A081185 -id:A081185 - cp's OEIS frontend

A081179 3rd binomial transform of (0,1,0,2,0,4,0,8,0,16,...).

0, 1, 6, 29, 132, 589, 2610, 11537, 50952, 224953, 993054, 4383653, 19350540, 85417669, 377052234, 1664389721, 7346972688, 32431108081, 143157839670, 631929281453, 2789470811028, 12313319895997, 54353623698786

Offset: 0

Paul Barry, Mar 11 2003

Binomial transform of 0, 1, 4, 14, 48, ... (A007070 with offset 1) and second binomial transform of A000129. - R. J. Mathar, Dec 10 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Sergio Falcon, Iterated Binomial Transforms of the k-Fibonacci Sequence, British Journal of Mathematics & Computer Science, 4 (22): 2014.
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. A081180, A081182, A081183, A081184, A081185, A153593.

I:=[0, 1]; [n le 2 select I[n] else 6*Self(n-1)-7*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Aug 06 2013

f:= gfun:-rectoproc({a(n) = 6*a(n-1)-7*a(n-2), a(0)=0, a(1)=1},a(n),remember):
map(f, [$0..50]); # Robert Israel, Mar 15 2016

CoefficientList[Series[x/(1-6 x +7 x^2), {x,0,30}], x] (* Vincenzo Librandi, Aug 06 2013 *)
LinearRecurrence[{6,-7}, {0,1}, 41] (* G. C. Greubel, Jan 14 2024 *)

[lucas_number1(n,6,7) for n in range(0, 23)] # Zerinvary Lajos, Apr 22 2009

a(n) = 6*a(n-1) - 7*a(n-2), a(0)=0, a(1)=1.
G.f.: x/(1-6*x+7*x^2).
a(n) = ((3+sqrt(2))^n - (3-sqrt(2))^n)/(2*sqrt(2)). [Corrected by Al Hakanson (hawkuu(AT)gmail.com), Dec 27 2008]
a(n) = 3^(n-1) Sum_{i>=0} binomial(n, 2i+1) * (2/9)^i. - Sergio Falcon, Mar 15 2016
a(n) = 2^(-1/2)*7^(n/2)*sinh(n*arcsinh(sqrt(2/7))). - Robert Israel, Mar 15 2016
E.g.f.: exp(3*x)*sinh(sqrt(2)*x)/sqrt(2). - Ilya Gutkovskiy, Aug 12 2017
a(n) = 7^((n-1)/2)*ChebyshevU(n-1, 3/sqrt(7)). - G. C. Greubel, Jan 14 2024

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)

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

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)

A081184 7th binomial transform of (0,1,0,2,0,4,0,8,0,16,...).

Original entry on oeis.org

0, 1, 14, 149, 1428, 12989, 114730, 995737, 8548008, 72872473, 618458246, 5233409213, 44200191420, 372832446869, 3142245259426, 26468308629121, 222870793614672, 1876180605036721, 15791601170624510, 132901927952017253

Offset: 0

Paul Barry, Mar 11 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..200
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 (14,-47).

Binomial transform of A081183.

Cf. A081185.

[n le 2 select n-1 else 14*Self(n-1)-47*Self(n-2): n in [1..25]]; // Vincenzo Librandi, Aug 07 2013

CoefficientList[Series[x/(1-14*x+47*x^2), {x,0,30}], x] (* Vincenzo Librandi, Aug 07 2013 *)
LinearRecurrence[{14,-47},{0,1},30] (* Harvey P. Dale, Nov 12 2013 *)

A081184=BinaryRecurrenceSequence(14,-47,0,1)
[A081184(n) for n in range(31)] # G. C. Greubel, Jan 14 2024

a(n) = 14*a(n-1) - 47*a(n-2), a(0)=0, a(1)=1.
G.f.: x/(1 - 14*x + 47*x^2). [Corrected by Georg Fischer, May 15 2019]
a(n) = ((7 + sqrt(2))^n - (7 - 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(7*x)*sinh(sqrt(2)*x)/sqrt(2). - Ilya Gutkovskiy, Aug 12 2017

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

A081179 3rd binomial transform of (0,1,0,2,0,4,0,8,0,16,...).

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

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

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

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

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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.

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

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.