A162255 -id:A162255 - cp's OEIS frontend

A164053 Partial sums of A162255.

3, 5, 11, 15, 27, 35, 59, 75, 123, 155, 251, 315, 507, 635, 1019, 1275, 2043, 2555, 4091, 5115, 8187, 10235, 16379, 20475, 32763, 40955, 65531, 81915, 131067, 163835, 262139, 327675, 524283, 655355, 1048571, 1310715, 2097147, 2621435, 4194299

Offset: 1

Klaus Brockhaus, Aug 08 2009

nonn

Apparently a(n) = A094958(n+4)-5.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1, 2, -2).

Cf. A162255, A094958.

T:=[ n le 2 select 4-n else 2*Self(n-2): n in [1..39] ]; [ n eq 1 select T[1] else Self(n-1)+T[n]: n in [1..#T]];

Accumulate[LinearRecurrence[{0,2},{3,2},50]] (* or *) LinearRecurrence[ {1,2,-2},{3,5,11},50] (* Harvey P. Dale, Aug 28 2012 *)

x='x+O('x^50); Vec(x*(3+2*x)/(1-x-2*x^2+2*x^3)) \\ G. C. Greubel, Sep 09 2017

a(n) = 2*a(n-2) + 5 for n > 2; a(1) = 3, a(2) = 5.
a(n) = (13 - 3*(-1)^n)*2^(1/4*(2*n -1 +(-1)^n))/2 - 5.
G.f.: x*(3+2*x)/(1-x-2*x^2+2*x^3).
a(1)=3, a(2)=5, a(3)=11, a(n)=a(n-1)+2*a(n-2)-2*a(n-3). - Harvey P. Dale, Aug 28 2012

A078343 a(0) = -1, a(1) = 2; a(n) = 2*a(n-1) + a(n-2).

Original entry on oeis.org

-1, 2, 3, 8, 19, 46, 111, 268, 647, 1562, 3771, 9104, 21979, 53062, 128103, 309268, 746639, 1802546, 4351731, 10506008, 25363747, 61233502, 147830751, 356895004, 861620759, 2080136522, 5021893803, 12123924128, 29269742059, 70663408246, 170596558551, 411856525348

Offset: 0

Benoit Cloitre, Nov 22 2002

Inverse binomial transform of -1, 1, 6, 22, 76, 260, ... (see A111566). Binomial transform of -1, 3, -2, 6, -4, 12, -8, 24, -16, ... (see A162255). - R. J. Mathar, Oct 02 2012

			G.f. = -1 + 2*x + 3*x^2 + 8*x^3 + 19*x^4 + 46*x^5 + 111*x^6 + ... - _Michael Somos_, Jun 30 2022

H. S. M. Coxeter, 1998, Numerical distances among the circles in a loxodromic sequence, Nieuw Arch. Wisk, 16, pp. 1-9.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
H. S. M. Coxeter, Numerical distances among the spheres in a loxodromic sequence, Math. Intell. 19(4) 1997 pp. 41-47. See page 41. See pp. 46-47.
Tanya Khovanova, Recursive Sequences.
José L. Ramírez, Gustavo N. Rubiano, and Rodrigo de Castro, A Generalization of the Fibonacci Word Fractal and the Fibonacci Snowflake, arXiv:1212.1368 [cs.DM], 2012.
Index entries for linear recurrences with constant coefficients, signature (2,1).

Cf. A000129, A048654, A048655, A048746, A111566, A162255, A221172, A221173, A221174, A221175.

a078343 n = a078343_list !! n
a078343_list = -1 : 2 : zipWith (+)
                        (map (* 2) $ tail a078343_list) a078343_list
-- Reinhard Zumkeller, Jan 04 2013

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-4*x)/(-1+2*x+x^2))); // G. C. Greubel, Jul 26 2018

f:=proc(n) option remember; if n=0 then RETURN(-1); fi; if n=1 then RETURN(2); fi; 2*f(n-1)+f(n-2); end;

