A162766 -id:A162766 - cp's OEIS frontend

A164265 Partial sums of A162766.

4, 7, 19, 28, 64, 91, 199, 280, 604, 847, 1819, 2548, 5464, 7651, 16399, 22960, 49204, 68887, 147619, 206668, 442864, 620011, 1328599, 1860040, 3985804, 5580127, 11957419, 16740388, 35872264, 50221171, 107616799, 150663520, 322850404

Offset: 1

Klaus Brockhaus, Aug 11 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,3,-3).

Cf. A162766, A164123 (partial sums of A162436).

T:=[ n le 2 select 5-n else 3*Self(n-2): n in [1..33] ]; [ n eq 1 select T[1] else Self(n-1)+T[n]: n in [1..#T]];

LinearRecurrence[{1,3,-3},{4,7,19},40] (* Harvey P. Dale, Aug 28 2016 *)

a(n)=if(n%2,15,7)*3^(n\2)\2-3 \\ Charles R Greathouse IV, Jul 15 2011

a(n) = 3*a(n-2)+7 for n > 2; a(1) = 4, a(2) = 7.
a(n) = (11-4*(-1)^n)*3^(1/4*(2*n-1+(-1)^n))/2-7/2.
G.f.: x*(4+3*x)/((1-x)*(1-3*x^2)).

A153594 a(n) = ((4 + sqrt(3))^n - (4 - sqrt(3))^n)/(2*sqrt(3)).

Original entry on oeis.org

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

Offset: 1

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

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....

G. C. Greubel, Table of n, a(n) for n = 1..500
Index entries for linear recurrences with constant coefficients, signature (8,-13).

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

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

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

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 *)

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

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

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

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

Edited by Klaus Brockhaus, Oct 11 2009

A074324 a(2n+1) = 3^n, a(2n) = 4*3^(n-1) except for a(0) = 1.

Original entry on oeis.org

1, 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

Offset: 0

Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002

Also: Coefficient of the highest power of q in the expansion of nu(0)=1, nu(1)=b and for n>=2, nu(n)=b*nu(n-1)+lambda*(n-1)_q*nu(n-2) with (b,lambda)=(1,3), where (n)_q=(1+q+...+q^(n-1)) and q is a root of unity.
Instead of listing the coefficients of the highest power of q in each nu(n), if we list the coefficients of the smallest power of q (i.e., constant terms), we get a weighted Fibonacci numbers described by f(0)=1, f(1)=1, for n>=2, f(n)=f(n-1)+3f(n-2).
Sequences A162766, A166552 are essentially the same. - M. F. Hasler, Dec 03 2014

			nu(0)=1;
nu(1)=1;
nu(2)=4;
nu(3)=7+3q;
nu(4)=19+15q+12q^2;
nu(5)=40+45q+42q^2+30q^3+9q^4;
nu(6)=97+147q+180q^2+168q^3+147q^4+81q^5+36q^6;
by listing the coefficients of the highest power in each nu(n), we get, 1,1,4,3,12,9,36,....

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
M. Beattie, S. Dăscălescu and S. Raianu, Lifting of Nichols Algebras of Type B_2, arXiv:math/0204075 [math.QA], 2002.
Index entries for linear recurrences with constant coefficients, signature (0,3).

Cf. A006130.

[1] cat [(1/6)*(7+(-1)^n)*3^Floor(n/2):n in [1..40]]; // Vincenzo Librandi, Jul 20 2013

CoefficientList[Series[-(1 + x + x^2) / (-1 + 3 x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 20 2013 *)
LinearRecurrence[{0,3},{1,1,4},40] (* Harvey P. Dale, Mar 13 2016 *)

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

For given b and lambda, the recurrence relation is given by; t(0)=1, t(1)=b, t(2)=b^2+lambda and for n>=3, t(n) = lambda*t(n-2).
G.f.: -(1+x+x^2)/(-1+3*x^2). - R. J. Mathar, Dec 05 2007
a(n) = 3*a(n-2) for n>2. - Ralf Stephan, Jul 19 2013
a(n) = (1/6)*(7+(-1)^n)*3^floor(n/2) for n>0. - Ralf Stephan, Jul 19 2013

More terms from R. J. Mathar, Dec 05 2007

Simpler definition from M. F. Hasler, Dec 03 2014

A162813 a(n) = 3*a(n-2) for n > 2; a(1) = 5, a(2) = 3.

