A164299 -id:A164299 - cp's OEIS frontend

A081185 8th binomial transform of (0,1,0,2,0,4,0,8,0,16,...).

0, 1, 16, 194, 2112, 21764, 217280, 2127112, 20562432, 197117968, 1879016704, 17842953248, 168988216320, 1597548359744, 15083504344064, 142288071200896, 1341431869882368, 12641049503662336, 119088016125890560

Offset: 0

Paul Barry, Mar 11 2003

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

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (16,-62).

Sequences in the class a(n, m): A164298 (m=1), A164299 (m=2), A164300 (m=3), A164301 (m=4), A164598 (m=5), A164599 (m=6), this sequence (m=7), A164600 (m=8).
Binomial transform of A081184.
Cf. A081185, A147959.

[n le 2 select n-1 else 16*Self(n-1)-62*Self(n-2): n in [1..25]]; // Vincenzo Librandi, Aug 07 2013

m:=30; S:=series( x/(1-16*x+62*x^2), x, m+1):
seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Mar 12 2021

Join[{a=0,b=1},Table[c=16*b-62*a;a=b;b=c,{n,40}]] (* Vladimir Joseph Stephan Orlovsky, Feb 08 2011 *)
CoefficientList[Series[x/(1-16x+62x^2), {x, 0, 20}], x] (* Vincenzo Librandi, Aug 07 2013 *)
LinearRecurrence[{16,-62},{0,1},30] (* Harvey P. Dale, Sep 24 2013 *)

[( x/(1-16*x+62*x^2) ).series(x,n+1).list()[n] for n in (0..30)] # G. C. Greubel, Mar 12 2021

a(n) = 16*a(n-1) - 62*a(n-2), a(0)=0, a(1)=1.
G.f.: x/(1 - 16*x + 62*x^2).
a(n) = ((8 + sqrt(2))^n - (8 - sqrt(2))^n)/(2*sqrt(2)).
a(n) = Sum_{k=0..n} C(n,2*k+1) * 2^k * 7^(n-2*k-1).
E.g.f.: exp(8*x)*sinh(sqrt(2)*x)/sqrt(2). - Ilya Gutkovskiy, Aug 12 2017
From G. C. Greubel, Mar 12 2021: (Start)
a(n) = 2*A147959(n) + 8*A081185(n).
a(n) = (1/2)*Sum_{k=0..n-1} binomial(n-1,k)*7^(n-k-1)*(5*Q(k) + 4*Q(k-1)), where Q(n) = Pell-Lucas(n) = A002203(n). (End)

A164298 a(n) = ((1+4sqrt(2))(2+sqrt(2))^n + (1-4sqrt(2))(2-sqrt(2))^n)/2.

Original entry on oeis.org

1, 10, 38, 132, 452, 1544, 5272, 18000, 61456, 209824, 716384, 2445888, 8350784, 28511360, 97343872, 332352768, 1134723328, 3874187776, 13227304448, 45160842240, 154188760064, 526433355776, 1797355902976, 6136556900352, 20951515795456, 71532949381120

Offset: 0

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

Binomial transform of A048696. Second binomial transform of A164587. Inverse binomial transform of A164299.

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 (4,-2).

Sequences in the class a(n, m): this sequence (m=1), A164299 (m=2), A164300 (m=3), A164301 (m=4), A164598 (m=5), A164599 (m=6), A081185 (m=7), A164600 (m=8).
Cf. A007070, A016921, A048696, A056236, A112032, A164587, A241204.
Cf. A016116(n+1).

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((1+4*r)*(2+r)^n+(1-4*r)*(2-r)^n)/2: n in [0..27] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 17 2009

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1+6*x)/(1-4*x+2*x^2) )); // G. C. Greubel, Dec 14 2018

a:=n->((1+4*sqrt(2))*(2+sqrt(2))^n+(1-4*sqrt(2))*(2-sqrt(2))^n)/2: seq(floor(a(n)),n=0..25); # Muniru A Asiru, Dec 15 2018

