqrt(2) + 3*8*2 - 2*sqrt(2))/2
     =  8^3                 + 3*8*2
     =  560.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (16,-62).

Crossrefs

Cf. A077957, A147958, A147960.

Programs

Magma

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((8+r2)^n+(8-r2)^n)/2: n in [0..18] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Nov 19 2008

Mathematica

LinearRecurrence[{16, -62}, {1, 8}, 50] (* G. C. Greubel, Aug 17 2018 *)

PARI

x='x+O('x^30); Vec((1-8*x)/(1-16*x+62*x^2)) \\ G. C. Greubel, Aug 17 2018

Formula

From Philippe Deléham, Nov 19 2008: (Start)

a(n) = 16*a(n-1) - 62*a(n-2), n > 1; a(0)=1, a(1)=8.

G.f.: (1 - 8*x)/(1 - 16*x + 62*x^2).

a(n) = (Sum_{k=0..n} A098158(n,k)*8^(2k)*2^(n-k))/8^n. (End)

E.g.f.: exp(8*x)*cosh(sqrt(2)*x). - Ilya Gutkovskiy, Aug 11 2017

Extensions

Extended beyond a(6) by Klaus Brockhaus, Nov 19 2008