A081179 -id:A081179 - cp's OEIS frontend

A190958 a(n) = 2a(n-1) - 10a(n-2), with a(0) = 0, a(1) = 1.

0, 1, 2, -6, -32, -4, 312, 664, -1792, -10224, -2528, 97184, 219648, -532544, -3261568, -1197696, 30220288, 72417536, -157367808, -1038910976, -504143872, 9380822016, 23803082752, -46202054656, -330434936832, -198849327104, 2906650714112, 7801794699264

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 24 2011

For the difference equation a(n) = c*a(n-1) - d*a(n-2), with a(0) = 0, a(1) = 1, the solution is a(n) = d^((n-1)/2) * ChebyshevU(n-1, c/(2*sqrt(d))) and has the alternate form a(n) = ( ((c + sqrt(c^2 - 4*d))/2)^n - ((c - sqrt(c^2 - 4*d))/2)^n )/sqrt(c^2 - 4*d). In the case c^2 = 4*d then the solution is a(n) = n*d^((n-1)/2). The generating function is x/(1 - c*x + d^2) and the exponential generating function takes the form (2/sqrt(c^2 - 4*d))*exp(c*x/2)*sinh(sqrt(c^2 - 4*d)*x/2) for c^2 > 4*d, (2/sqrt(4*d - c^2))*exp(c*x/2)*sin(sqrt(4*d - c^2)*x/2) for 4*d > c^2, and x*exp(sqrt(d)*x) if c^2 = 4*d. - G. C. Greubel, Jun 10 2022

Vincenzo Librandi, Table of n, a(n) for n = 0..190
Index entries for linear recurrences with constant coefficients, signature (2,-10).

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
.A128834,A107920,A106852,A106853,A106854,A145934,A145976,A145978,A146078,A146080
.A000027,A009545,A088137,A088138,A045873,A088139,A089181,A090591,A127357,A190958
.A001906,A000225,A057083,A049072,A190959,A190960,A190961,A190962,A190963,A190964
.A001353,A007070,A003462,A001787,A099456,A190965,A168175,A190966,A190967,A190968
.A004254,A107839,A116415,A002450,A030191,A001047,A099450,A190969,A190970,A190971
.A001109,A154244,A138395,A084326,A003463,A030192,A081179,A006516,A027471,A106392
.A004187,A186446,A190972,A190973,A190974,A003464,A030240,A152268,A099459,A016127
.A001090,A190975,A190976,A099156,A190977,A190978,A023000,A057084,A154245,A154235
.A018913,A190979,A190980,A190981,A190982,A190983,A190984,A023001,A057085,A178869
A004189,A190869,A190985,A190986,A190987,A190988,A190989,A190990,A002452,A057086

I:=[0,1]; [n le 2 select I[n] else 2*Self(n-1)-10*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Sep 17 2011

Mathematica
```
LinearRecurrence[{2,-10}, {0,1}, 50]
```

a(n)=([0,1; -10,2]^n*[0;1])[1,1] \\ Charles R Greathouse IV, Apr 08 2016

[lucas_number1(n,2,10) for n in (0..50)] # G. C. Greubel, Jun 10 2022

G.f.: x / ( 1 - 2*x + 10*x^2 ). - R. J. Mathar, Jun 01 2011
E.g.f.: (1/3)*exp(x)*sin(3*x). - Franck Maminirina Ramaharo, Nov 13 2018
a(n) = 10^((n-1)/2) * ChebyshevU(n-1, 1/sqrt(10)). - G. C. Greubel, Jun 10 2022
a(n) = (1/3)*10^(n/2)*sin(n*arctan(3)) = Sum_{k=0..floor(n/2)} (-1)^k*3^(2*k)*binomial(n,2*k+1). - Gerry Martens, Oct 15 2022

A367211 Triangular array read by rows: T(n, k) = binomial(n, k) * A000129(n - k) for 0 <= k < n.