LinearRecurrence[{4,-2}, {1,10}, 50] (* or *) CoefficientList[Series[(1 + 6*x)/(1 - 4*x + 2*x^2), {x,0,50}], x] (* G. C. Greubel, Sep 12 2017 *)

my(x='x+O('x^50)); Vec((1+6*x)/(1-4*x+2*x^2)) \\ G. C. Greubel, Sep 12 2017

[( (1+6*x)/(1-4*x+2*x^2) ).series(x,n+1).list()[n] for n in (0..30)] # G. C. Greubel, Dec 14 2018; Mar 12 2021

a(n) = 4*a(n-1) - 2*a(n-2) for n > 1; a(0)=1, a(1)=10.
G.f.: (1+6*x)/(1-4*x+2*x^2).
E.g.f.: (cosh(sqrt(2)*x) + 4*sqrt(2)*sinh(sqrt(2)*x))*exp(2*x). - G. C. Greubel, Sep 12 2017
From G. C. Greubel, Mar 12 2021: (Start)
a(n) = A056236(n) + 8*A007070(n-1).
a(n) = (1/2)*Sum_{k=0..n} binomial(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

A164600 a(n) = 18a(n-1) - 79a(n-2) for n > 1; a(0) = 1, a(1) = 17.

Original entry on oeis.org

1, 17, 227, 2743, 31441, 349241, 3802499, 40854943, 434991553, 4602307457, 48477201539, 509007338599, 5332433173201, 55772217368297, 582637691946467, 6081473282940943, 63438141429166081, 661450156372654961

Offset: 0

Klaus Brockhaus, Aug 17 2009

nonn

Binomial transform of A081185 without initial term 0. Ninth binomial transform of A164587.

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..975 (terms 0..100 from Vincenzo Librandi)
Index entries for linear recurrences with constant coefficients, signature (18, -79).

Sequences in the class a(n, m): A164298 (m=1), A164299 (m=2), A164300 (m=3), A164301 (m=4), A164598 (m=5), A164599 (m=6), A081185 (m=7), this sequence (m=8).

Cf. A147960, A153593, A164587.

[ n le 2 select 16*n-15 else 18*Self(n-1)-79*Self(n-2): n in [1..18] ];

m:=30; S:=series( (1-x)/(1-18*x+79*x^2), x, m+1):
seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Mar 12 2021

LinearRecurrence[{18,-79},{1,17},30] (* Harvey P. Dale, Oct 30 2013 *)

my(x='x+O('x^50)); Vec((1-x)/(1-18*x+79*x^2)) \\ G. C. Greubel, Aug 11 2017

[( (1-x)/(1-18*x+79*x^2) ).series(x,n+1).list()[n] for n in (0..30)] # G. C. Greubel, Mar 12 2021

a(n) = ((1+4*sqrt(2))*(9+sqrt(2))^n + (1-4*sqrt(2))*(9-sqrt(2))^n)/2.
G.f.: (1-x)/(1-18*x+79*x^2).
E.g.f.: exp(9*x)*(cosh(sqrt(2)*x) + 4*sqrt(2)*sinh(sqrt(2)*x)). - G. C. Greubel, Aug 11 2017
From G. C. Greubel, Mar 12 2021: (Start)
a(n) = 2*A147960(n) + 8*A153593(n).
a(n) = (1/2)*Sum_{k=0..n} binomial(n,k)*8^(n-k)*(5*Q(k) + 4*Q(k-1)), where Q(n) = Pell-Lucas(n) = A002203(n). (End)

A164300 a(n) = ((1+4sqrt(2))(4+sqrt(2))^n + (1-4sqrt(2))(4-sqrt(2))^n)/2.

Original entry on oeis.org

1, 12, 82, 488, 2756, 15216, 83144, 452128, 2453008, 13294272, 72012064, 389976704, 2111644736, 11433484032, 61904845952, 335169991168, 1814692086016, 9825156811776, 53195565289984, 288012326955008

Offset: 0

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

Binomial transform of A164299. Fourth binomial transform of A164587. Inverse binomial transform of A164301.

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 (8,-14).

Sequences in the class a(n, m): A164298 (m=1), A164299 (m=2), this sequence (m=3), A164301 (m=4), A164598 (m=5), A164599 (m=6), A081185 (m=7), A164600 (m=8).

Cf. A081180, A083879, A164587.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((1+4*r)*(4+r)^n+(1-4*r)*(4-r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 17 2009

LinearRecurrence[{8,-14},{1,12},30] (* Harvey P. Dale, Apr 13 2012 *)

my(x='x+O('x^50)); Vec((1+4*x)/(1-8*x+14*x^2)) \\ G. C. Greubel, Sep 13 2017

[( (1+4*x)/(1-8*x+14*x^2) ).series(x,n+1).list()[n] for n in (0..30)] # G. C. Greubel, Mar 12 2021

a(n) = 8*a(n-1) - 14*a(n-2) for n > 1; a(0) = 1, a(1) = 12.
G.f.: (1+4*x)/(1-8*x+14*x^2).
E.g.f.: (4*sqrt(2)*sinh(sqrt(2)*x) + cosh(sqrt(2)*x))*exp(4*x). - Ilya Gutkovskiy, Jun 24 2016
From G. C. Greubel, Mar 12 2021: (Start)
a(n) = 2*A083879(n) + 8*A081180(n).
a(n) = (1/2)*Sum_{k=0..n} binomial(n,k)*3^(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

A164301 a(n) = ((1+4sqrt(2))(5+sqrt(2))^n + (1-4sqrt(2))(5-sqrt(2))^n)/2.

