A057087 -id:A057087 - cp's OEIS frontend

A218134 Norm of coefficients in the expansion of 1/(1 - 2x - ix^2), where i is the imaginary unit.

1, 4, 17, 80, 369, 1700, 7841, 36160, 166753, 768996, 3546289, 16354000, 75417809, 347795396, 1603886913, 7396455680, 34109360321, 157298104900, 725393076049, 3345209499600, 15426707209777, 71141522037604, 328074947492321, 1512944453384000, 6977067089461281

Offset: 0

Paul D. Hanna, Oct 21 2012

nonn

The radius of convergence of g.f. equals 1 + sqrt(2) - sqrt(2)*sqrt(1 + sqrt(2)) = 0.216845335...

The following remarks assume an offset of 1. This sequence is a divisibility sequence, i.e., a(n) divides a(m) whenever n divides m. The sequence satisfies a linear recurrence of order 4. It is the case P1 = 4, P2 = -4, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Mar 25 2014

			G.f.: A(x) = 1 + 4*x + 17*x^2 + 80*x^3 + 369*x^4 + 1700*x^5 + 7841*x^6 +...
The terms equal the norm of the complex coefficients in the expansion:
1/(1 - 2*x - i*x^2) = 1 + 2*x + (4 + i)*x^2 + (8 + 4*i)*x^3 + (15 + 12*i)*x^4 + (26 + 32*i)*x^5 + (40 + 79*i)*x^6 + (48 + 184*i)*x^7 +...
so that
a(1) = 2^2, a(2) = 4^2 + 1, a(3) = 8^2 + 4^2, a(4) = 15^2 + 12^2, a(5) = 26^2 + 32^2, ...

Peter Bala, Linear divisibility sequences and Chebyshev polynomials
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume
Index entries for linear recurrences with constant coefficients, signature (4,2,4,-1).

Cf. A105309, A218135.

Cf. also A000129, A057087, A100047.

LinearRecurrence[{4, 2, 4, -1}, {1, 4, 17, 80}, 25] (* Jean-François Alcover, Nov 02 2019 *)

{a(n)=norm(polcoeff(1/(1-2*x-I*x^2+x*O(x^n)), n))}
for(n=0,31,print1(a(n),", "))

G.f.: (1 - x^2)/(1 - 4*x - 2*x^2 - 4*x^3 + x^4).
From Peter Bala, Mar 25 2014: (Start)
The following formulas assume an offset of 1.
a(n) = 1/(2*sqrt(2))*(T(n,1 + sqrt(2)) - T(n,1 - sqrt(2))), where T(n,x) denotes the Chebyshev polynomial of the first kind.
a(n) = the bottom left entry of the 2 X 2 matrix T(n, M), where M is the 2 X 2 matrix [0, 1; 1, 2]. Note, the bottom left element of the matrix M^n gives the Lucas sequence A000129.
a(n) = U(n-1,exp(2*i*Pi/8))*U(n-1,exp(-2*i*Pi/8)) = U(n-1,(1 + i)/sqrt(2))*U(n-1,(1 - i)/sqrt(2)), where U(n,x) denotes the Chebyshev polynomial of the second kind.
The o.g.f. is the Chebyshev transform of the rational function x/(1 - 4*x - 4*x^2), where the Chebyshev transform takes the function A(x) to the function (1 - x^2)/(1 + x^2)*A(x/(1 + x^2)). See the remarks in A100047 for the general connection between Chebyshev polynomials and 4th-order linear divisibility sequences. (End)

A099582 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k-1)*4^(n-k-1).

Original entry on oeis.org

0, 0, 1, 4, 24, 112, 560, 2688, 13056, 62976, 304384, 1469440, 7096320, 34263040, 165441536, 798818304, 3857055744, 18623496192, 89922273280, 434183077888, 2096421666816, 10122418978816, 48875363631104, 235991130439680

Offset: 0

Paul Barry, Oct 23 2004

In general a(n) = Sum{k=0..floor(n/2)} binomial(n-k, k-1) * r^(n-k-1) has g.f. x^2/((1-r*x^2)*(1-r*x-r*x^2)) and satisfies a(n) = r*a(n-1) + 2*r*a(n-2) - r^2*a(n-3) - r^2*a(n-4).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,8,-16,-16).

