A080879 -id:A080879 - cp's OEIS frontend

A102591 a(n) = Sum_{k=0..n} binomial(2n+1, 2k)*3^(n-k).

1, 6, 44, 328, 2448, 18272, 136384, 1017984, 7598336, 56714752, 423324672, 3159738368, 23584608256, 176037912576, 1313964867584, 9807567290368, 73204678852608, 546407161659392, 4078438577864704, 30441879976280064

Offset: 0

Paul Barry, Jan 22 2005

In general, Sum_{k=0..n} binomial(2n+1,2k)*r^(n-k) has g.f. (1-(r-1)x)/(1-2(r+1)+(r-1)^2x^2) and a(n) = ((sqrt(r)-1)^(2n+1) + (sqrt(r)+1)^(2n+1))/(2*sqrt(r)).

Index entries for linear recurrences with constant coefficients, signature (8,-4).

Bisection of A080879 or A002605.

Cf. A066443, A079935, A099156, A102592, A107903.

LinearRecurrence[{8,-4},{1,6},20] (* Harvey P. Dale, Sep 28 2021 *)

G.f.: (1-2x)/(1-8x+4x^2);
a(n) = 8*a(n-1) - 4*a(n-2);
a(n) = sqrt(3)*(sqrt(3)-1)^(2n+1)/6 + sqrt(3)*(sqrt(3)+1)^(2n+1)/6.
a(n) = 2^n*A079935(n). - R. J. Mathar, Sep 20 2012
a(n) = 2^(2*n+1)*Sum_{k >= n} binomial(2*k,2*n)*(1/3)^(k+1). Cf. A099156. - Peter Bala, Nov 29 2021
3*a(n)^2 = A107903(n)^2 + 2^(2*n+1). - Philippe Deléham, Mar 21 2023

A080881 a(n)a(n+3) - a(n+1)a(n+2) = 2^n, given a(0)=1, a(1)=2, a(2)=10.

Original entry on oeis.org

1, 2, 10, 21, 106, 223, 1126, 2369, 11962, 25167, 127078, 267361, 1350010, 2840303, 14341798, 30173889, 152359738, 320551567, 1618589926, 3405371681, 17195050234, 36176882223, 182671192870, 384324217729, 1940602920634

Offset: 0

Paul D. Hanna, Feb 22 2003

nonn

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,11,0,-4).

Cf. A080876, A080877, A080878, A080879, A080880, A080882.

CoefficientList[Series[(-x^3-x^2+2x+1)/(4x^4-11x^2+1),{x,0,30}],x] (* or *) LinearRecurrence[ {0,11,0,-4},{1,2,10,21},30] (* Harvey P. Dale, Jun 10 2024 *)

G.f.: (-x^3 - x^2 + 2*x + 1)/(4*x^4 - 11*x^2 + 1)
a(n + 4) = 11*a(n + 2) - 4*a(n) [From Richard Choulet, Dec 06 2008]
a(n) = (3/140*15^(1/2)*7^(1/2) + 1/4 + 3/56*7^(1/2) + 1/24*15^(1/2))*sqrt((11 + sqrt(105))/2)^n + ( - 3/140*15^(1/2)*7^(1/2) + 1/4 - 3/56*7^(1/2) + 1/24*15^(1/2))*sqrt((11 - sqrt(105))/2)^n + (3/140*15^(1/2)*7^(1/2) - 1/24*15^(1/2) - 3/56*7^(1/2) + 1/4)*( - sqrt((11 + sqrt(105))/2))^n + (1/4 + 3/56*7^(1/2) - 1/24*15^(1/2) - 3/140*15^(1/2)*7^(1/2))*( - sqrt((11 - sqrt(105))/2))^n [From Richard Choulet, Dec 07 2008]

A080882 a(n)a(n+3) - a(n+1)a(n+2) = 2^n, given a(0)=1, a(1)=3, a(2)=7.