Table[4 Fibonacci[n, 2] - Fibonacci[n + 1, 2], {n, 0, 30}] (* Vladimir Reshetnikov, Sep 27 2016 *)
LinearRecurrence[{2,1},{-1,2},40] (* Harvey P. Dale, Apr 15 2019 *)

a(n)=([0,1;1,2]^n*[-1;2])[1,1] \\ Charles R Greathouse IV, Jun 11 2015

For the unsigned version: a(1)=1; a(2)=2; a(n) = Sum_{k=2..n-1} (a(k) + a(k-1)).
a(n) is asymptotic to (1/4)*(-2+3*sqrt(2))*(1+sqrt(2))^n.
a(n) = A048746(n-3) + 2, for n > 2. - Ralf Stephan, Oct 17 2003
a(n) = 2*A000129(n) - A000129(n-1) if n > 0; abs(a(n)) = Sum_{k=0..floor(n/2)} (C(n-k-1, k) - C(n-k-1, k-1))2^(n-2k). - Paul Barry, Dec 23 2004
O.g.f.: (1-4*x)/(-1 + 2*x + x^2). - R. J. Mathar, Feb 15 2008
a(n) = 4*Pell(n) - Pell(n+1), where Pell = A000129. - Vladimir Reshetnikov, Sep 27 2016
a(n) = -(-1)^n * A048654(-n) = ( (-2+3*sqrt(2))*(1+sqrt(2))^n + (-2-3*sqrt(2))*(1-sqrt(2))^n )/4 for all n in Z. - Michael Somos, Jun 30 2022
2*a(n+1)^2 = A048655(n)^2 + (-1)^n*7. - Philippe Deléham, Mar 07 2023
E.g.f.: 3*exp(x)*sinh(sqrt(2)*x)/sqrt(2) - exp(x)*cosh(sqrt(2)*x). - Stefano Spezia, May 26 2024

Entry revised by N. J. A. Sloane, Apr 29 2004

A162396 a(n) = 2*a(n-2) for n > 2; a(1) = 5, a(2) = 2.

Original entry on oeis.org

5, 2, 10, 4, 20, 8, 40, 16, 80, 32, 160, 64, 320, 128, 640, 256, 1280, 512, 2560, 1024, 5120, 2048, 10240, 4096, 20480, 8192, 40960, 16384, 81920, 32768, 163840, 65536, 327680, 131072, 655360, 262144, 1310720, 524288, 2621440, 1048576, 5242880

Offset: 1

Klaus Brockhaus, Jul 02 2009

Binomial transform is A162268. Fifth binomial transform is A083880 without initial 1.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,2).

Cf. A083880, A162255, A162268.

[ n le 2 select 8-3*n else 2*Self(n-2): n in [1..41] ];

[Floor((3/2-(-1)^n)*2^(1/4*(2*n+3+(-1)^n))):  n in [1..50]]; // Vincenzo Librandi, Oct 09 2017

A162396:=n->(3/2-(-1)^n)*2^(1/4*(2*n+3+(-1)^n)): seq(A162396(n), n=1..60); # Wesley Ivan Hurt, Oct 08 2017

CoefficientList[Series[(5 + 2*x)/(1 - 2*x^2), {x, 0, 40}], x] (* Wesley Ivan Hurt, Oct 08 2017 *)
RecurrenceTable[{a[1]==5, a[2]==2, a[n]==2 a[n-2]}, a, {n, 40}] (* Vincenzo Librandi, Oct 09 2017 *)

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

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

G.f. corrected, formula simplified, comment added by Klaus Brockhaus, Sep 18 2009

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

A164682 a(n) = 2*a(n-2) for n > 2; a(1) = 5, a(2) = 8.

Original entry on oeis.org

5, 8, 10, 16, 20, 32, 40, 64, 80, 128, 160, 256, 320, 512, 640, 1024, 1280, 2048, 2560, 4096, 5120, 8192, 10240, 16384, 20480, 32768, 40960, 65536, 81920, 131072, 163840, 262144, 327680, 524288, 655360, 1048576, 1310720, 2097152, 2621440, 4194304

Offset: 1

Klaus Brockhaus, Aug 21 2009

Interleaving of A020714 and A000079 without initial terms 1, 2, 4.
First differences are in A162255.
Binomial transform is A135532 without initial terms -1, 3. Fourth binomial transform is A164537.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0, 2).

Equals A094958 (numbers of the form 2^n or 5*2^n) without initial terms 1, 2, 4.

Cf. A020714 (5*2^n), A000079 (powers of 2), A162255, A135532, A164537.

[ n le 2 select 2+3*n else 2*Self(n-2): n in [1..40] ];

LinearRecurrence[{0,2},{5,8},60] (* Harvey P. Dale, Jul 20 2022 *)

a(n) = (9-(-1)^n)*2^(1/4*(2*n-5+(-1)^n)).

G.f.: x*(5+8*x)/(1-2*x^2).

A072946 Coefficient of the highest power of q in the expansion of nu(0)=1, nu(1)=b and for n>=2, nu(n)=bnu(n-1)+lambda(n-1)_q*nu(n-2) with (b,lambda)=(2,2), where (n)_q=(1+q+...+q^(n-1)) and q is a root of unity.

