A098648 -id:A098648 - cp's OEIS frontend

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

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

A108404 Expansion of (1-4x)/(1-8x+11x^2).

Original entry on oeis.org

1, 4, 21, 124, 761, 4724, 29421, 183404, 1143601, 7131364, 44471301, 277325404, 1729418921, 10784771924, 67254567261, 419404046924, 2615432135521, 16310012568004, 101710347053301, 634272638178364, 3955367287840601

Offset: 0

Philippe Deléham, Jul 04 2005

Binomial transform of A098648. Second binomial transform of A001077. Third binomial transform of A084057. 4th binomial transform of (1, 0, 5, 0, 25, 0, 125, 0, 625, 0, 3125, ...).

Seiichi Manyama, Table of n, a(n) for n = 0..1258
Index entries for linear recurrences with constant coefficients, signature (8,-11).

Cf. A001077, A084057, A098648.

CoefficientList[Series[(1-4x)/(1-8x+11x^2),{x,0,30}],x] (* or *) LinearRecurrence[{8,-11},{1,4},30] (* Harvey P. Dale, Jan 03 2012 *)

E.g.f.: exp(4x)cosh(sqrt(5)x).
a(n) = 8a(n-1) - 11a(n-2), a(0) = 1, a(1) = 4.
a(n) = ((4+sqrt(5))^n + (4-sqrt(5))^n)/2.
a(n+1)/a(n) converges to 4 + sqrt(5) = 6.2360679774997896964... = 4+A002163.
a(n) = A091870(n+1)-4*A091870(n). - R. J. Mathar, Nov 10 2013

A163063 Lucas(3n+2) = Fibonacci(3n+1) + Fibonacci(3n+3).

Original entry on oeis.org

3, 11, 47, 199, 843, 3571, 15127, 64079, 271443, 1149851, 4870847, 20633239, 87403803, 370248451, 1568397607, 6643838879, 28143753123, 119218851371, 505019158607, 2139295485799, 9062201101803, 38388099893011

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Jul 20 2009

Binomial transform of A163062. Second binomial transform of A163114. Inverse binomial transform of A098648 without initial 1.

Nathaniel Johnston, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (4,1).

Cf. A000032, A000045, A163062, A163114, A098648, A001077 (L(3*n)/L(2)), A048876 (L(3*n+1)).

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-5); S:=[ ((3+r)*(2+r)^n+(3-r)*(2-r)^n)/2: n in [0..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 21 2009

[Lucas(3*n+2): n in [0..30]]; // Vincenzo Librandi, Apr 18 2011

with(combinat):A163063:=proc(n)return fibonacci(3*n+1) + fibonacci(3*n+3): end:seq(A163063(n), n=0..21); # Nathaniel Johnston, Apr 18 2011

Table[Fibonacci[3n + 1] + Fibonacci[3n + 3], {n, 0, 21}] (* Alonso del Arte, Nov 29 2010 *)
LinearRecurrence[{4,1},{3,11},30] (* Harvey P. Dale, Apr 14 2021 *)

Vec((3-x)/(1-4*x-x^2) + O(x^100)) \\ Altug Alkan, Dec 10 2015

a(n) = 4*a(n-1)+a(n-2) for n > 1; a(0) = 3, a(1) = 11.
G.f.: (3-x)/(1-4*x-x^2).
a(n) = A033887(n) + A014445(n+1).
a(n) = ((3+sqrt(5))*(2+sqrt(5))^n+(3-sqrt(5))*(2-sqrt(5))^n)/2.
a(n) = A000032(3*n+2), n>=0, (Lucas trisection). - Wolfdieter Lang, Mar 09 2011.
a(n) = 5*F(n)*F(n+1)*L(n+1) + L(n+2)*(-1)^n with F(n)=A000045(n) and L(n)=A000032(n). - J. M. Bergot, Dec 10 2015

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 21 2009

A163114 a(n) = 5*a(n-2) for n > 2; a(1) = 3, a(2) = 5.

Original entry on oeis.org

3, 5, 15, 25, 75, 125, 375, 625, 1875, 3125, 9375, 15625, 46875, 78125, 234375, 390625, 1171875, 1953125, 5859375, 9765625, 29296875, 48828125, 146484375, 244140625, 732421875, 1220703125, 3662109375, 6103515625, 18310546875

Offset: 1

Klaus Brockhaus, Jul 21 2009

Binomial transform is A163062, second binomial transform is A163063, third binomial transform is A098648 without initial 1, fourth binomial transform is A163064, fifth binomial transform is A163065.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,5).

Cf. A056487, A098648, A163062, A163063, A163064, A163065, A111386.