Cf. A099177, A099581.

I:=[0,0,1,4]; [n le 4 select I[n] else 4*(Self(n-1) +2*Self(n-2) -4*Self(n-3) -4*Self(n-4)): n in [1..41]]; // G. C. Greubel, Jul 22 2022

Table[Sum[Binomial[n-k,k-1]*4^(n-k-1),{k,0,Floor[n/2]}],{n,0,30}] (* or *) LinearRecurrence[{4,8,-16,-16},{0,0,1,4},30] (* Harvey P. Dale, Jul 19 2012 *)

[2^(n-3)*(lucas_number1(n,2,-1) - (n%2)) for n in (0..40)] # G. C. Greubel, Jul 22 2022

G.f.: x^2/((1-4*x^2)*(1-4*x-4*x^2)).
a(n) = 4*a(n-1) + 8*a(n-2) - 16*a(n-3) - 16*a(n-4) with a(n) = (n^3-n)/6 for n<5.
From G. C. Greubel, Jul 22 2022: (Start)
a(n) = 2^(n-4)*(2*A000129(n) - (1 - (-1)^n)).
a(n) = (1/4)*(A057087(n-1) - 2^(n-2)*(1 - (-1)^n)).
E.g.f.: (exp(2*x)*sinh(2*sqrt(2)*x) - sqrt(2)*sinh(2*x))/(8*sqrt(2)). (End)

A189800 a(n) = 6a(n-1) + 8a(n-2), with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 6, 44, 312, 2224, 15840, 112832, 803712, 5724928, 40779264, 290475008, 2069084160, 14738305024, 104982503424, 747801460736, 5326668791808, 37942424436736, 270267896954880, 1925146777223168, 13713023838978048, 97679317251653632, 695780094221746176

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 24 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,8).

Sequences of the form a(n) = c*a(n-1) + d*a(n-2), with a(0)=0, a(1)=1:
c/d...1.......2.......3.......4.......5.......6.......7.......8.......9......10
.A000045,A001045,A006130,A006131,A015440,A015441,A015442,A015443,A015445,A015446
.A000129,A002605,A015518,A063727,A002532,A083099,A015519,A003683,A002534,A083102
.A006190,A007482,A030195,A015521,A015523,A083858,A015524,A015525,A099012,A015528
.A001076,A090017,A015530,A057087,A015531,A085939,A015532,A180222,A015533,A180226
.A052918,A015535,A015536,A015537,A057088,A015540,A015541,A015544,A015545,A180250
.A005668,A135030,A090018,A135032,A015551,A057089,A015552,A189800,A189801,A190005
.A054413,A015555,A015559,A015561,A015562,A015564,A057090,A015565,A015566,A015568
.A041025,A190331,A015574,A190510,A015575,A190560,A015576,A057091,A015577,A190953
.A099371,A015579,A181353,A015580,A015581,A153191,A015583,A015584,A057092,A015585
A041041,A191014,A015588,A190954,A190955,A190956,A015589,A190957,A015591,A057093

I:=[0,1]; [n le 2 select I[n] else 6*Self(n-1)+8*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 14 2011

LinearRecurrence[{6, 8}, {0, 1}, 50]
CoefficientList[Series[-(x/(-1+6 x+8 x^2)),{x,0,50}],x] (* Harvey P. Dale, Jul 26 2011 *)

a(n)=([0,1; 8,6]^n*[0;1])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

G.f.: x/(1 - 2*x*(3+4*x)). - Harvey P. Dale, Jul 26 2011

A218988 Power floor sequence of 2+sqrt(8).