Original entry on oeis.org

1, 2, 2, 5, 6, 3, 12, 20, 12, 4, 29, 60, 50, 20, 5, 70, 174, 180, 100, 30, 6, 169, 490, 609, 420, 175, 42, 7, 408, 1352, 1960, 1624, 840, 280, 56, 8, 985, 3672, 6084, 5880, 3654, 1512, 420, 72, 9, 2378, 9850, 18360, 20280, 14700, 7308, 2520, 600, 90, 10

Offset: 1

Clark Kimberling, Nov 13 2023

T(n, k) are the coefficients of the polynomials p(1, x) = 1, p(2, x) = 2 + 2*x, p(n, x) = u*p(n-1, x) + v*p(n-2, x) for n >= 3, where u = p(2, x), v = 1 - 2*x - x^2.

Because (p(n, x)) is a strong divisibility sequence, for each integer k, the sequence (p(n, k)) is a strong divisibility sequence of integers.

			First nine rows:
  [n\k] 0     1     2     3     4     5    6   7  8
  [1]   1;
  [2]   2     2;
  [3]   5     6    3;
  [4]  12    20    12     4;
  [5]  29    60    50    20     5;
  [6]  70   174   180   100    30     6;
  [7] 169   490   609   420   175    42   7;
  [8] 408  1352  1960  1624   840   280   56   8;
  [9] 985  3672  6084  5880  3654  1512  420  72  9;
.
Row 4 represents the polynomial p(4,x) = 12 + 20 x + 12 x^2 + 4 x^3, so that (T(4,k)) = (12, 20, 12, 4), k = 0..3.

Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), Paper No. A14.

Cf. A000129 (column 1, Pell numbers), A361732 (column 2), A000027 (T(n,n-1)), A007070 (row sums, p(n,1)), A077957 (alternating row sums, p(n,-1)), A081179 (p(n,2)), A077985 (p(n,-2)), A081180 (p(n,3)), A007070 (p(n,-3)), A081182 (p(n,4)), A094440, A367208, A367209, A367210.

P := proc(n) option remember; ifelse(n <= 1, n, 2*P(n - 1) + P(n - 2)) end:
T := (n, k) -> P(n - k) * binomial(n, k):
for n from 1 to 9 do [n], seq(T(n, k), k = 0..n-1) od;
# (after Werner Schulte)  Peter Luschny, Nov 24 2023

p[1, x_] := 1; p[2, x_] := 2 + 2 x; u[x_] := p[2, x]; v[x_] := 1 - 2 x - x^2;
p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
(* Or: *)
T[n_, k_] := Module[{P},
  P[m_] := P[m] = If[m <= 1, m, 2*P[m - 1] + P[m - 2]];
  P[n - k] * Binomial[n, k] ];
Table[T[n, k], {n, 1, 9}, {k, 0, n - 1}]  (* Peter Luschny, Mar 07 2025 *)

p(n, x) = u*p(n-1, x) + v*p(n-2, x) for n >= 3, where p(1, x) = 1, p(2, x) = 2 + 2*x, u = p(2, x), and v = 1 - 2*x - x^2.
p(n, x) = k*(b^n - c^n), where k = sqrt(1/8), b = x + 1 - sqrt(2), c = x + 1 + sqrt(2).
From Werner Schulte, Nov 24 2023 and Nov 25 2023: (Start)
The row polynomials p(n, x) = Sum_{k=0..n-1} T(n, k) * x^k satisfy the equation p'(n, x) = n * p(n-1, x) where p' is the first derivative of p.
T(n, k) = T(n-1, k-1) * n / k for 0 < k < n and T(n, 0) = A000129(n) for n > 0.
T(n, k) = A000129(n-k) * binomial(n, k) for 0 <= k < n.
G.f.: t / (1 - (2+2*x) * t - (1-2*x-x^2) * t^2). (End)

New name using a formula of Werner Schulte by Peter Luschny, Mar 07 2025

A081180 4th binomial transform of (0,1,0,2,0,4,0,8,0,16,...).

