A163614 -id:A163614 - cp's OEIS frontend

A163864 a(n) = 2*a(n-2) for n > 2; a(1) = 1, a(2) = 6.

1, 6, 2, 12, 4, 24, 8, 48, 16, 96, 32, 192, 64, 384, 128, 768, 256, 1536, 512, 3072, 1024, 6144, 2048, 12288, 4096, 24576, 8192, 49152, 16384, 98304, 32768, 196608, 65536, 393216, 131072, 786432, 262144, 1572864, 524288, 3145728, 1048576, 6291456

Offset: 1

Klaus Brockhaus, Aug 05 2009

nonn

Interleaving of A000079 and A007283 without initial 3.

Binomial transform is A048694, second binomial transform is A163613, third binomial transform is A163614, fourth binomial transform is A163615, fifth binomial transform is A163616, sixth binomial transform is A081183 without initial 0.

Vincenzo Librandi, Table of n, a(n) for n = 1..2000
Index entries for linear recurrences with constant coefficients, signature (0,2).

Cf. A000079 (powers of 2), A007283 (3*2^n), A048694, A163613, A163614, A163615, A163616, A081183.

[ n le 2 select 5*n-4 else 2*Self(n-2): n in [1..42] ];

LinearRecurrence[{0, 2}, {1, 6, 2, 12}, 50] (* G. C. Greubel, Aug 06 2017 *)

x='x+O('x^50); Vec(x*(1+6*x)/(1-2*x^2)) \\ G. C. Greubel, Aug 06 2017

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

G.f.: x*(1+6*x)/(1-2*x^2).

A163615 a(n) = ((1 + 3sqrt(2))(4 + sqrt(2))^n + (1 - 3sqrt(2))(4 - sqrt(2))^n)/2.

Original entry on oeis.org

1, 10, 66, 388, 2180, 12008, 65544, 356240, 1932304, 10471072, 56716320, 307135552, 1663055936, 9004549760, 48753614976, 263965223168, 1429171175680, 7737856281088, 41894453789184, 226825642378240, 1228082785977344

Offset: 0

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

nonn

Binomial transform of A163614. Fourth binomial transform of A163864. Inverse binomial transform of A163616.

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

Cf. A163614, A163864, A163616.

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((1+3*r)*(4+r)^n+(1-3*r)*(4-r)^n)/2: n in [0..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 06 2009

LinearRecurrence[{8,-14},{1,10},30] (* Harvey P. Dale, Jun 11 2014 *)

x='x+O('x^50); Vec((1+2*x)/(1-8*x+14*x^2)) \\ G. C. Greubel, Jul 30 2017

a(n) = 8*a(n-1) - 14*a(n-2) for n > 1; a(0) = 1, a(1) = 10.
G.f.: (1+2*x)/(1-8*x+14*x^2).
E.g.f.: exp(4*x)*( cosh(sqrt(2)*x) + 3*sqrt(2)*sinh(sqrt(2)*x) ). - G. C. Greubel, Jul 30 2017

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

A163613 a(n) = ((1 + 3sqrt(2))(2 + sqrt(2))^n + (1 - 3sqrt(2))(2 - sqrt(2))^n)/2.

Original entry on oeis.org

1, 8, 30, 104, 356, 1216, 4152, 14176, 48400, 165248, 564192, 1926272, 6576704, 22454272, 76663680, 261746176, 893657344, 3051137024, 10417233408, 35566659584, 121432171520, 414595366912, 1415517124608, 4832877764608

Offset: 0

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

nonn

Binomial transform of A048694. Second binomial transform of A163864. Inverse binomial transform of A163614.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-2).

Cf. A048694, A163864 (1, 6, 2, 12, 4, 24, ...), A163614.

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((1+3*r)*(2+r)^n+(1-3*r)*(2-r)^n)/2: n in [0..23] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 06 2009

LinearRecurrence[{4, -2}, {1, 8}, 50] (* G. C. Greubel, Jul 30 2017 *)

x='x+O('x^50); Vec((1+4*x)/(1-4*x+2*x^2)) \\ G. C. Greubel, Jul 30 2017

a(n) = 4*a(n-1) - 2*a(n-2) for n > 1; a(0) = 1, a(1) = 8.
G.f.: (1+4*x)/(1-4*x+2*x^2).
E.g.f.: exp(2*x)*( cosh(sqrt(2)*x) + 3*sqrt(2)*sinh(sqrt(2)*x) ). - G. C. Greubel, Jul 30 2017

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

cp's OEIS Frontend

A163864 a(n) = 2*a(n-2) for n > 2; a(1) = 1, a(2) = 6.

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A163615 a(n) = ((1 + 3sqrt(2))(4 + sqrt(2))^n + (1 - 3sqrt(2))(4 - sqrt(2))^n)/2.

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A163613 a(n) = ((1 + 3sqrt(2))(2 + sqrt(2))^n + (1 - 3sqrt(2))(2 - sqrt(2))^n)/2.

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cp's OEIS Frontend

A163864 a(n) = 2*a(n-2) for n > 2; a(1) = 1, a(2) = 6.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A163615 a(n) = ((1 + 3*sqrt(2))*(4 + sqrt(2))^n + (1 - 3*sqrt(2))*(4 - sqrt(2))^n)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A163613 a(n) = ((1 + 3*sqrt(2))*(2 + sqrt(2))^n + (1 - 3*sqrt(2))*(2 - sqrt(2))^n)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A163615 a(n) = ((1 + 3sqrt(2))(4 + sqrt(2))^n + (1 - 3sqrt(2))(4 - sqrt(2))^n)/2.

A163613 a(n) = ((1 + 3sqrt(2))(2 + sqrt(2))^n + (1 - 3sqrt(2))(2 - sqrt(2))^n)/2.