[ n le 2 select 2*n+1 else 5*Self(n-2): n in [1..29] ];

CoefficientList[Series[x*(3 + 5*x)/(1 - 5*x^2), {x, 0, 50}], x] (* G. C. Greubel, Dec 21 2017 *)
LinearRecurrence[{0,5},{3,5},30] (* Harvey P. Dale, Aug 01 2021 *)

x='x+O('x^30); Vec(x*(3+5*x)/(1-5*x^2)) \\ G. C. Greubel, Dec 21 2017

a(n) = (2-(-1)^n)*5^(1/4*(2*n-1+(-1)^n)).
G.f.: x*(3+5*x)/(1-5*x^2).
a(n) = A056487(n), n>=1.
E.g.f.: cosh(sqrt(5)*x) + 3*sinh(sqrt(5)*x)/sqrt(5) - 1. - Stefano Spezia, Nov 19 2023

A098647 Trace sequence associated to the 4 X 4 Krawtchouk matrix and its transpose.

Original entry on oeis.org

1, 12, 224, 4608, 96256, 2015232, 42205184, 883949568, 18513657856, 387755016192, 8121246285824, 170093589823488, 3562486393470976, 74613683694600192, 1562729279488262144, 32730226951263879168

Offset: 0

Paul Barry, Sep 18 2004

Let A=[1,1,1,1;3,1,-1,-3;3,-1,-1,3;1,-1,1,-1], the 4 X 4 Krawtchouk matrix. Then a(n)=trace((A*A')^n)/4.

Twelfth binomial transform of ((4*sqrt(5))^n +(-4*sqrt(5))^n)/2, with g.f. 1/(1-80*x^2).

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (24,-64).

Cf. A098646.

G.f.: (1-12*x)/(1-24*x+64*x^2).
a(n) = ((12+4*sqrt(5))^n+(12-4*sqrt(5))^n)/2.
a(n) = 2^(n-1)*((sqrt(5)-1)^(2*n)+(sqrt(5)+1)^(2*n)).
a(n) = 4^n*A098648(n). - R. J. Mathar, Nov 11 2013

A163064 a(n) = ((3+sqrt(5))(4+sqrt(5))^n + (3-sqrt(5))(4-sqrt(5))^n)/2.

Original entry on oeis.org

3, 17, 103, 637, 3963, 24697, 153983, 960197, 5987763, 37339937, 232854103, 1452093517, 9055353003, 56469795337, 352149479663, 2196028088597, 13694580432483, 85400334485297, 532562291125063, 3321094649662237

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Jul 20 2009

nonn

Binomial transform of A098648 without initial 1. Fourth binomial transform of A163114. Inverse binomial transform of A163065.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-11).

Cf. A098648, A163114, A163065.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-5); S:=[ ((3+r)*(4+r)^n+(3-r)*(4-r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 21 2009

I:=[3,17]; [n le 2 select I[n] else 8*Self(n-1) - 11*Self(n-2): n in [1..30]]; // G. C. Greubel, Dec 22 2017

CoefficientList[Series[(3-7*x)/(1-8*x+11*x^2), {x,0,50}], x] (* or *) LinearRecurrence[{8,-11}, {3,17}, 30] (* G. C. Greubel, Dec 22 2017 *)

x='x+O('x^30); Vec((3-7*x)/(1-8*x+11*x^2)) \\ G. C. Greubel, Dec 22 2017

a(n) = 8*a(n-1) - 11*a(n-2) for n > 1; a(0) = 3, a(1) = 17.

G.f.: (3-7*x)/(1-8*x+11*x^2).

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 21 2009

A191348 Array read by antidiagonals: ((ceiling(sqrt(n)) + sqrt(n))^k + (ceiling(sqrt(n)) - sqrt(n))^k)/2 for columns k >= 0 and rows n >= 0.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 4, 6, 2, 1, 0, 8, 20, 7, 2, 1, 0, 16, 68, 26, 8, 3, 1, 0, 32, 232, 97, 32, 14, 3, 1, 0, 64, 792, 362, 128, 72, 15, 3, 1, 0, 128, 2704, 1351, 512, 376, 81, 16, 3, 1, 0