Original entry on oeis.org

4, 19, 91, 439, 2119, 10231, 49399, 238519, 1151671, 5560759, 26849719, 129641911, 625966519, 3022433719, 14593600951, 70464138679, 340230958519, 1642780388791, 7932045389239, 38299303112119, 184925394005431, 892898788470199, 4311296729902519

Offset: 0

Clark Kimberling, Nov 11 2012

See A214992 for a discussion of power floor sequence and the power floor function, p1(x) = limit of a(n,x)/x^n. The present sequence is a(n,r), where r = 2+sqrt(8), and the limit p1(r) = 3.8983688904482395322594950087206...

See A218989 for the power floor function, p4. For comparison with p1, limit(p4(r)/p1(r) = 4/3.

			a(0) = [r] = 4, where r = 2+sqrt(8).
a(1) = [4*r] = 19; a(2) = [19*r] = 91.

Clark Kimberling, Table of n, a(n) for n = 0..250
Index entries for linear recurrences with constant coefficients, signature (5,0,-4).

Cf. A214992, A057087, A086347, A218989.

x = 2 + Sqrt[8]; z = 30; (* z = # terms in sequences *)
f[x_] := Floor[x]; c[x_] := Ceiling[x];
p1[0] = f[x]; p2[0] = f[x]; p3[0] = c[x]; p4[0] = c[x];
p1[n_] := f[x*p1[n - 1]]
p2[n_] := If[Mod[n, 2] == 1, c[x*p2[n - 1]], f[x*p2[n - 1]]]
p3[n_] := If[Mod[n, 2] == 1, f[x*p3[n - 1]], c[x*p3[n - 1]]]
p4[n_] := c[x*p4[n - 1]]
t1 = Table[p1[n], {n, 0, z}]  (* this sequence *)
t2 = Table[p2[n], {n, 0, z}]  (* A057087 *)
t3 = Table[p3[n], {n, 0, z}]  (* A086347 *)
t4 = Table[p4[n], {n, 0, z}]  (* A218989 *)

Vec((4 - x - 4*x^2) / ((1 - x)*(1 - 4*x - 4*x^2)) + O(x^40)) \\ Colin Barker, Nov 13 2017

a(n) = floor(x*a(n-1)), where x=2+sqrt(8), a(0) = floor(x).
a(n) = 5*a(n-1) - 4*a(n-3).
G.f.: (4 - x - 4*x^2) / ((1 - x)*(1 - 4*x - 4*x^2)). [Corrected by Colin Barker, Nov 13 2017]
a(n) = (1/28)*(4 + (54-39*sqrt(2))*(2-2*sqrt(2))^n + (2*(1+sqrt(2)))^n*(54+39*sqrt(2))). - Colin Barker, Nov 13 2017
From Philippe Deléham, Mar 18 2024: (Start)
a(n) = 4*a(n-1) + 4*a(n-2) - 1.
a(n-1) = Sum_{k = 0..n} A370174(n,k)*3^k. (End)

A218989 Power ceiling sequence of 2+sqrt(8).

Original entry on oeis.org

5, 25, 121, 585, 2825, 13641, 65865, 318025, 1535561, 7414345, 35799625, 172855881, 834622025, 4029911625, 19458134601, 93952184905, 453641278025, 2190373851721, 10576060518985, 51065737482825, 246567192007241, 1190531717960265, 5748395639870025

Offset: 0

Clark Kimberling, Nov 11 2012

See A214992 for a discussion of power ceiling sequence and the power ceiling function, p4(x) = limit of a(n,x)/x^n. The present sequence is a(n,r), where r = 2+sqrt(8), and the limit p4(r) = (18 + 13*sqrt(2))/2 = 5.1978251872643193763459933449608678602008191971286...

See A218988 for the power floor function, p1(x); for comparison of p1 and p4, we have limit(p4(r)/p1(r) = 4 - sqrt(7).

			a(0) = ceiling(r) = 5, where r = 2+sqrt(8);
a(1) = ceiling(5*r) = 25; a(2) = ceiling(25*r) = 121.

