A074324 -id:A074324

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A162766

a(n) = 3*a(n-2) for n > 2; a(1) = 4, a(2) = 3.

+10
9

4, 3, 12, 9, 36, 27, 108, 81, 324, 243, 972, 729, 2916, 2187, 8748, 6561, 26244, 19683, 78732, 59049, 236196, 177147, 708588, 531441, 2125764, 1594323, 6377292, 4782969, 19131876, 14348907, 57395628, 43046721, 172186884, 129140163

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

OFFSET

1,1

COMMENTS

Binomial transform is A162559. Second binomial transform is A077236.

LINKS

Table of n, a(n) for n=1..34.

Index entries for linear recurrences with constant coefficients, signature (0, 3).

FORMULA

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

G.f.: x*(4+3*x)/(1-3*x^2).

a(n) = A074324(n+1) = A166552(n+1) = 3^floor(n/2)*4^(n%2), where n%2 = 0 for n even, 1 for n odd. - M. F. Hasler, Dec 03 2014

PROG

(Magma) [ n le 2 select 5-n else 3*Self(n-2): n in [1..34] ];

(PARI) a(n)=3^(n\2)*4^(n%2) \\ M. F. Hasler, Dec 03 2014

CROSSREFS

Cf. A074324, A166552, A162559, A077236, A162436.

KEYWORD

nonn

AUTHOR

Klaus Brockhaus, Jul 13 2009

EXTENSIONS

G.f. corrected by Klaus Brockhaus, Sep 18 2009

STATUS

approved

A153594

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

+10
6

1, 8, 51, 304, 1769, 10200, 58603, 336224, 1927953, 11052712, 63358307, 363181200, 2081791609, 11932977272, 68400527259, 392075513536, 2247397253921, 12882196355400, 73841406542227, 423262699717616, 2426163312691977, 13906891405206808

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

OFFSET

1,2

COMMENTS

Second binomial transform of A054491. Fourth binomial transform of 1 followed by A162766 and of A074324 without initial term 1.

First differences are in A161728.

Lim_{n -> infinity} a(n)/a(n-1) = 4 + sqrt(3) = 5.73205080756887729....

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..500

Index entries for linear recurrences with constant coefficients, signature (8,-13).

FORMULA

G.f.: x/(1 - 8*x + 13*x^2). - Klaus Brockhaus, Dec 31 2008, corrected Oct 11 2009

a(n) = 8*a(n-1) - 13*a(n-2) for n > 1; a(0)=0, a(1)=1. - Philippe Deléham, Jan 01 2009

E.g.f.: sinh(sqrt(3)*x)*exp(4*x)/sqrt(3). - Ilya Gutkovskiy, Aug 23 2016

a(n) = Sum_{k=0..n-1} A027907(n,2k+1)*3^k. - J. Conrad, Aug 30 2016

a(n) = Sum_{k=0..n-1} A083882(n-1-k)*4^k. - J. Conrad, Sep 03 2016

MATHEMATICA

Join[{a=1, b=8}, Table[c=8*b-13*a; a=b; b=c, {n, 60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 19 2011 *)

LinearRecurrence[{8, -13}, {1, 8}, 40] (* Harvey P. Dale, Aug 16 2012 *)

PROG

(Magma) Z<x>:= PolynomialRing(Integers()); N<r>:=NumberField(x^2-3); S:=[ ((4+r)^n-(4-r)^n)/(2*r): n in [1..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Dec 31 2008

(Sage) [lucas_number1(n, 8, 13) for n in range(1, 22)] # Zerinvary Lajos, Apr 23 2009

(Magma) I:=[1, 8]; [n le 2 select I[n] else 8*Self(n-1)-13*Self(n-2): n in [1..25]]; // Vincenzo Librandi, Aug 23 2016

(PARI) a(n)=([0, 1; -13, 8]^(n-1)*[1; 8])[1, 1] \\ Charles R Greathouse IV, Sep 04 2016

CROSSREFS

Cf. A002194 (decimal expansion of sqrt(3)), A054491, A074324, A161728, A162766.