Original entry on oeis.org

1, 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

Offset: 0

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

Instead of listing the coefficients of the highest power of q in each nu(n), if we listed 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)=2f(n-1)+2f(n-2).

			nu(0)=1,
nu(1)=2,
nu(2)=6,
nu(3)=16+4q,
nu(4)=44+20q+12q^2,
nu(5)=120+80q+64q^2+40q^3+8q^4,
nu(6)=328+288q+280q^2+232q^3+168q^4+64q^5+24q^6.
By listing the coefficients of the highest power in each nu(n) we get 1,2,6,4,12,8,24,...

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, 2).

Essentially the same as A162255 and A164073.

Cf. A002605.

LinearRecurrence[{0,2},{1,2,6},50] (* Harvey P. Dale, Dec 31 2015 *)

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).

O.g.f.: (1+2*x+4*x^2)/(1-2*x^2). - R. J. Mathar, Dec 05 2007

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

A161938 a(n) = ((3+sqrt(2))(2+sqrt(2))^n + (3-sqrt(2))(2-sqrt(2))^n)/2.

Original entry on oeis.org

3, 8, 26, 88, 300, 1024, 3496, 11936, 40752, 139136, 475040, 1621888, 5537472, 18906112, 64549504, 220385792, 752444160, 2569005056, 8771131904, 29946517504, 102243806208, 349082189824, 1191841146880, 4069200207872, 13893118537728, 47434073735168

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Jun 22 2009, Jun 29 2009

Second binomial transform of A162255.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-2).

Cf. A162255, A135532, A083878.

a:=[3,8];; for n in [3..25] do a[n]:=4*a[n-1]-2*a[n-2]; od; a; # Muniru A Asiru, Sep 28 2018

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((3+r)*(2+r)^n+(3-r)*(2-r)^n)/2: n in [0..23] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 01 2009

I:=[3,8]; [n le 2 select I[n] else 4*Self(n-1) - 2*Self(n-2): n in [1..30]]; // G. C. Greubel, Sep 28 2018

seq(coeff(series((3-4*x)/(1-4*x+2*x^2),x,n+1), x, n), n = 0..25); # Muniru A Asiru, Sep 28 2018

CoefficientList[Series[(3-4*x)/(1-4*x+2*x^2), {x,0,50}], x] (* G. C. Greubel, Sep 28 2018 *)

my(x='x+O('x^50)); Vec((3-4*x)/(1-4*x+2*x^2)) \\ G. C. Greubel, Sep 28 2018

a(n) = 4*a(n-1) - 2*a(n-2) for n>1; a(0) = 3; a(1) = 8.
G.f.: (3-4*x)/(1-4*x+2*x^2).
From G. C. Greubel, Sep 28 2018: (Start)
a(2*n) = 2^(n-1) * (Q(2*n +1) + 2*Q(2*n)), Q(m) = Pell-Lucas numbers = A002203(m).
a(2*n+1) = 2^(n-1) * (P(2*n+2) + 2*P(2*n+1)), P(m) = Pell numbers = A000129(m). (End)

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 01 2009

A161939 a(n) = ((3+sqrt(2))(4+sqrt(2))^n + (3-sqrt(2))(4-sqrt(2))^n)/2.

Original entry on oeis.org

3, 14, 70, 364, 1932, 10360, 55832, 301616, 1631280, 8827616, 47783008, 258677440, 1400457408, 7582175104, 41050997120, 222257525504, 1203346244352, 6515164597760, 35274469361152, 190983450520576, 1034025033108480

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Jun 22 2009

nonn

Fourth binomial transform of A162255.

Muniru A Asiru, Table of n, a(n) for n = 0..280
Index entries for linear recurrences with constant coefficients, signature (8, -14).

Cf. A162255, A161940 (Fifth binomial transform of A162255).

a := [3, 14];; for n in [3..10^2] do a[n] := 8*a[n-1] - 14*a[n-2]; od; a; # Muniru A Asiru, Feb 02 2018

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((3+r)*(4+r)^n+(3-r)*(4-r)^n)/2: n in [0..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 01 2009

seq(simplify(((3+sqrt(2))*(4+sqrt(2))^n+(3-sqrt(2))*(4-sqrt(2))^n)*1/2), n = 0 .. 20); # Emeric Deutsch, Jun 28 2009

LinearRecurrence[{8,-14},{3,14},30] (* Harvey P. Dale, May 10 2012 *)

x='x+O('x^30); Vec((3-10*x)/(1-8*x+14*x^2)) \\ G. C. Greubel, Aug 17 2018

a(n) = 8*a(n-1) - 14*a(n-2) for n > 1; a(0) = 3; a(1) = 14.

G.f.: (3-10*x)/(1-8*x+14*x^2).

