A163609 -id:A163609 - 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).

A163608 a(n) = ((5 + 2sqrt(2))(2 + sqrt(2))^n + (5 - 2sqrt(2))(2 - sqrt(2))^n)/2.

Original entry on oeis.org

5, 14, 46, 156, 532, 1816, 6200, 21168, 72272, 246752, 842464, 2876352, 9820480, 33529216, 114475904, 390845184, 1334428928, 4556025344, 15555243520, 53108923392, 181325206528, 619082979328, 2113681504256, 7216560058368

Offset: 0

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

Binomial transform of A163607. Inverse binomial transform of A163609.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-2).

Cf. A163607, A163609.

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((5+2*r)*(2+r)^n+(5-2*r)*(2-r)^n)/2: n in [0..23] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 06 2009

LinearRecurrence[{4,-2},{5,14},30] (* Harvey P. Dale, Jan 31 2017 *)

x='x+O('x^50); Vec((5-6*x)/(1-4*x+2*x^2)) \\ G. C. Greubel, Jul 29 2017

a(n) = 4*a(n-1) - 2*a(n-2) for n > 1; a(0) = 5, a(1) = 14.
G.f.: (5-6*x)/(1-4*x+2*x^2).
a(n) = 5*A007070(n) - 6*A007070(n-1). - R. J. Mathar, Nov 08 2013
E.g.f.: exp(2*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

A163610 a(n) = ((5 + 2sqrt(2))(4 + sqrt(2))^n + (5 - 2sqrt(2))(4 - sqrt(2))^n)/2.

Original entry on oeis.org

5, 24, 122, 640, 3412, 18336, 98920, 534656, 2892368, 15653760, 84736928, 458742784, 2483625280, 13446603264, 72802072192, 394164131840, 2134084044032, 11554374506496, 62557819435520, 338701312393216, 1833801027048448

Offset: 0

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

nonn

Binomial transform of A163609. Fourth binomial transform of A163888. Inverse binomial transform of A163611.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-14).

Cf. A163609, A163611, A163888.

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((5+2*r)*(4+r)^n+(5-2*r)*(4-r)^n)/2: n in [0..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 06 2009

LinearRecurrence[{8,-14},{5,24},30] (* Harvey P. Dale, May 29 2016 *)

x='x+O('x^50); Vec((5-16*x)/(1-8*x+14*x^2)) \\ G. C. Greubel, Jul 29 2017

a(n) = 8*a(n-1) - 14*a(n-2) for n > 1; a(0) = 5, a(1) = 24.
G.f.: (5-16*x)/(1-8*x+14*x^2).
E.g.f.: exp(4*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

cp's OEIS Frontend

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

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A163608 a(n) = ((5 + 2sqrt(2))(2 + sqrt(2))^n + (5 - 2sqrt(2))(2 - sqrt(2))^n)/2.

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A163610 a(n) = ((5 + 2sqrt(2))(4 + sqrt(2))^n + (5 - 2sqrt(2))(4 - 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.

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Author

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Programs

Magma

Mathematica

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Formula

A163608 a(n) = ((5 + 2*sqrt(2))*(2 + sqrt(2))^n + (5 - 2*sqrt(2))*(2 - sqrt(2))^n)/2.

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Author

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Programs

Magma

Mathematica

PARI

Formula

Extensions

A163610 a(n) = ((5 + 2*sqrt(2))*(4 + sqrt(2))^n + (5 - 2*sqrt(2))*(4 - sqrt(2))^n)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A163608 a(n) = ((5 + 2sqrt(2))(2 + sqrt(2))^n + (5 - 2sqrt(2))(2 - sqrt(2))^n)/2.

A163610 a(n) = ((5 + 2sqrt(2))(4 + sqrt(2))^n + (5 - 2sqrt(2))(4 - sqrt(2))^n)/2.