Original entry on oeis.org

5, 3, 15, 9, 45, 27, 135, 81, 405, 243, 1215, 729, 3645, 2187, 10935, 6561, 32805, 19683, 98415, 59049, 295245, 177147, 885735, 531441, 2657205, 1594323, 7971615, 4782969, 23914845, 14348907, 71744535, 43046721, 215233605, 129140163, 645700815, 387420489

Offset: 1

Klaus Brockhaus, Jul 20 2009

Binomial transform is A162562.

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

Cf. A162562, A162436, A162766.

[ n le 2 select 7-2*n else 3*Self(n-2): n in [1..34] ];

nxt[{a_,b_}]:={b,3a}; NestList[nxt,{5,3},40][[All,1]] (* or *) LinearRecurrence[ {0,3},{5,3},40] (* Harvey P. Dale, May 29 2021 *)

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

G.f.: x*(5+3*x)/(1-3*x^2)

a(35)-a(36) from Yifan Xie, Jul 20 2022

A162852 a(n) = 3*a(n-2) for n > 2; a(1) = 3, a(2) = -1.

Original entry on oeis.org

3, -1, 9, -3, 27, -9, 81, -27, 243, -81, 729, -243, 2187, -729, 6561, -2187, 19683, -6561, 59049, -19683, 177147, -59049, 531441, -177147, 1594323, -531441, 4782969, -1594323, 14348907, -4782969, 43046721, -14348907, 129140163

Offset: 1

Klaus Brockhaus, Jul 14 2009

sign

Third binomial transform is A162560.

Equivalently, 3^n followed by -3^(n-1), n > 0. - Muniru A Asiru, Oct 25 2018

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,3).

Cf. A162560, A162436, A162766, A162813.

a:=[3,-1];; for n in [3..25] do a[n]:=3*a[n-2]; od; a; # Muniru A Asiru, Oct 25 2018

[ n le 2 select 7-4*n else 3*Self(n-2): n in [1..34] ];

seq(op([3^n,-3^(n-1)]),n=1..18); # Muniru A Asiru, Oct 25 2018

Rest[CoefficientList[Series[x*(3-x)/(1-3*x^2), {x, 0, 40}], x]] (* or *) LinearRecurrence[{0,3}, {3,-1}, 40] (* G. C. Greubel, Oct 24 2018 *)

x='x+O('x^40); Vec(x*(3-x)/(1-3*x^2)) \\ G. C. Greubel, Oct 24 2018

a(n) = ((4-5*(-1)^n)*3^(1/4*(2*n-1+(-1)^n)))/3.
G.f.: x*(3-x)/(1-3*x^2). [corrected by Klaus Brockhaus, Sep 18 2009]
E.g.f.: (1 - cosh(sqrt(3)*x) + 3*sqrt(3)*sinh(sqrt(3)*x))/3. - G. C. Greubel, Oct 24 2018

A166552 a(n) = 3*a(n-2) for n > 2; a(1) = 1; a(2) = 4.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Oct 16 2009

nonn

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.

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)

Equals A162766 preceded by 1.

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

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

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

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

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

A162559 a(n) = ((4+sqrt(3))(1+sqrt(3))^n + (4-sqrt(3))(1-sqrt(3))^n)/2.

Original entry on oeis.org

4, 7, 22, 58, 160, 436, 1192, 3256, 8896, 24304, 66400, 181408, 495616, 1354048, 3699328, 10106752, 27612160, 75437824, 206099968, 563075584, 1538351104, 4202853376, 11482408960, 31370524672, 85705867264, 234152783872, 639717302272, 1747740172288, 4774914949120

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Jul 06 2009

nonn

Binomial transform of A162766. Inverse binomial transform of A077236.

Index entries for linear recurrences with constant coefficients, signature (2,2).

Cf. A162766, A077236.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-3); S:=[ ((4+r)*(1+r)^n+(4-r)*(1-r)^n)/2: n in [0..25] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 13 2009

seq((1/2)*simplify((4+sqrt(3))*(1+sqrt(3))^n+(4-sqrt(3))*(1-sqrt(3))^n), n = 0 .. 27); # Emeric Deutsch, Jul 16 2009

LinearRecurrence[{2,2},{4,7},30] (* Harvey P. Dale, Sep 21 2018 *)

a(n) = 2*a(n-1) + 2*a(n-2) for n > 1; a(0) = 4, a(1) = 7.