Original entry on oeis.org

1, 3, 7, 22, 52, 164, 388, 1224, 2896, 9136, 21616, 68192, 161344, 508992, 1204288, 3799168, 8988928, 28357376, 67094272, 211662336, 500798464, 1579869184, 3738010624, 11792304128, 27900891136, 88018956288, 208255086592

Offset: 0

Paul D. Hanna, Feb 22 2003

nonn

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0, 8, 0, -4).

Cf. A080876, A080877, A080878, A080879, A080880, A080881.

a:= n-> (Matrix([[22,7,3,1]]). Matrix(4, (i,j)-> if (i=j-1) then 1 elif j=1 then [0,8,0,-4][i] else 0 fi)^(n))[1,4]: seq(a(n), n=0..26); # Alois P. Heinz, Aug 23 2008

a[0]=1; a[1]=3; a[2]=7; a[3]=22; a[n_] := a[n] = 8*a[n-2] - 4*a[n-4]; Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Jun 15 2015, after Richard Choulet *)
LinearRecurrence[{0,8,0,-4},{1,3,7,22},30] (* Harvey P. Dale, Mar 23 2025 *)

a(2n)=A080879(2n+1)=A080876(2n+4)/4, a(2n+1)=A080879(2n+2)/2=A080876(2n+5)/4.
G.f.: (-2*x^3 - x^2 + 3*x + 1)/(4*x^4 - 8*x^2 + 1).
a(n + 4) = 8*a(n + 2) - 4*a(n). - Richard Choulet, Dec 06 2008
a(n) = (7/24*3^(1/2) + 1/2)*((1 + sqrt(3)))^n + ( - 7/24*3^(1/2) + 1/2)*((1 - sqrt(3)))^n + ( - 1/24*3^(1/2))*( - (1 + sqrt(3)))^n + (1/24*3^(1/2))*( - ((1 - sqrt(3))))^n. - Richard Choulet, Dec 06 2008

A080876 a(n)a(n+3) - a(n+1)a(n+2) = 2^n, given a(0) = 1, a(1) = 1, and a(2) = 1.

Original entry on oeis.org

1, 1, 1, 2, 4, 12, 28, 88, 208, 656, 1552, 4896, 11584, 36544, 86464, 272768, 645376, 2035968, 4817152, 15196672, 35955712, 113429504, 268377088, 846649344, 2003193856, 6319476736, 14952042496, 47169216512, 111603564544

Offset: 0

Paul D. Hanna, Feb 22 2003

nonn

Index entries for linear recurrences with constant coefficients, signature (0, 8, 0, -4).

Cf. A080871, A080877, A080878, A080879, A080880, A080881, A080882.

CoefficientList[Series[(-6x^3-7x^2+x+1)/(4x^4-8x^2+1),{x,0,40}],x]  (* Harvey P. Dale, Mar 04 2011 *)

G.f.: (-6*x^3 - 7*x^2 + x + 1)/(4*x^4 - 8*x^2 + 1)
a(n + 4) = 8*a(n + 2) - 4*a(n). - Richard Choulet, Dec 06 2008
a(n) = (1/24 * 3^(1/2)) * (1 + sqrt(3))^n - (1/24 * 3^(1/2)) * (1 - sqrt(3))^n + (1/2 - 7/24 * 3^(1/2)) * (-(1 + sqrt(3)))^n + (1/2 + 7/24 * 3^(1/2))*(-(1 - sqrt(3)))^n. - Richard Choulet, Dec 06 2008

A080877 a(n)a(n+3) - a(n+1)a(n+2) = 2^n, given a(0)=1, a(1)=1, a(2)=2.

Original entry on oeis.org

1, 1, 2, 3, 8, 14, 40, 72, 208, 376, 1088, 1968, 5696, 10304, 29824, 53952, 156160, 282496, 817664, 1479168, 4281344, 7745024, 22417408, 40553472, 117379072, 212340736, 614604800, 1111830528, 3218112512, 5821620224, 16850255872

Offset: 0

Paul D. Hanna, Feb 22 2003

nonn

Index entries for linear recurrences with constant coefficients, signature (0, 6, 0, -4).