KEYWORD

nonn,easy

AUTHOR

Al Hakanson (hawkuu(AT)gmail.com), Dec 29 2008

EXTENSIONS

Extended beyond a(7) by Klaus Brockhaus, Dec 31 2008

Edited by Klaus Brockhaus, Oct 11 2009

STATUS

approved

A166552

a(n) = 3*a(n-2) for n > 2; a(1) = 1; a(2) = 4.

+10
3

1, 4, 3, 12, 9, 36, 27, 108, 81, 324, 243, 972, 729, 2916, 2187, 8748, 6561, 26244, 19683, 78732, 59049, 236196, 177147, 708588, 531441, 2125764, 1594323, 6377292, 4782969, 19131876, 14348907, 57395628, 43046721, 172186884, 129140163

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

OFFSET

1,2

COMMENTS

Interleaving of A000244 (powers of 3) and 4*A000244.

a(n) = A074324(n); A074324 has the additional term a(0)=1.

First differences are in A162852.

Second binomial transform is A054491. Fourth binomial transform is A153594.

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1000[Terms 1 through 300 were computed by Vincenzo Librandi; Terms 301 through 1000 by G. C. Greubel, May 17 2016]

Index entries for linear recurrences with constant coefficients, signature (0,3)

FORMULA

a(n) = (7+(-1)^n)*3^(1/4*(2*n-5+(-1)^n))/2.

G.f.: x*(1+4*x)/(1-3*x^2).

a(n+3) = a(n+2)*a(n+1)/a(n). - Reinhard Zumkeller, Mar 04 2011

a(n) = 3^floor((n-1)/2)*4^(1-n%2). - M. F. Hasler, Dec 03 2014

E.g.f.: (sqrt(3)*sinh(sqrt(3)*x) + 4*cosh(sqrt(3)*x) - 4)/3. - Ilya Gutkovskiy, May 17 2016

MATHEMATICA

LinearRecurrence[{0, 3}, {1, 4}, 50] (* G. C. Greubel, May 17 2016 *)

PROG

(Magma) [ n le 2 select 3*n-2 else 3*Self(n-2): n in [1..35] ];

(PARI) a(n)=3^(n\2)*(4/3)^!bittest(n, 0) \\ M. F. Hasler, Dec 03 2014

CROSSREFS

Equals A162766 preceded by 1.

Cf. A000244 (powers of 3), A074324, A162852, A054491, A153594.

KEYWORD

nonn

AUTHOR

Klaus Brockhaus, Oct 16 2009

STATUS

approved

A162466

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

+10
1

1, 8, 12, 96, 144, 1152, 1728, 13824, 20736, 165888, 248832, 1990656, 2985984, 23887872, 35831808, 286654464, 429981696, 3439853568, 5159780352, 41278242816, 61917364224, 495338913792, 743008370688, 5944066965504

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

OFFSET

1,2

COMMENTS

Eighth binomial transform is A161729.

LINKS

Table of n, a(n) for n=1..24.

Index entries for linear recurrences with constant coefficients, signature (0,12).

FORMULA

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

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

a(n) = 2^(n-1)*A074324(n). - M. F. Hasler, Dec 03 2014

MATHEMATICA

LinearRecurrence[{0, 12}, {1, 8}, 30] (* Harvey P. Dale, Sep 17 2020 *)

PROG

(PARI) {m=24; v=concat([1, 8], vector(m-2)); for(n=3, m, v[n]=12*v[n-2]); v}

(PARI) Vec(x*(1+8*x)/(1-12*x^2)+O(x^29)) \\ M. F. Hasler, Dec 03 2014

CROSSREFS

Cf. A161729, A161728, A162436, A162272, A074324.

KEYWORD

nonn

AUTHOR

Klaus Brockhaus, Jul 04 2009

EXTENSIONS

G.f. and comment corrected, formula added by Klaus Brockhaus, Sep 18 2009

STATUS

approved

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