Clark Kimberling, Table of n, a(n) for n = 0..250
Index entries for linear recurrences with constant coefficients, signature (5,0,-4).

Cf. A214992, A057087, A086347, A218988.

Mathematica
```
(See A218988.)
```

Vec((5 - 4*x^2) / ((1 - x)*(1 - 4*x - 4*x^2)) + O(x^40)) \\ Colin Barker, Nov 13 2017

a(n) = ceiling(x*a(n-1)), where x=2+sqrt(8), a(0) = ceiling(x).
a(n) = 5*a(n-1) - 4*a(n-3).
G.f.: (5 - 4*x^2) / ((1 - x)*(1 - 4*x - 4*x^2)). Corrected by Colin Barker, Nov 13 2017
a(n) = (1/7)*(-1 + (18-13*sqrt(2))*(2-2*sqrt(2))^n + (2*(1+sqrt(2)))^n*(18+13*sqrt(2))). - Colin Barker, Nov 13 2017

A342120 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of g.f. 1/(1 - kx - kx^2).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 6, 3, 0, 1, 4, 12, 16, 5, 0, 1, 5, 20, 45, 44, 8, 0, 1, 6, 30, 96, 171, 120, 13, 0, 1, 7, 42, 175, 464, 648, 328, 21, 0, 1, 8, 56, 288, 1025, 2240, 2457, 896, 34, 0, 1, 9, 72, 441, 1980, 6000, 10816, 9315, 2448, 55, 0

Offset: 0

Seiichi Manyama, Feb 28 2021

			Square array begins:
  1, 1,   1,   1,    1,    1, ...
  0, 1,   2,   3,    4,    5, ...
  0, 2,   6,  12,   20,   30, ...
  0, 3,  16,  45,   96,  175, ...
  0, 5,  44, 171,  464, 1025, ...
  0, 8, 120, 648, 2240, 6000, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Index entries for sequences related to Chebyshev polynomials.

Columns 0..10 give A000007, A000045(n+1), A002605(n+1), A030195(n+1), A057087, A057088, A057089, A057090, A057091, A057092, A057093.
Rows 0..2 give A000012, A001477, A002378.
Main diagonal gives A109516(n+1).
Cf. A342129, A342133, A342134.

T:= (n, k)-> (<<0|1>, >^(n+1))[1, 2]:
seq(seq(T(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Mar 01 2021

T[n_, k_] := Sum[If[k == j == 0, 1, k^j] * Binomial[j, n - j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Apr 28 2021 *)

T(n, k) = sum(j=0, n\2, k^(n-j)*binomial(n-j, j));

T(n, k) = sum(j=0, n, k^j*binomial(j, n-j));

T(n, k) = round((-sqrt(k)*I)^n*polchebyshev(n, 2, sqrt(k)*I/2));

T(0,k) = 1, T(1,k) = k and T(n,k) = k*(T(n-1,k) + T(n-2,k)) for n > 1.
T(n,k) = Sum_{j=0..floor(n/2)} k^(n-j) * binomial(n-j,j) = Sum_{j=0..n} k^j * binomial(j,n-j).
T(n,k) = (-sqrt(k)*i)^n * S(n, sqrt(k)*i) with S(n, x) := U(n, x/2), Chebyshev's polynomials of the 2nd kind.

A015541 Expansion of x/(1 - 5x - 7x^2).

Original entry on oeis.org

0, 1, 5, 32, 195, 1199, 7360, 45193, 277485, 1703776, 10461275, 64232807, 394392960, 2421594449, 14868722965, 91294775968, 560554940595, 3441838134751, 21133075257920, 129758243232857, 796722742969725, 4891921417478624, 30036666288181195

Offset: 0

Olivier Gérard

Pisano period lengths: 1, 3, 8, 6, 8, 24, 6, 6, 24, 24, 5, 24, 12, 6, 8, 12, 16, 24, 120, 24, ... - R. J. Mathar, Aug 10 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,7).