Original entry on oeis.org

1, 13, 107, 771, 5249, 34757, 226843, 1469019, 9472801, 60940573, 391531307, 2513679891, 16131578849, 103501150997, 663985196443, 4259325491499, 27321595396801, 175251467663533, 1124117982508907, 7210396068827811, 46249247090573249, 296653361322692837

Offset: 0

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

nonn

Binomial transform of A164300. Fifth binomial transform of A164587. Inverse binomial transform of A164598.

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 (terms 0..100 from Vincenzo Librandi)
Index entries for linear recurrences with constant coefficients, signature (10,-23).

Sequences in the class a(n, m): A164298 (m=1), A164299 (m=2), A164300 (m=3), this sequence (m=4), A164598 (m=5), A164599 (m=6), A081185 (m=7), A164600 (m=8).

Cf. A081182, A083880, A164587.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((1+4*r)*(5+r)^n+(1-4*r)*(5-r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 17 2009

LinearRecurrence[{10,-23},{1,13},20] (* Harvey P. Dale, Oct 15 2015 *)

my(x='x+O('x^50)); Vec((1+3*x)/(1-10*x+23*x^2)) \\ G. C. Greubel, Sep 13 2017

[( (1+3*x)/(1-10*x+23*x^2) ).series(x,n+1).list()[n] for n in (0..30)] # G. C. Greubel, Mar 12 2021

a(n) = 10*a(n-1) - 23*a(n-2) for n > 1; a(0) = 1, a(1) = 13.
G.f.: (1+3*x)/(1-10*x+23*x^2).
E.g.f.: ( cosh(sqrt(2)*x) + 4*sqrt(2)*sinh(sqrt(2)*x) )*exp(5*x). - G. C. Greubel, Sep 13 2017
From G. C. Greubel, Mar 12 2021: (Start)
a(n) = 2*A083880(n) + 8*A081182(n).
a(n) = (1/2)*Sum_{k=0..n} binomial(n,k)*4^(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

A164598 a(n) = 12a(n-1) - 34a(n-2), for n > 1, with a(0) = 1, a(1) = 14.

Original entry on oeis.org

1, 14, 134, 1132, 9028, 69848, 531224, 3999856, 29936656, 223244768, 1661090912, 12342768832, 91636134976, 679979479424, 5044125163904, 37410199666432, 277422140424448, 2057118896434688, 15253073982785024, 113094845314640896

Offset: 0

Klaus Brockhaus, Aug 17 2009

Binomial transform of A164301. Sixth binomial transform of A164587. Inverse binomial transform of A164599.

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 have 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 11 2021

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..100 from Vincenzo Librandi)
Index entries for linear recurrences with constant coefficients, signature (12,-34).

Sequences in the class a(n, m): A164298 (m=1), A164299 (m=2), A164300 (m=3), A164301 (m=4), this sequence (m=5), A164599 (m=6), A081185 (m=7), A164600 (m=8).

Cf. A081183, A147957, A164587.

[ n le 2 select 13*n-12 else 12*Self(n-1)-34*Self(n-2): n in [1..30] ];

m:=30; S:=series( (1+2*x)/(1-12*x+34*x^2), x, m+1):
seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Mar 11 2021

LinearRecurrence[{12,-34}, {1,14}, 30] (* G. C. Greubel, Aug 11 2017 *)

my(x='x+O('x^30)); Vec((1+2*x)/(1-12*x+34*x^2)) \\ G. C. Greubel, Aug 11 2017

[( (1+2*x)/(1-12*x+34*x^2) ).series(x,n+1).list()[n] for n in (0..30)] # G. C. Greubel, Mar 11 2021

a(n) = ((1+4*sqrt(2))*(6+sqrt(2))^n + (1-4*sqrt(2))*(6-sqrt(2))^n)/2.
G.f.: (1+2*x)/(1-12*x+34*x^2).
E.g.f.: exp(6*x)*(cosh(sqrt(2)*x) + 4*sqrt(2)*sinh(sqrt(2)*x)). - G. C. Greubel, Aug 11 2017
From G. C. Greubel, Mar 11 2021: (Start)
a(n) = A147957(n) + 8*A081183(n).
a(n) = (1/2)*Sum_{k=0..n} binomial(n,k)*5^(n-k)*(5*Q(k) + 4*Q(k-1)), where Q(n) = Pell-Lucas(n) = A002203(n). (End)

A164599 a(n) = 14a(n-1) - 47a(n-2), for n > 1, with a(0) = 1, a(1) = 15.