Offset: 0

Charles L. Hohn, May 31 2011

			1, 0,  0,   0,    0,     0,      0,      0,       0,        0,         0, ...
1, 1,  2,   4,    8,    16,     32,     64,     128,      256,       512, ...
1, 2,  6,  20,   68,   232,    792,   2704,    9232,    31520,    107616, ...
1, 2,  7,  26,   97,   362,   1351,   5042,   18817,    70226,    262087, ...
1, 2,  8,  32,  128,   512,   2048,   8192,   32768,   131072,    524288, ...
1, 3, 14,  72,  376,  1968,  10304,  53952,  282496,  1479168,   7745024, ...
1, 3, 15,  81,  441,  2403,  13095,  71361,  388881,  2119203,  11548575, ...
1, 3, 16,  90,  508,  2868,  16192,  91416,  516112,  2913840,  16450816, ...
1, 3, 17,  99,  577,  3363,  19601, 114243,  665857,  3880899,  22619537, ...
1, 3, 18, 108,  648,  3888,  23328, 139968,  839808,  5038848,  30233088, ...
1, 4, 26, 184, 1316,  9424,  67496, 483424, 3462416, 24798784, 177615776, ...
1, 4, 27, 196, 1433, 10484,  76707, 561236, 4106353, 30044644, 219825387, ...
1, 4, 28, 208, 1552, 11584,  86464, 645376, 4817152, 35955712, 268377088, ...
1, 4, 29, 220, 1673, 12724,  96773, 736012, 5597777, 42574180, 323800109, ...
1, 4, 30, 232, 1796, 13904, 107640, 833312, 6451216, 49943104, 386642400, ...
...

Row 1 is A000007, row 2 is A011782, row 3 is A006012, row 4 is A001075, row 5 is A081294, row 6 is A098648, row 7 is A084120, row 8 is A146963, row 9 is A001541, row 10 is A081341, row 11 is A084134, row 13 is A090965.
Row 3*2 is A056236, row 4*2 is A003500, row 5*2 is A155543, row 9*2 is A003499.
Cf. A191347 which uses floor() in place of ceiling().

T(n, k) = if (k==0, 1, if (k==1, ceil(sqrt(n)), T(n,k-2)*(n-T(n,1)^2) + T(n,k-1)*T(n,1)*2));
matrix(9, 9, n, k, T(n-1, k-1)) \\ Charles L. Hohn, Aug 23 2019

For each row n >= 0 let T(n,0)=1 and T(n,1) = ceiling(sqrt(n)), then for each column k >= 2: T(n,k) = T(n,k-2)*(n-T(n,1)^2) + T(n,k-1)*T(n,1)*2. - Charles L. Hohn, Aug 23 2019

A228842 Binomial transform of A014448.

Original entry on oeis.org

2, 6, 28, 144, 752, 3936, 20608, 107904, 564992, 2958336, 15490048, 81106944, 424681472, 2223661056, 11643240448, 60964798464, 319215828992, 1671435780096, 8751751364608, 45824765067264, 239941584945152, 1256350449401856, 6578336356630528, 34444616342175744

Offset: 0

R. J. Mathar, Nov 10 2013

The binomial transform of this sequence is 2, 8, 42, 248,... = 2*A108404(n).

C. Smith, A Treatise on Algebra, Macmillan, London, 5th ed., 1950, p. 360, Example 44.

Colin Barker, Table of n, a(n) for n = 0..1000
P. Bhadouria, D. Jhala, and B. Singh, Binomial Transforms of the k-Lucas Sequences and its Properties, The Journal of Mathematics and Computer Science (JMCS), Volume 8, Issue 1, Pages 81-92; sequence B_4.
Takao Komatsu, Asymmetric Circular Graph with Hosoya Index and Negative Continued Fractions, arXiv:2105.08277 [math.CO], 2021.
Index entries for linear recurrences with constant coefficients, signature (6,-4).

Cf. A014448, A108404, A098648.

When divided by 2^n this becomes(essentially) A005248.

CoefficientList[Series[2*(1 - 3 x)/(1 - 6 x + 4 x^2), {x, 0, 23}], x] (* Michael De Vlieger, Aug 26 2021 *)
LinearRecurrence[{6,-4},{2,6},30] (* Harvey P. Dale, Jun 30 2024 *)

Vec(2*(1 - 3*x) / (1 - 6*x + 4*x^2) + O(x^30)) \\ Colin Barker, Sep 21 2017

G.f.: 2*( 1-3*x ) / ( 1-6*x+4*x^2 ).
a(n) = 2*A098648(n).
From Colin Barker, Sep 21 2017: (Start)
a(n) = (3-sqrt(5))^n + (3+sqrt(5))^n.
a(n) = 6*a(n-1) - 4*a(n-2) for n>1.
(End)

cp's OEIS Frontend

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A108404 Expansion of (1-4x)/(1-8x+11x^2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A163063 Lucas(3n+2) = Fibonacci(3n+1) + Fibonacci(3n+3).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Magma

Maple

Mathematica

PARI

Formula

Extensions

A163114 a(n) = 5*a(n-2) for n > 2; a(1) = 3, a(2) = 5.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A098647 Trace sequence associated to the 4 X 4 Krawtchouk matrix and its transpose.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A163064 a(n) = ((3+sqrt(5))*(4+sqrt(5))^n + (3-sqrt(5))*(4-sqrt(5))^n)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Magma

Mathematica

PARI

Formula

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.

A163064 a(n) = ((3+sqrt(5))(4+sqrt(5))^n + (3-sqrt(5))(4-sqrt(5))^n)/2.