Definition corrected by Emeric Deutsch, Jun 28 2009
Edited and extended beyond a(5) by Klaus Brockhaus, Jul 01 2009
Extended by Emeric Deutsch, Jun 28 2009

A161940 a(n) = ((3+sqrt(2))(5+sqrt(2))^n + (3-sqrt(2))(5-sqrt(2))^n)/2.

Original entry on oeis.org

3, 17, 101, 619, 3867, 24433, 155389, 991931, 6345363, 40639217, 260448821, 1669786219, 10707539307, 68670310033, 440429696269, 2824879831931, 18118915305123, 116216916916817, 745434117150341, 4781352082416619

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Jun 22 2009

nonn

Fifth binomial transform of A162255.

Muniru A Asiru, Table of n, a(n) for n = 0..256
Index entries for linear recurrences with constant coefficients, signature (10, -23).

Cf. A162255, A161939 (fourth binomial transform of A162255).

a := [3, 17];; for n in [3..10^2] do a[n] := 10*a[n-1] - 23*a[n-2]; od; a; # Muniru A Asiru, Feb 02 2018

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((3+r)*(5+r)^n+(3-r)*(5-r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 01 2009

a[0] := 3: a[1] := 17: for n from 2 to 20 do a[n] := 10*a[n-1]-23*a[n-2] end do: seq(a[n], n = 0 .. 20); # Emeric Deutsch, Jun 27 2009

LinearRecurrence[{10,-23},{3,17},30] (* Harvey P. Dale, Oct 05 2012 *)

x='x+O('x^30); Vec((3-13*x)/(1-10*x+23*x^2)) \\ G. C. Greubel, Aug 17 2018

a(n) = 10*a(n-1) - 23*a(n-2) for n > 1; a(0) = 3, a(1) = 17.

G.f.: (3-13*x)/(1-10*x+23*x^2).

Edited and extended beyond a(4) by Klaus Brockhaus, Jul 01 2009

Extended by Emeric Deutsch, Jun 27 2009

A164120 Partial sums of A162396.

Original entry on oeis.org

5, 7, 17, 21, 41, 49, 89, 105, 185, 217, 377, 441, 761, 889, 1529, 1785, 3065, 3577, 6137, 7161, 12281, 14329, 24569, 28665, 49145, 57337, 98297, 114681, 196601, 229369, 393209, 458745, 786425, 917497, 1572857, 1835001, 3145721, 3670009

Offset: 1

Klaus Brockhaus, Aug 10 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2).

Cf. A162396, A164053 (partial sums of A162255).

T:=[ n le 2 select 8-3*n else 2*Self(n-2): n in [1..38] ]; [ n eq 1 select T[1] else Self(n-1)+T[n]: n in [1..#T]];

Rest[CoefficientList[Series[x*(5 + 2*x)/((1 - x)*(1 - 2*x^2)), {x,0,50}], x]] (* or *) LinearRecurrence[{1,2,-2}, {5,7,17}, 50] (* G. C. Greubel, Sep 12 2017 *)

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

a(n) = 2*a(n-2) + 7 for n > 2; a(1) = 5, a(2) = 7.
a(n) = (19 - 5*(-1)^n)*2^((2*n-1+(-1)^n)/4)/2 - 7.
G.f.: x*(5+2*x)/((1-x)*(1-2*x^2)).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3). - G. C. Greubel, Sep 12 2017

cp's OEIS Frontend

A164053 Partial sums of A162255.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A078343 a(0) = -1, a(1) = 2; a(n) = 2*a(n-1) + a(n-2).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

Formula

Extensions

A162396 a(n) = 2*a(n-2) for n > 2; a(1) = 5, a(2) = 2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Magma

Maple

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

A164682 a(n) = 2*a(n-2) for n > 2; a(1) = 5, a(2) = 8.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A072946 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)=(2,2), where (n)_q=(1+q+...+q^(n-1)) and q is a root of unity.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A072946 Coefficient of the highest power of q in the expansion of nu(0)=1, nu(1)=b and for n>=2, nu(n)=bnu(n-1)+lambda(n-1)_q*nu(n-2) with (b,lambda)=(2,2), where (n)_q=(1+q+...+q^(n-1)) and q is a root of unity.

A161938 a(n) = ((3+sqrt(2))(2+sqrt(2))^n + (3-sqrt(2))(2-sqrt(2))^n)/2.

A161939 a(n) = ((3+sqrt(2))(4+sqrt(2))^n + (3-sqrt(2))(4-sqrt(2))^n)/2.

A161940 a(n) = ((3+sqrt(2))(5+sqrt(2))^n + (3-sqrt(2))(5-sqrt(2))^n)/2.