A164539 -id:A164539

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A164675

a(n) = 8*a(n-2) for n > 2; a(1) = 1, a(2) = 12.

+10
6

1, 12, 8, 96, 64, 768, 512, 6144, 4096, 49152, 32768, 393216, 262144, 3145728, 2097152, 25165824, 16777216, 201326592, 134217728, 1610612736, 1073741824, 12884901888, 8589934592, 103079215104, 68719476736, 824633720832

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Interleaving of A001018 and 12*A001018.

Binomial transform is A164539.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..500

Index entries for linear recurrences with constant coefficients, signature (0,8).

FORMULA

a(n) = (5+(-1)^n)*2^(1/4*(6*n-11+3*(-1)^n)).

G.f.: x*(1+12*x)/(1-8*x^2).

MATHEMATICA

Riffle[#, 12*#] & [8^Range[0, 14]] (* or *)

LinearRecurrence[{0, 8}, {1, 12}, 30] (* Paolo Xausa, Apr 22 2024 *)

PROG

(Magma) [ n le 2 select 11*n-10 else 8*Self(n-2): n in [1..26] ];

CROSSREFS

Cf. A001018 (powers of 8), A164539.

KEYWORD

nonn,easy

AUTHOR

Klaus Brockhaus, Aug 20 2009

STATUS

approved

A164540

a(n) = 4*a(n-1) + 4*a(n-2) for n > 1; a(0) = 1, a(1) = 14.

+10
3

1, 14, 60, 296, 1424, 6880, 33216, 160384, 774400, 3739136, 18054144, 87173120, 420909056, 2032328704, 9812951040, 47381118976, 228776280064, 1104629596160, 5333623504896, 25753012404224, 124346543636480, 600398224162816

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Binomial transform of A164539. Second binomial transform of A164675. Inverse binomial transform of A164541.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..164

Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.

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) = 14.

G.f.: (1+10*x)/(1-4*x-4*x^2).

a(n) = ((1+3*sqrt(2))*(2+2*sqrt(2))^n + (1-3*sqrt(2))*(2-2*sqrt(2))^n)/2.

MATHEMATICA

LinearRecurrence[{4, 4}, {1, 14}, 30] (* Harvey P. Dale, Jul 18 2024 *)

PROG

(Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((1+3*r)*(2+2*r)^n+(1-3*r)*(2-2*r)^n)/2: n in [0..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 20 2009

CROSSREFS

Cf. A164539, A164675, A164541.

KEYWORD

nonn,easy

AUTHOR

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

EXTENSIONS

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 20 2009

STATUS

approved

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