A083880 -id:A083880 - cp's OEIS frontend

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

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

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

A147957 a(n) = ((6 + sqrt(2))^n + (6 - sqrt(2))^n)/2.

Original entry on oeis.org

1, 6, 38, 252, 1732, 12216, 87704, 637104, 4663312, 34298208, 253025888, 1870171584, 13839178816, 102484311936, 759279663488, 5626889356032, 41707163713792, 309171726460416, 2292017151256064, 16992367115418624, 125979822242317312, 934017384983574528, 6924894663564105728

Offset: 0

Al Hakanson (hawkuu(AT)blogspot.com), Nov 17 2008

6th binomial transform of A077957. Binomial transform of A083880. Inverse binomial transform of A147958. - Philippe Deléham, Nov 30 2008

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

Cf. A077957, A083880, A098158, A147958.

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((6+r2)^n+(6-r2)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Nov 19 2008

LinearRecurrence[{12, -34}, {1, 6}, 50] (* G. C. Greubel, Aug 17 2018 *)

my(x='x+O('x^50)); Vec((1-6*x)/(1-12*x+34*x^2)) \\ G. C. Greubel, Aug 17 2018

From Philippe Deléham, Nov 19 2008: (Start)
a(n) = 12*a(n-1) - 34*a(n-2), n > 1; a(0)=1, a(1)=6.
G.f.: (1 - 6*x)/(1 - 12*x + 34*x^2).
a(n) = (Sum_{k=0..n} A098158(n,k)*6^(2k)*2^(n-k))/6^n. (End)
E.g.f.: exp(6*x)*cosh(sqrt(2)*x). - Ilya Gutkovskiy, Aug 11 2017

Extended beyond a(6) by Klaus Brockhaus, Nov 19 2008

A083879 a(0)=1, a(1)=4, a(n) = 8a(n-1) - 14a(n-2), n >= 2.

Original entry on oeis.org

1, 4, 18, 88, 452, 2384, 12744, 68576, 370192, 2001472, 10829088, 58612096, 317289536, 1717746944, 9299922048, 50350919168, 272608444672, 1475954689024, 7991119286784, 43265588647936, 234249039168512, 1268274072276992

Offset: 0

Paul Barry, May 08 2003

Binomial transform of A083878.
4th binomial transform of A077957. Inverse binomial transform of A083880. - Philippe Deléham, Nov 30 2008
From L. Edson Jeffery, Apr 26 2011: (Start)
Let G be the Gram matrix
  G =
  (4  1  0  1)
  (1  4  1  0)
  (0  1  4 -1)
  (1  0 -1  4)
of A028997. Then a(n) = (1/4)*Trace(G^n). (End)

Index entries for linear recurrences with constant coefficients, signature (8,-14).

Cf. A028997, A083880.

LinearRecurrence[{8,-14},{1,4},30] (* Harvey P. Dale, May 08 2013 *)

a(n) = 2^((n-2)/2)*(2*sqrt(2)-1)^n + 2^((n-2)/2)*(2*sqrt(2)+1)^n;
a(n) = Sum_{k=0..n} C(n, 2k)*5^(n-2k)2^k.
G.f.: (1-4x)/(1-8x+14x^2).
E.g.f.: exp(4x)cosh(x*sqrt(2)).
((4+sqrt(2))^n + (4-sqrt(2))^n)/2. Offset=0. a(3)=88. - Al Hakanson (hawkuu(AT)gmail.com), Oct 15 2008
a(n) = Sum_{k=0..n} A098158(n,k)*2^(3*k-n). - Philippe Deléham, Nov 30 2008

cp's OEIS Frontend

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

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

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A147957 a(n) = ((6 + sqrt(2))^n + (6 - sqrt(2))^n)/2.

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Magma

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A083879 a(0)=1, a(1)=4, a(n) = 8*a(n-1) - 14*a(n-2), n >= 2.

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Formula

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

A083879 a(0)=1, a(1)=4, a(n) = 8a(n-1) - 14a(n-2), n >= 2.