Original entry on oeis.org

0, 1, 8, 50, 288, 1604, 8800, 47944, 260352, 1411600, 7647872, 41420576, 224294400, 1214467136, 6575615488, 35602384000, 192760455168, 1043650265344, 5650555750400, 30593342288384, 165638957801472, 896804870374400

Offset: 0

Paul Barry, Mar 11 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..300
S. Falcon, Iterated Binomial Transforms of the k-Fibonacci Sequence, British Journal of Mathematics & Computer Science, 4 (22): 2014.
Index entries for linear recurrences with constant coefficients, signature (8,-14).

Binomial transform of A081179.

Cf. A081182.

I:=[0, 1]; [n le 2 select I[n] else 8*Self(n-1)-14*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Aug 06 2013

Join[{a=0,b=1},Table[c=8*b-14*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 19 2011 *)
CoefficientList[Series[x / (1 - 8 x + 14 x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 06 2013 *)
LinearRecurrence[{8,-14},{0,1},30] (* Harvey P. Dale, Aug 17 2019 *)

[lucas_number1(n,8,14) for n in range(0, 22)] # Zerinvary Lajos, Apr 23 2009

a(n) = 8a(n-1) - 14a(n-2), a(0)=0, a(1)=1.
G.f.: x/(1 - 8x + 14x^2).
a(n) = ((4 + sqrt(2))^n - (4 - sqrt(2))^n)/(2*sqrt(2)).
a(n) = Sum_{k=0..n} C(n,2k+1) 2^k*4^(n-2k-1).
If shifted once left, fourth binomial transform of A143095. - Al Hakanson (hawkuu(AT)gmail.com), Jul 25 2009, R. J. Mathar, Oct 15 2009
E.g.f.: exp(4*x)*sinh(sqrt(2)*x)/sqrt(2). - Ilya Gutkovskiy, Aug 12 2017

Modified the completing comment on the fourth binomial transform - R. J. Mathar, Oct 15 2009

A083878 a(0)=1, a(1)=3, a(n) = 6a(n-1) - 7a(n-2), n >= 2.

Original entry on oeis.org

1, 3, 11, 45, 193, 843, 3707, 16341, 72097, 318195, 1404491, 6199581, 27366049, 120799227, 533233019, 2353803525, 10390190017, 45864515427, 202455762443, 893682966669, 3944907462913, 17413664010795, 76867631824379

Offset: 0

Paul Barry, May 08 2003

Binomial transform of A006012. Second binomial transform of A001333.

Third binomial transform of A077957. Inverse binomial transform of A083879. - Philippe Deléham, Dec 01 2008

Michael De Vlieger, Table of n, a(n) for n = 0..1551
Yassine Otmani, The 2-Pascal Triangle and a Related Riordan Array, J. Int. Seq. (2025) Vol. 28, Issue 3, Art. No. 25.3.5. See p. 12.
Index entries for linear recurrences with constant coefficients, signature (6,-7).

Cf. A083879, A056241, A081179, A098158.

f[n_] := Simplify[(3 + Sqrt@2)^n + (3 - Sqrt@2)^n]/2; Array[f, 23, 0] (* Robert G. Wilson v, Oct 31 2010 *)

a(n) = ((3 - sqrt(2))^n + (3 + sqrt(2))^n)/2;
a(n) = Sum_{k=0..n} C(n, 2k)*3^(n-2k)*2^k;
G.f.: (1-3x)/(1-6x+7x^2);
E.g.f.: exp(3x)*cosh(x*sqrt(2)).
a(n) = Sum_{k=0..n} C(n, k)*2^((n-k)/2)(1+(-1)^(n-k))*3^k/2. - Paul Barry, Jan 22 2005
a(n) = Sum_{k=0..n} A098158(n,k)*3^(2k-n)*2^(n-k). - Philippe Deléham, Dec 01 2008
a(n) = A081179(n+1) - 3*A081179(n). - R. J. Mathar, Nov 10 2013
a(n) = Sum_{k=1..n} A056241(n, k) * 2^(k-1). - J. Conrad, Nov 23 2022

A090018 a(n) = 6a(n-1) + 3a(n-2) for n > 2, a(0)=1, a(1)=6.