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

Edited by Klaus Brockhaus, Paolo P. Lava and Emeric Deutsch, Jul 13 2009

Two different extensions were received. This version was rechecked by N. J. A. Sloane, Jul 19 2009

A162561 a(n) = ((4+sqrt(3))(5+sqrt(3))^nv+v(4-sqrt(3))(5-sqrt(3))^n)/2.

Original entry on oeis.org

4, 23, 142, 914, 6016, 40052, 268168, 1800536, 12105664, 81444848, 548123872, 3689452064, 24835795456, 167190009152, 1125512591488, 7576945713536, 51008180122624, 343388995528448, 2311709992586752, 15562542024241664

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Jul 06 2009

nonn

Third binomial transform of A077236. Fourth binomial transform of A162559. Fifth binomial transform of A162766.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10, -22).

Cf. A077236, A162559, A162766.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-3); S:=[ ((4+r)*(5+r)^n+(4-r)*(5-r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 14 2009

LinearRecurrence[{10,-22},{4,23},30] (* Harvey P. Dale, Mar 27 2013 *)

a(n) = 10*a(n-1) - 22*a(n-2) for n > 2; a(0) = 4, a(1) = 23.

G.f.: (4-17*x)/(1-10*x+22*x^2).

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 14 2009

A164311 a(n) = 12a(n-1) - 33a(n-2) for n > 1; a(0) = 4, a(1) = 27.

Original entry on oeis.org

4, 27, 192, 1413, 10620, 80811, 619272, 4764501, 36738036, 283627899, 2191179600, 16934434533, 130904287596, 1012015111563, 7824339848088, 60495579495477, 467743738958820, 3616570744155099, 27963305544220128, 216212831973523269

Offset: 0

Klaus Brockhaus, Aug 12 2009

Binomial transform of A162561. Sixth binomial transform of A162766.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-33).

Cf. A162561, A162766.

[ n le 2 select 23*n-19 else 12*Self(n-1)-33*Self(n-2): n in [1..20] ];

LinearRecurrence[{12,-33}, {4,27}, 50] (* or *) CoefficientList[Series[(4 - 21*x)/(1 - 12*x + 33*x^2), {x,0,50}], x] (* G. C. Greubel, Sep 13 2017 *)

x='x+O('x^50); Vec((4-21*x)/(1-12*x+33*x^2)) \\ G. C. Greubel, Sep 13 2017

a(n) = ((4+sqrt(3))*(6+sqrt(3))^n + (4-sqrt(3))*(6-sqrt(3))^n)/2.
G.f.: (4-21*x)/(1-12*x+33*x^2).
E.g.f.: (4*cosh(sqrt(3)*x) + sqrt(3)*sinh(sqrt(3)*x))*exp(6*x). - G. C. Greubel, Sep 13 2017

cp's OEIS Frontend

A164265 Partial sums of A162766.

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A153594 a(n) = ((4 + sqrt(3))^n - (4 - sqrt(3))^n)/(2*sqrt(3)).

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A074324 a(2n+1) = 3^n, a(2n) = 4*3^(n-1) except for a(0) = 1.

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A162813 a(n) = 3*a(n-2) for n > 2; a(1) = 5, a(2) = 3.

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A162852 a(n) = 3*a(n-2) for n > 2; a(1) = 3, a(2) = -1.

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A166552 a(n) = 3*a(n-2) for n > 2; a(1) = 1; a(2) = 4.

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A162559 a(n) = ((4+sqrt(3))*(1+sqrt(3))^n + (4-sqrt(3))*(1-sqrt(3))^n)/2.

A162559 a(n) = ((4+sqrt(3))(1+sqrt(3))^n + (4-sqrt(3))(1-sqrt(3))^n)/2.

A162561 a(n) = ((4+sqrt(3))(5+sqrt(3))^nv+v(4-sqrt(3))(5-sqrt(3))^n)/2.

A164311 a(n) = 12a(n-1) - 33a(n-2) for n > 1; a(0) = 4, a(1) = 27.