A163608 -id:A163608 - cp's OEIS frontend

A163888 a(n) = 2*a(n-2) for n > 2; a(1) = 5, a(2) = 4.

5, 4, 10, 8, 20, 16, 40, 32, 80, 64, 160, 128, 320, 256, 640, 512, 1280, 1024, 2560, 2048, 5120, 4096, 10240, 8192, 20480, 16384, 40960, 32768, 81920, 65536, 163840, 131072, 327680, 262144, 655360, 524288, 1310720, 1048576, 2621440, 2097152, 5242880

Offset: 1

Klaus Brockhaus, Aug 06 2009

nonn

Interleaving of A020714 and A000079 without initial terms 1 and 2.

Binomial transform is A163607, second binomial transform is A163608, third binomial transform is A163609, fourth binomial transform is A163610, fifth binomial transform is A163611.

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

Cf. A020714 (5*2^n), A000079 (powers of 2), A163607, A163608, A163609, A163610, A163611.

[ n le 2 select 6-n else 2*Self(n-2): n in [1..41] ];

Transpose[NestList[{Last[#],2First[#]}&,{5,4},40]] [[1]]  (* Harvey P. Dale, Mar 14 2011 *)
LinearRecurrence[{0, 2},{5, 4},41] (* Ray Chandler, Aug 14 2015 *)

x='x+O('x^50); vec(x*(5+4*x)/(1-2*x^2)) \\ G. C. Greubel, Aug 07 2017

a(n) = (7 - 3*(-1)^n)*2^((2*n-5+(-1)^n)/4).

G.f.: x*(5+4*x)/(1-2*x^2).

A163609 a(n) = ((5 + 2sqrt(2))(3 + sqrt(2))^n + (5 - 2sqrt(2))(3 - sqrt(2))^n)/2.

Original entry on oeis.org

5, 19, 79, 341, 1493, 6571, 28975, 127853, 564293, 2490787, 10994671, 48532517, 214232405, 945666811, 4174374031, 18426576509, 81338840837, 359047009459, 1584910170895, 6996131959157, 30882420558677, 136321599637963

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Aug 01 2009

Binomial transform of A163608. Third binomial transform of A163888. Inverse binomial transform of A163610.

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

Cf. A163608, A163610, A163888.

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((5+2*r)*(3+r)^n+(5-2*r)*(3-r)^n)/2: n in [0..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 06 2009

LinearRecurrence[{6, -7}, {5, 19}, 50] (* G. C. Greubel, Jul 29 2017 *)

x='x+O('x^50); Vec((5-11*x)/(1-6*x+7*x^2)) \\ G. C. Greubel, Jul 29 2017

a(n) = 6*a(n-1) - 7*a(n-2) for n > 1; a(0) = 5, a(1) = 19.
G.f.: (5-11*x)/(1-6*x+7*x^2).
a(n) = 5*A081179(n+1) - 11*A081179(n). - R. J. Mathar, Nov 08 2013
E.g.f.: exp(3*x)*( 5*cosh(sqrt(2)*x) + 2*sqrt(2)*sinh(sqrt(2)*x) ). - G. C. Greubel, Jul 29 2017

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 06 2009

A163607 a(n) = ((5 + 2sqrt(2))(1 + sqrt(2))^n + (5 - 2sqrt(2))(1 - sqrt(2))^n)/2.

Original entry on oeis.org

5, 9, 23, 55, 133, 321, 775, 1871, 4517, 10905, 26327, 63559, 153445, 370449, 894343, 2159135, 5212613, 12584361, 30381335, 73347031, 177075397, 427497825, 1032071047, 2491639919, 6015350885, 14522341689, 35060034263, 84642410215

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Aug 01 2009

Binomial transform of A163888. Inverse binomial transform of A163608.

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

Cf. A163608, A163888.

Cf. A000129, A001333.

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((5+2*r)*(1+r)^n+(5-2*r)*(1-r)^n)/2: n in [0..27] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 06 2009

LinearRecurrence[{2, 1}, {5, 9}, 50] (* G. C. Greubel, Jul 29 2017 *)

x='x+O('x^50); Vec((5-x)/(1-2*x-x^2)) \\ G. C. Greubel, Jul 29 2017

a(n) = 2*a(n-1) + a(n-2) for n > 1; a(0) = 5, a(1) = 9.
G.f.: (5-x)/(1-2*x-x^2).
a(n) = 5*A000129(n+1) - A000129(n). - R. J. Mathar, Nov 08 2013
E.g.f.: exp(x)*( 5*cosh(sqrt(2)*x) + 2*sqrt(2)*sinh(sqrt(2)*x) ). - G. C. Greubel, Jul 29 2017
a(n) = 2*A001333(n) + A001333(n+2). - Philippe Deléham, Mar 06 2023

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 06 2009

cp's OEIS Frontend

A163888 a(n) = 2*a(n-2) for n > 2; a(1) = 5, a(2) = 4.

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A163609 a(n) = ((5 + 2sqrt(2))(3 + sqrt(2))^n + (5 - 2sqrt(2))(3 - sqrt(2))^n)/2.

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A163607 a(n) = ((5 + 2sqrt(2))(1 + sqrt(2))^n + (5 - 2sqrt(2))(1 - sqrt(2))^n)/2.

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cp's OEIS Frontend

A163888 a(n) = 2*a(n-2) for n > 2; a(1) = 5, a(2) = 4.

Views

Author

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Programs

Magma

Mathematica

PARI

Formula

A163609 a(n) = ((5 + 2*sqrt(2))*(3 + sqrt(2))^n + (5 - 2*sqrt(2))*(3 - sqrt(2))^n)/2.

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Author

Keywords

Comments

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Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A163607 a(n) = ((5 + 2*sqrt(2))*(1 + sqrt(2))^n + (5 - 2*sqrt(2))*(1 - sqrt(2))^n)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A163609 a(n) = ((5 + 2sqrt(2))(3 + sqrt(2))^n + (5 - 2sqrt(2))(3 - sqrt(2))^n)/2.

A163607 a(n) = ((5 + 2sqrt(2))(1 + sqrt(2))^n + (5 - 2sqrt(2))(1 - sqrt(2))^n)/2.