Cf. A001076, A006190, A007482, A015520, A015521, A015523, A015524, A015525, A015528, A015529, A015530, A015531, A015532, A015533, A015534, A015535, A015536, A015537, A015443, A015447, A030195, A053404, A057087, A057088, A083858, A085939, A090017, A091914, A099012, A180222, A180226.

[n le 2 select n-1 else 5*Self(n-1) + 7*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 13 2012

Join[{a=0,b=1},Table[c=5*b+7*a;a=b;b=c,{n,100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 16 2011 *)
LinearRecurrence[{5, 7}, {0, 1}, 30] (* Vincenzo Librandi, Nov 13 2012 *)

x='x+O('x^30); concat([0], Vec(x/(1-5*x-7*x^2))) \\ G. C. Greubel, Jan 24 2018

[lucas_number1(n,5,-7) for n in range(0, 21)] # Zerinvary Lajos, Apr 24 2009

a(n) = 5*a(n-1) + 7*a(n-2).

A015544 Lucas sequence U(5,-8): a(n+1) = 5a(n) + 8a(n-1), a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 5, 33, 205, 1289, 8085, 50737, 318365, 1997721, 12535525, 78659393, 493581165, 3097180969, 19434554165, 121950218577, 765227526205, 4801739379641, 30130517107845, 189066500576353, 1186376639744525, 7444415203333449, 46713089134623445

Offset: 0

Olivier Gérard

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,8).

Cf. A001076, A006190, A007482, A015520, A015521, A015523, A015524, A015525, A015528, A015529, A015530, A015531, A015532, A015533, A015534, A015535, A015536, A015537, A015441, A015443, A015447, A030195, A053404, A057087, A057088, A083858, A085939, A090017, A091914, A099012, A180222, A180226, A015555 (binomial transform).

[n le 2 select n-1 else 5*Self(n-1) + 8*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 13 2012

a[n_]:=(MatrixPower[{{1,2},{1,-6}},n].{{1},{1}})[[2,1]]; Table[Abs[a[n]],{n,-1,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
LinearRecurrence[{5, 8}, {0, 1}, 30] (* Vincenzo Librandi, Nov 13 2012 *)

A015544(n)=imag((2+quadgen(57))^n) \\ M. F. Hasler, Mar 06 2009

x='x+O('x^30); concat([0], Vec(x/(1 - 5*x - 8*x^2))) \\ G. C. Greubel, Jan 01 2018

[lucas_number1(n,5,-8) for n in range(0, 21)] # Zerinvary Lajos, Apr 24 2009

a(n) = 5*a(n-1) + 8*a(n-2).

G.f.: x/(1 - 5*x - 8*x^2). - M. F. Hasler, Mar 06 2009

More precise definition by M. F. Hasler, Mar 06 2009

A110047 Expansion of (1+2x-4x^2)/((2x+1)(2x-1)(4x^2+4x-1)).

Original entry on oeis.org

1, 6, 28, 144, 688, 3360, 16192, 78336, 378112, 1826304, 8817664, 42577920, 205582336, 992649216, 4792926208, 23142334464, 111741042688, 539533639680, 2605098729472, 12578530000896, 60734514921472, 293252181786624, 1415946786832384, 6836795882864640

Offset: 0

Creighton Dement, Jul 10 2005

Note (see program code): ibaseseq[A*B] = A057087, basejseq[A*B] = A099582, tesseq[A*B] = A110046.

Matthew House, Table of n, a(n) for n = 0..1454
Robert Munafo, Sequences Related to Floretions
Index entries for linear recurrences with constant coefficients, signature (4,8,-16,-16).