Cf. A080876, A080878, A080879, A080880, A080881, A080882.

Cf. A154626, A098648 (bisections). [From R. J. Mathar, Oct 26 2009]

LinearRecurrence[{0,6,0,-4},{1,1,2,3},50] (* or *) CoefficientList[ Series[ (-3x^3-4x^2+x+1)/(4x^4-6x^2+1),{x,0,50}],x] (* Harvey P. Dale, May 02 2011 *)

G.f.: (-3*x^3 - 4*x^2 + x + 1)/(4*x^4 - 6*x^2 + 1)
a(n + 4) = 6*a(n + 2) - 4*a(n) [From Richard Choulet, Dec 06 2008]
a(n) = ( - 1/20*5^(1/2) + 1/16*5^(1/2)*2^(1/2) - 1/16*2^(1/2) + 1/4)*(sqrt(3 + sqrt(5)))^n + (1/20*5^(1/2) + 1/16*5^(1/2)*2^(1/2) + 1/16*2^(1/2) + 1/4)*(sqrt(3 - sqrt(5)))^n + ( - 1/20*5^(1/2) - 1/16*5^(1/2)*2^(1/2) + 1/16*2^(1/2) + 1/4)*( - (sqrt(3 + sqrt(5))))^n + (1/20*5^(1/2) - 1/16*5^(1/2)*2^(1/2) - 1/16*2^(1/2) + 1/4)*( - (sqrt(3 - sqrt(5))))^n [From Richard Choulet, Dec 07 2008]

A080878 a(n)a(n+3) - a(n+1)a(n+2) = 2^n, given a(0)=1, a(1)=1, a(2)=3.

Original entry on oeis.org

1, 1, 3, 4, 14, 20, 72, 104, 376, 544, 1968, 2848, 10304, 14912, 53952, 78080, 282496, 408832, 1479168, 2140672, 7745024, 11208704, 40553472, 58689536, 212340736, 307302400, 1111830528, 1609056256, 5821620224, 8425127936, 30482399232

Offset: 0

Paul D. Hanna, Feb 22 2003

nonn

			G.f. = 1 + x + 3*x^2 + 4*x^3 + 14*x^4 + 20*x^5 + 72*x^6 + 104*x^7 + 376*x^8 + ...

Index entries for linear recurrences with constant coefficients, signature (0, 6, 0, -4).

Cf. A080876, A080877, A080879, A080880, A080881, A080882.

Cf. A000079, A082761, A098648.

a[ n_] := If[ n < 0, 2^n, 1] SeriesCoefficient[ (1 + x - 3*x^2 - 2*x^3)/(1 - 6*x^2 + 4*x^4), {x, 0, Abs@n}]; (* Michael Somos, May 25 2014 *)
a[ n_] := 2^Quotient[ n - 1, 2] If[ OddQ@n, Fibonacci@n, LucasL@n]; (* Michael Somos, May 25 2014 *)
LinearRecurrence[{0,6,0,-4},{1,1,3,4},40] (* Harvey P. Dale, Dec 07 2014 *)

{a(n) = if( n<0, 2^n, 1) * polcoeff( (1 + x - 3*x^2 - 2*x^3) / (1 - 6*x^2 + 4*x^4) + x * O(x^abs(n)), abs(n))}; /* Michael Somos, May 25 2014 */

{a(n) = 2^((n - 1)\2) * if( n%2, fibonacci(n), fibonacci(n-1) + fibonacci(n+1))}; /* Michael Somos, May 25 2014 */