Original entry on oeis.org

1, 6, 39, 252, 1629, 10530, 68067, 439992, 2844153, 18384894, 118841823, 768205620, 4965759189, 32099171994, 207492309531, 1341251373168, 8669985167601, 56043665125110, 362271946253463, 2341762672896108, 15137391876137037, 97849639275510546, 632510011281474387

Offset: 0

Paul Barry, Nov 19 2003

From Johannes W. Meijer, Aug 09 2010: (Start)

a(n) represents the number of n-move routes of a fairy chess piece starting in a given corner or side square on a 3 X 3 chessboard. This fairy chess piece behaves like a white queen on the eight side and corner squares but on the central square the queen explodes with fury and turns into a red queen, see A180032. The central square leads to A180028. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (6,3).

Cf. A141041, A180028, A180032.

Sequences with g.f. of the form 1/(1 - 6*x - k*x^2): A106392 (k=-10), A027471 (k=-9), A006516 (k=-8), A081179 (k=-7), A030192 (k=-6), A003463 (k=-5), A084326 (k=-4), A138395 (k=-3), A154244 (k=-2), A001109 (k=-1), A000400 (k=0), A005668 (k=1), A135030 (k=2), this sequence (k=3), A135032 (k=4), A015551 (k=5), A057089 (k=6), A015552 (k=7), A189800 (k=8), A189801 (k=9), A190005 (k=10), A015553 (k=11).

[n le 2 select 6^(n-1) else 6*Self(n-1)+3*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 15 2011

a:= n-> (<<0|1>, <3|6>>^n. <<1,6>>)[1,1]:
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 17 2011

Join[{a=1,b=6},Table[c=6*b+3*a;a=b;b=c,{n,100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 16 2011 *)
LinearRecurrence[{6,3}, {1,6}, 41] (* G. C. Greubel, Oct 10 2022 *)

my(x='x+O('x^30)); Vec(1/(1-6*x-3*x^2)) \\ G. C. Greubel, Jan 24 2018

[lucas_number1(n,6,-3) for n in range(1, 31)] # Zerinvary Lajos, Apr 24 2009

a(n) = (3+2*sqrt(3))^n*(sqrt(3)/4+1/2) + (1/2-sqrt(3)/4)*(3-2*sqrt(3))^n.
a(n) = (-i*sqrt(3))^n * ChebyshevU(n, isqrt(3)), i^2=-1.
From Johannes W. Meijer, Aug 09 2010: (Start)
G.f.: 1/(1 - 6*x - 3*x^2).
Limit_{k->oo} a(n+k)/a(k) = A141041(n) + A090018(n-1)*sqrt(12) for n >= 1.
Limit_{n->oo} A141041(n)/A090018(n-1) = sqrt(12). (End)
a(n) = Sum_{k=0..n} A099089(n,k)*3^k. - Philippe Deléham, Nov 21 2011
E.g.f.: exp(3*x)*(2*cosh(2*sqrt(3)*x) + sqrt(3)*sinh(2*sqrt(3)*x))/2. - Stefano Spezia, Apr 23 2025

Typo in Mathematica program corrected by Vincenzo Librandi, Nov 15 2011

A164299 a(n) = ((1+4sqrt(2))(3+sqrt(2))^n + (1-4sqrt(2))(3-sqrt(2))^n)/2.

Original entry on oeis.org

1, 11, 59, 277, 1249, 5555, 24587, 108637, 479713, 2117819, 9348923, 41268805, 182170369, 804140579, 3549650891, 15668921293, 69165971521, 305313380075, 1347718479803, 5949117218293, 26260673951137, 115920223178771

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Aug 12 2009

nonn

Binomial transform of A164298. Third binomial transform of A164587. Inverse binomial transform of A164300.

This sequence is part of a class of sequences defined by the recurrence a(n,m) = 2*(m+1)*a(n-1,m) - ((m+1)^2 - 2)*a(n-2,m) with a(0) = 1 and a(1) = m+9. The generating function is Sum_{n>=0} a(n,m)*x^n = (1 - (m-7)*x)/(1 - 2*(m+1)*x + ((m+1)^2 - 2)*x^2) and has a series expansion in terms of Pell-Lucas numbers defined by a(n, m) = (1/2)*Sum_{k=0..n} binomial(n,k)*m^(n-k)*(5*Q(k) + 4*Q(k-1)). - G. C. Greubel, Mar 12 2021

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-7).