Cf. A110046, A110048, A110049.

seriestolist(series((1+2*x-4*x^2)/((2*x+1)*(2*x-1)*(4*x^2+4*x-1)), x=0,25)); -or- Floretion Algebra Multiplication Program, FAMP Code: -kbasekseq[A*B] with A = + 'i - .5'j + .5'k - .5j' + .5k' - 'ii' - .5'ij' - .5'ik' - .5'ji' - .5'ki' and B = - .5'i + .5'j + 'k - .5i' + .5j' - 'kk' - .5'ik' - .5'jk' - .5'ki' - .5'kj'

CoefficientList[Series[(1 + 2 x - 4 x^2)/((2 x + 1)(2 x - 1)(4 x^2 + 4 x - 1)), {x, 0, 21}], x] (* or *)
LinearRecurrence[{4, 8, -16, -16}, {1, 6, 28, 144}, 22] (* Michael De Vlieger, Feb 17 2017 *)

Vec((1+2*x-4*x^2) / ((2*x+1)*(2*x-1)*(4*x^2+4*x-1)) + O(x^30)) \\ Colin Barker, Feb 17 2017

a(n) = 4*a(n-1) + 8*a(n-2) - 16*a(n-3) - 16*a(n-4). - Matthew House, Feb 17 2017

a(n) = (-3*(2-2*sqrt(2))^n*(-2+sqrt(2)) + 2^n*(-2*(1+(-1)^n)+3*(1+sqrt(2))^n*(2+sqrt(2)))) / 8. - Colin Barker, Feb 17 2017

Definition corrected by Matthew House, Feb 17 2017

A196468 a(1)=11; a(n) = floor((2 + sqrt(8))*a(n-1)) for n > 1.

Original entry on oeis.org

11, 53, 255, 1231, 5943, 28695, 138551, 668983, 3230135, 15596471, 75306423, 363611575, 1755671991, 8477134263, 40931225015, 197633437111, 954258648503, 4607568342455, 22247307963831, 107419505225143, 518667252755895, 2504347031924151, 12092057138720183

Offset: 1

Philippe Deléham, Oct 02 2011

Contains only odd numbers.

Vincenzo Librandi, Table of n, a(n) for n = 1..150
Index entries for linear recurrences with constant coefficients, signature (5,0,-4).

Cf. A057087, A086843, A086844.

I:=[11,53,255]; [n le 3 select I[n] else 5*Self(n-1)-4*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Oct 05 2011

NestList[Floor[(2+Sqrt[8])#]&,11,30] (* or *) LinearRecurrence[{5,0,-4},{11,53,255},30] (* Harvey P. Dale, Nov 18 2013 *)

G.f.: -x*(-11 + 2*x + 10*x^2) / ( (x-1)*(4*x^2 + 4*x - 1) ).

a(n) = (2 + 33*A057087(n) + 20*A057087(n-1))/14. - R. J. Mathar, Oct 04 2011

cp's OEIS Frontend

A218134 Norm of coefficients in the expansion of 1/(1 - 2*x - i*x^2), where i is the imaginary unit.

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A099582 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k-1)*4^(n-k-1).

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A189800 a(n) = 6*a(n-1) + 8*a(n-2), with a(0)=0, a(1)=1.

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A218988 Power floor sequence of 2+sqrt(8).

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A218989 Power ceiling sequence of 2+sqrt(8).

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A342120 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of g.f. 1/(1 - k*x - k*x^2).

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A015541 Expansion of x/(1 - 5*x - 7*x^2).

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A218134 Norm of coefficients in the expansion of 1/(1 - 2x - ix^2), where i is the imaginary unit.

A189800 a(n) = 6a(n-1) + 8a(n-2), with a(0)=0, a(1)=1.

A342120 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of g.f. 1/(1 - kx - kx^2).

A015541 Expansion of x/(1 - 5x - 7x^2).

A015544 Lucas sequence U(5,-8): a(n+1) = 5a(n) + 8a(n-1), a(0)=0, a(1)=1.

A110047 Expansion of (1+2x-4x^2)/((2x+1)(2x-1)(4x^2+4x-1)).