G.f.: (1 + x - 3*x^2 - 2*x^3) / (1 - 6*x^2 + 4*x^4). a(n) = 6*a(n-2) - 4*a(n-4). - Michael Somos, Mar 05 2003
a(2n) = A080877(2n+1), a(2n+1) = A080877(2n+2)/2.
a(n) = (1/20*10^(1/2) + 1/4)*(sqrt(3 + sqrt(5)))^n + (1/20*10^(1/2) + 1/4)*(sqrt(3 - sqrt(5)))^n + ( - 1/20*10^(1/2) + 1/4)*( - (sqrt(3 + sqrt(5))))^n + ( - 1/20*10^(1/2) + 1/4)*( - (sqrt(3 - sqrt(5))))^n. - Richard Choulet, Dec 07 2008
a(-n) = a(n) / 2^n. a(2*n) = A098648(n). a(2*n + 1) = A082761(n). - Michael Somos, May 25 2014
0 = a(n)*(+2*a(n+2)) + a(n+1)*(+2*a(n+1) - 7*a(n+2) + a(n+3)) + a(n+2)*(+a(n+2)) for all n in Z. - Michael Somos, May 25 2014

A080880 a(n)a(n+3) - a(n+1)a(n+2) = 2^n, given a(0)=1, a(1)=2, a(2)=2.

Original entry on oeis.org

1, 2, 2, 5, 6, 17, 22, 65, 86, 257, 342, 1025, 1366, 4097, 5462, 16385, 21846, 65537, 87382, 262145, 349526, 1048577, 1398102, 4194305, 5592406, 16777217, 22369622, 67108865, 89478486, 268435457, 357913942, 1073741825, 1431655766, 4294967297

Offset: 0

Paul D. Hanna, Feb 22 2003

nonn

Index entries for linear recurrences with constant coefficients, signature (0, 5, 0, -4).

Cf. A080876, A080877, A080878, A080879, A080881, A080882.

LinearRecurrence[{0,5,0,-4},{1,2,2,5},40] (* Harvey P. Dale, Nov 30 2012 *)

a(2n)=(4^n+2)/3, a(2n+1)=4^n+1.
G.f.: (-5*x^3 - 3*x^2 + 2*x + 1)/(4*x^4 - 5*x^2 + 1)
a(n) = 5/6 + 5/12*2^n - 1/6*( - 1)^n - 1/12*( - 2)^n [From Richard Choulet, Dec 07 2008]
a(n + 4) = 5*a(n + 2) - 4*a(n) [From Richard Choulet, Dec 06 2008]

More terms from Ralf Stephan, Jul 25 2003

cp's OEIS Frontend

A102591 a(n) = Sum_{k=0..n} binomial(2n+1, 2k)*3^(n-k).

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A080881 a(n)*a(n+3) - a(n+1)*a(n+2) = 2^n, given a(0)=1, a(1)=2, a(2)=10.

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A080882 a(n)*a(n+3) - a(n+1)*a(n+2) = 2^n, given a(0)=1, a(1)=3, a(2)=7.

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A080876 a(n)*a(n+3) - a(n+1)*a(n+2) = 2^n, given a(0) = 1, a(1) = 1, and a(2) = 1.

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A080877 a(n)*a(n+3) - a(n+1)*a(n+2) = 2^n, given a(0)=1, a(1)=1, a(2)=2.

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A080878 a(n)*a(n+3) - a(n+1)*a(n+2) = 2^n, given a(0)=1, a(1)=1, a(2)=3.

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A080880 a(n)*a(n+3) - a(n+1)*a(n+2) = 2^n, given a(0)=1, a(1)=2, a(2)=2.

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A080881 a(n)a(n+3) - a(n+1)a(n+2) = 2^n, given a(0)=1, a(1)=2, a(2)=10.

A080882 a(n)a(n+3) - a(n+1)a(n+2) = 2^n, given a(0)=1, a(1)=3, a(2)=7.

A080876 a(n)a(n+3) - a(n+1)a(n+2) = 2^n, given a(0) = 1, a(1) = 1, and a(2) = 1.

A080877 a(n)a(n+3) - a(n+1)a(n+2) = 2^n, given a(0)=1, a(1)=1, a(2)=2.

A080878 a(n)a(n+3) - a(n+1)a(n+2) = 2^n, given a(0)=1, a(1)=1, a(2)=3.

A080880 a(n)a(n+3) - a(n+1)a(n+2) = 2^n, given a(0)=1, a(1)=2, a(2)=2.