Sequences in the class a(n, m): A164298 (m=1), this sequence (m=2), A164300 (m=3), A164301 (m=4), A164598 (m=5), A164599 (m=6), A081185 (m=7), A164600 (m=8).

Cf. A081179, A083878, A164587.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((1+4*r)*(3+r)^n+(1-4*r)*(3-r)^n)/2: n in [0..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 17 2009

LinearRecurrence[{6,-7}, {1,11}, 50] (* or *) CoefficientList[Series[(1 + 5*x)/(1 - 6*x + 7*x^2), {x,0,50}], x] (* G. C. Greubel, Sep 12 2017 *)

my(x='x+O('x^50)); Vec((1+5*x)/(1-6*x+7*x^2)) \\ G. C. Greubel, Sep 12 2017

[( (1+5*x)/(1-6*x+7*x^2) ).series(x,n+1).list()[n] for n in (0..30)] # G. C. Greubel, Mar 12 2021

a(n) = 6*a(n-1) - 7*a(n-2) for n > 1; a(0) = 1, a(1) = 11.
G.f.: (1+5*x)/(1-6*x+7*x^2).
E.g.f.: (cosh(sqrt(2)*x) + 4*sqrt(2)*sinh(sqrt(2)*x))*exp(3*x). - G. C. Greubel, Sep 12 2017
From G. C. Greubel, Mar 12 2021: (Start)
a(n) = 2*A083878(n) + 8*A081179(n).
a(n) = (1/2)*Sum_{k=0..n} binomial(n,k)*2^(n-k)*(5*Q(k) + 4*Q(k-1)), where Q(n) = Pell-Lucas(n) = A002203(n). (End)

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 17 2009

A086351 T(n,3) of A086350.

Original entry on oeis.org

1, 4, 17, 74, 325, 1432, 6317, 27878, 123049, 543148, 2397545, 10583234, 46716589, 206216896, 910285253, 4018193246, 17737162705, 78295623508, 345613602113, 1525612248122, 6734378273941, 29726983906792, 131221255523165

Offset: 0

Paul Barry, Jul 18 2003

Binomial transform of A007052. Second binomial transform of Pell numbers A000129 (without leading zero).

Index entries for linear recurrences with constant coefficients, signature (6,-7).

a(n)=my(sqrt2=quadgen(8)); simplify(((1+sqrt2)*(3+sqrt2)^n-(1-sqrt2)*(3-sqrt2)^n)/sqrt2^3) \\ Charles R Greathouse IV, Oct 24 2014

G.f.: (1-2x)/(1-6x+7x^2);
a(n)=((1+sqrt(2))(3+sqrt(2))^n-(1-sqrt(2))(3-sqrt(2))^n)/(sqrt(8)).
a(n) = A081179(n+1)-2*A081179(n). - R. J. Mathar, Dec 05 2022

A102285 G.f. (1-x)/(7x^2-6x+1).

Original entry on oeis.org

1, 5, 23, 103, 457, 2021, 8927, 39415, 174001, 768101, 3390599, 14966887, 66067129, 291634565, 1287337487, 5682582967, 25084135393, 110726731589, 488771441783, 2157541529575, 9523849084969, 42040303802789

Offset: 0

Creighton Dement, Feb 19 2005

A floretion-generated sequence relating to the second binomial transform of Pell numbers A000129.

Floretion Algebra Multiplication Program, FAMP Code: (a(n)) = jesforseq[ + .5'i + .5i' + 2'jj' + .5'ij' + .5'ji' ]; A000004 = vesforseq.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-7).

Cf. A086351, A027649, A007070 (inverse binomial transform), A081179, A163350 (binomial transform).