Original entry on oeis.org

1, 15, 163, 1577, 14417, 127719, 1110467, 9543745, 81420481, 691330719, 5851867459, 49433600633, 417032638289, 3515077706295, 29610553888547, 249339102243793, 2099051398651393, 17667781775661231, 148693529122641763

Offset: 0

Klaus Brockhaus, Aug 17 2009

nonn

Binomial transform of A164598. Seventh binomial transform of A164587. Inverse binomial transform of A081185 without initial term 0.

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 have 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 11 2021

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..100 from Vincenzo Librandi)
Index entries for linear recurrences with constant coefficients, signature (14,-47).

Sequences in the class a(n, m): A164298 (m=1), A164299 (m=2), A164300 (m=3), A164301 (m=4), A164598 (m=5), this sequence (m=6), A081185 (m=7), A164600 (m=8).

Cf. A002203, A081184, A081185, A147958, A164587.

[ n le 2 select 14*n-13 else 14*Self(n-1)-47*Self(n-2): n in [1..30] ];

m:=30; S:=series( (1+x)/(1-14*x+47*x^2), x, m+1):
seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Mar 11 2021

LinearRecurrence[{14,-47}, {1,15}, 30] (* G. C. Greubel, Aug 11 2017 *)

my(x='x+O('x^30)); Vec((1+x)/(1-14*x+47*x^2)) \\ G. C. Greubel, Aug 11 2017

def A164599_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)/(1-14*x+47*x^2) ).list()
A164599_list(30) # G. C. Greubel, Mar 11 2021

a(n) = ((1+4*sqrt(2))*(7+sqrt(2))^n + (1-4*sqrt(2))*(7-sqrt(2))^n)/2.
G.f.: (1+x)/(1-14*x+47*x^2).
E.g.f.: exp(7*x)*(cosh(sqrt(2)*x) + 4*sqrt(2)*sinh(sqrt(2)*x)). - G. C. Greubel, Aug 11 2017
From G. C. Greubel, Mar 11 2021: (Start)
a(n) = A147958(n) + 8*A081184(n).
a(n) = (1/2)*Sum_{k=0..n} binomial(n,k)*6^(n-k)*(5*Q(k) + 4*Q(k-1)), where Q(n) = Pell-Lucas(n) = A002203(n). (End)

cp's OEIS Frontend

A081185 8th binomial transform of (0,1,0,2,0,4,0,8,0,16,...).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Sage

Formula

A164298 a(n) = ((1+4*sqrt(2))*(2+sqrt(2))^n + (1-4*sqrt(2))*(2-sqrt(2))^n)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A164600 a(n) = 18*a(n-1) - 79*a(n-2) for n > 1; a(0) = 1, a(1) = 17.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

A164300 a(n) = ((1+4*sqrt(2))*(4+sqrt(2))^n + (1-4*sqrt(2))*(4-sqrt(2))^n)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

Extensions

A164301 a(n) = ((1+4*sqrt(2))*(5+sqrt(2))^n + (1-4*sqrt(2))*(5-sqrt(2))^n)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

Extensions

A164598 a(n) = 12*a(n-1) - 34*a(n-2), for n > 1, with a(0) = 1, a(1) = 14.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

A164298 a(n) = ((1+4sqrt(2))(2+sqrt(2))^n + (1-4sqrt(2))(2-sqrt(2))^n)/2.

A164600 a(n) = 18a(n-1) - 79a(n-2) for n > 1; a(0) = 1, a(1) = 17.

A164300 a(n) = ((1+4sqrt(2))(4+sqrt(2))^n + (1-4sqrt(2))(4-sqrt(2))^n)/2.

A164301 a(n) = ((1+4sqrt(2))(5+sqrt(2))^n + (1-4sqrt(2))(5-sqrt(2))^n)/2.

A164598 a(n) = 12a(n-1) - 34a(n-2), for n > 1, with a(0) = 1, a(1) = 14.

A164599 a(n) = 14a(n-1) - 47a(n-2), for n > 1, with a(0) = 1, a(1) = 15.