A163141 -id:A163141 - cp's OEIS frontend

A163070 a(n) = ((4+sqrt(5))(2+sqrt(5))^n + (4-sqrt(5))(2-sqrt(5))^n)/2.

4, 13, 56, 237, 1004, 4253, 18016, 76317, 323284, 1369453, 5801096, 24573837, 104096444, 440959613, 1867934896, 7912699197, 33518731684, 141987625933, 601469235416, 2547864567597, 10792927505804, 45719574590813

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Jul 20 2009

nonn

Binomial transform of A163069. Second binomial transform of A163141. Inverse binomial transform of A163071.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,1).

Cf. A000032, A000045, A001076, A014448, A163069, A163071, A163141.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-5); S:=[ ((4+r)*(2+r)^n+(4-r)*(2-r)^n)/2: n in [0..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 21 2009

LinearRecurrence[{4,1},{4,13},30] (* Harvey P. Dale, Sep 19 2011 *)

x='x+O('x^30); Vec((4-3*x)/(1-4*x-x^2)) \\ G. C. Greubel, Jan 08 2018

a(n) = 4*a(n-1) + a(n-2) for n > 1; a(0) = 4, a(1) = 13.
G.f.: (4-3*x)/(1-4*x-x^2).
a(n) = 2*A000032(3*n) + 5*A000045(3*n)/2 = 2*A014448(n) + 5*A001076(n). - Diego Rattaggi, Aug 09 2020

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 21 2009

A163069 a(n) = ((4+sqrt(5))(1+sqrt(5))^n + (4-sqrt(5))(1-sqrt(5))^n)/2.

Original entry on oeis.org

4, 9, 34, 104, 344, 1104, 3584, 11584, 37504, 121344, 392704, 1270784, 4112384, 13307904, 43065344, 139362304, 450985984, 1459421184, 4722786304, 15283257344, 49457659904, 160048349184, 517927337984, 1676048072704

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Jul 20 2009

nonn

Binomial transform A163141. Inverse binomial transform A163070.

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

Cf. A163141, A163070.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-5); S:=[ ((4+r)*(1+r)^n+(4-r)*(1-r)^n)/2: n in [0..23] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 21 2009

LinearRecurrence[{2, 4}, {4, 9}, 30] (* G. C. Greubel, Jan 08 2018 *)

x='x+O('x^30); Vec((4+x)/(1-2*x-4*x^2)) \\ G. C. Greubel, Jan 08 2018

a(n) = 2*a(n-1) + 4*a(n-2) for n > 1; a(0) = 4, a(1) = 9.
G.f.: (4+x)/(1-2*x-4*x^2).
a(n) = 2^(n+1) * A000032(n) + 5 * 2^(n-1) * A000045(n). - Diego Rattaggi, Jun 27 2020

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 21 2009

A163071 a(n) = ((4+sqrt(5))(3+sqrt(5))^n + (4-sqrt(5))(3-sqrt(5))^n)/2.

Original entry on oeis.org

4, 17, 86, 448, 2344, 12272, 64256, 336448, 1761664, 9224192, 48298496, 252894208, 1324171264, 6933450752, 36304019456, 190090313728, 995325804544, 5211593572352, 27288258215936, 142883175006208, 748146017173504

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Jul 20 2009

nonn

Binomial transform of A163070. Third binomial transform of A163141. Inverse binomial transform of A108404 without initial 1.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6, -4).

Cf. A163070, A163141, A108404.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-5); S:=[ ((4+r)*(3+r)^n+(4-r)*(3-r)^n)/2: n in [0..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 21 2009

LinearRecurrence[{6,-4},{4,17},40] (* Harvey P. Dale, Feb 12 2013 *)

a(n) = 6*a(n-1) - 4*a(n-2) for n > 1; a(0) = 4, a(1) = 17.
G.f.: (4-7*x)/(1-6*x+4*x^2).
a(n) = 2^(n+1) * A000032(2*n) + 5 * 2^(n-1) * A000045(2*n) = 2^(n+1) * A005248(n) + 5 * 2^(n-1) * A001906(n). - Diego Rattaggi, Aug 02 2020

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 21 2009

A163072 a(n) = ((4+sqrt(5))(5+sqrt(5))^n + (4-sqrt(5))(5-sqrt(5))^n)/2.

Original entry on oeis.org

4, 25, 170, 1200, 8600, 62000, 448000, 3240000, 23440000, 169600000, 1227200000, 8880000000, 64256000000, 464960000000, 3364480000000, 24345600000000, 176166400000000, 1274752000000000, 9224192000000000, 66746880000000000

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Jul 20 2009

nonn

Binomial transform of A108404 without initial 1. Fifth binomial transform of A163141.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-20).

Cf. A108404, A163141.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-5); S:=[ ((4+r)*(5+r)^n+(4-r)*(5-r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 21 2009

LinearRecurrence[{10, -20}, {4, 25}, 30] (* G. C. Greubel, Jan 08 2018 *)

x='x+O('x^30); Vec((4-15*x)/(1-10*x+20*x^2)) \\ G. C. Greubel, Jan 08 2018

a(n) = 10*a(n-1) - 20*a(n-2) for n > 1; a(0) = 4, a(1) = 25.

G.f.: (4-15*x)/(1-10*x+20*x^2).

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 21 2009

cp's OEIS Frontend

A163070 a(n) = ((4+sqrt(5))*(2+sqrt(5))^n + (4-sqrt(5))*(2-sqrt(5))^n)/2.

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A163069 a(n) = ((4+sqrt(5))*(1+sqrt(5))^n + (4-sqrt(5))*(1-sqrt(5))^n)/2.

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Magma

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A163071 a(n) = ((4+sqrt(5))*(3+sqrt(5))^n + (4-sqrt(5))*(3-sqrt(5))^n)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

Extensions

A163072 a(n) = ((4+sqrt(5))*(5+sqrt(5))^n + (4-sqrt(5))*(5-sqrt(5))^n)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A163070 a(n) = ((4+sqrt(5))(2+sqrt(5))^n + (4-sqrt(5))(2-sqrt(5))^n)/2.

A163069 a(n) = ((4+sqrt(5))(1+sqrt(5))^n + (4-sqrt(5))(1-sqrt(5))^n)/2.

A163071 a(n) = ((4+sqrt(5))(3+sqrt(5))^n + (4-sqrt(5))(3-sqrt(5))^n)/2.

A163072 a(n) = ((4+sqrt(5))(5+sqrt(5))^n + (4-sqrt(5))(5-sqrt(5))^n)/2.