[Floor(((1+Sqrt(2))*(3+Sqrt(2))^n+(1-Sqrt(2))*(3-Sqrt(2))^n)/2): n in [0..30]]; // Vincenzo Librandi, Oct 12 2011

CoefficientList[Series[(1-x)/(7x^2-6x+1),{x,0,30}],x] (* or *) LinearRecurrence[{6,-7},{1,5},30] (* Harvey P. Dale, Dec 10 2017 *)

a(n) = A086351(n+1) - 3*A086351(n) (FAMP result); Inversion gives A027649 (SuperSeeker result); Inverse binomial transform of A007070 (SuperSeeker result);
From Al Hakanson (hawkuu(AT)gmail.com), Jul 25 2009: (Start)
a(n) = ((1+sqrt(2))*(3+sqrt(2))^n + (1-sqrt(2))*(3-sqrt(2))^n)/2 offset 0.
Third binomial transform of 1,2,2,4,4. (End)
a(n) = 6*a(n-1) - 7*a(n-2) for n > 1; a(0)=1, a(1)=5. - Philippe Deléham, Sep 19 2009
a(n) = A081179(n) + A086351(n). - Joseph M. Shunia, Sep 09 2019
a(n) = A081179(n+1)-A081179(n). - R. J. Mathar, Sep 11 2019

A164073 a(n) = 2*a(n-2) for n > 2; a(1) = 1, a(2) = 3.

Original entry on oeis.org

1, 3, 2, 6, 4, 12, 8, 24, 16, 48, 32, 96, 64, 192, 128, 384, 256, 768, 512, 1536, 1024, 3072, 2048, 6144, 4096, 12288, 8192, 24576, 16384, 49152, 32768, 98304, 65536, 196608, 131072, 393216, 262144, 786432, 524288, 1572864, 1048576, 3145728, 2097152, 6291456

Offset: 1

Klaus Brockhaus, Aug 09 2009

Interleaving of A000079 and A007283.
Binomial transform is A048654. Second binomial transform is A111567. Third binomial transform is A081179 without initial 0. Fourth binomial transform is A164072. Fifth binomial transform is A164031.
Absolute second differences are the sequence itself. - Eric Angelini, Jul 30 2013
Least number having n - 1 Gaussian prime factors, counted with multiplicity, excluding units. See A239628 for a similar sequence. - T. D. Noe, Mar 31 2014
Writing the prime factorizations of the terms of this sequence, the exponents of prime factor 2 give the integers repeated (A004526), while the exponents of prime factor 3 give the sequence of alternating 0's and 1's (A000035). - Alonso del Arte, Nov 30 2016

Vincenzo Librandi, Table of n, a(n) for n = 1..2000
Index entries for linear recurrences with constant coefficients, signature (0, 2).

Cf. A000079 (powers of 2), A007283 (3*2^n), A074323, A162255, A048654, A111567, A081179, A164072, A164031, A239628.

[ n le 2 select 2*n-1 else 2*Self(n-2): n in [1..42] ];

terms = 50; CoefficientList[Series[x * (1 + 3 * x)/(1 - 2 * x^2), {x, 0, terms}], x] (* T. D. Noe, Mar 31 2014 *)
Flatten[Table[{2^n, 3 * 2^n}, {n, 0, 31}]] (* Alonso del Arte, Nov 30 2016 *)
CoefficientList[Series[x (1 + 3 x)/(1 - 2 x^2), {x, 0, 44}], x] (* Michael De Vlieger, Dec 13 2016 *)

PARI
```
a(n) = (5 + (-1)^n) * 2^((2*n-9)\/4)
```

Vec(x*(1+3*x)/(1-2*x^2)+O(x^99)) \\ Charles R Greathouse IV, Dec 13 2016

a(n) = (5 + (-1)^n) * 2^(1/4 * (2*n - 1 + (-1)^n))/4.
G.f.: x*(1 + 3 * x)/(1 - 2 * x^2).
a(n) = A074323(n), n>=1.
a(n) = A162255(n-1), n>=2.
a(n) = A072946(n-2), n > 2. - R. J. Mathar, Aug 17 2009
a(n+3) = a(n + 2) * a(n + 1)/a(n). - Reinhard Zumkeller, Mar 04 2011
a(n) = (2/3)a(n - 1) for odd n > 1, a(n) = 3a(n - 1) for even n. - Alonso del Arte, Nov 30 2016

A153593 a(n) = ((9 + sqrt(2))^n - (9 - sqrt(2))^n)/(2*sqrt(2)).

Original entry on oeis.org

1, 18, 245, 2988, 34429, 383670, 4186169, 45041112, 480032665, 5082340122, 53559541661, 562566880260, 5895000053461, 61667217421758, 644304909368225, 6725778192309168, 70163919621475249, 731614075994130210

Offset: 1

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

nonn

Preceded by zero, this is the eighth binomial transform of the Pell sequence A000129. - Sergio Falcon, Sep 21 2009; edited by Klaus Brockhaus, Oct 11 2009
Eighth binomial transform of A048697.
First differences are in A164600.
lim_{n -> infinity} a(n)/a(n-1) = 9 + sqrt(2) = 10.4142135623....

Vincenzo Librandi, Table of n, a(n) for n = 1..100
S. Falcon, Iterated Binomial Transforms of the k-Fibonacci Sequence, British Journal of Mathematics & Computer Science, 4 (22): 2014.
Index entries for linear recurrences with constant coefficients, signature (18,-79).

Cf. A000129 (Pell numbers), A007070, A081185, A081184, A081183, A081182, A081180, A081179. - Sergio Falcon, Sep 21 2009

Cf. A002193 (decimal expansion of sqrt(2)), A048697, A164600.

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((9+r)^n-(9-r)^n)/(2*r): n in [1..18] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Dec 31 2008

Join[{a=1,b=18},Table[c=18*b-79*a;a=b;b=c,{n,40}]] (* Vladimir Joseph Stephan Orlovsky, Feb 09 2011 *)
LinearRecurrence[{18,-79},{1,18},25] (* G. C. Greubel, Aug 22 2016 *)

a(n) = 18*a(n-1) - 79*a(n-2) for n>1; a(0)=0, a(1)=1. - Philippe Deléham, Jan 01 2009
G.f.: x/(1 - 18*x + 79*x^2). - Klaus Brockhaus, Dec 31 2008, corrected Oct 11 2009
a(n) = Sum[Binomial[n - 1 - i, i] (-1)^i * 18^(n - 1 - 2 i) * 79^i, {i, 0, Floor[(n - 1)/2]}]. - Sergio Falcon, Sep 21 2009
E.g.f.: exp(9*x)*sinh(sqrt(2)*x)/sqrt(2). - Ilya Gutkovskiy, Aug 12 2017

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

Edited by Klaus Brockhaus, Oct 11 2009

cp's OEIS Frontend

A190958 a(n) = 2*a(n-1) - 10*a(n-2), with a(0) = 0, a(1) = 1.

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A367211 Triangular array read by rows: T(n, k) = binomial(n, k) * A000129(n - k) for 0 <= k < n.

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A081180 4th binomial transform of (0,1,0,2,0,4,0,8,0,16,...).

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A083878 a(0)=1, a(1)=3, a(n) = 6*a(n-1) - 7*a(n-2), n >= 2.

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A090018 a(n) = 6*a(n-1) + 3*a(n-2) for n > 2, a(0)=1, a(1)=6.

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A164299 a(n) = ((1+4*sqrt(2))*(3+sqrt(2))^n + (1-4*sqrt(2))*(3-sqrt(2))^n)/2.

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A086351 T(n,3) of A086350.

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A190958 a(n) = 2a(n-1) - 10a(n-2), with a(0) = 0, a(1) = 1.

A083878 a(0)=1, a(1)=3, a(n) = 6a(n-1) - 7a(n-2), n >= 2.

A090018 a(n) = 6a(n-1) + 3a(n-2) for n > 2, a(0)=1, a(1)=6.

A164299 a(n) = ((1+4sqrt(2))(3+sqrt(2))^n + (1-4sqrt(2))(3-sqrt(2))^n)/2.

A102285 G.f. (1-x)/(7x^2-6x+1).