A162963 -id:A162963 - cp's OEIS frontend

A162773 a(n) = ((2+sqrt(5))(5+sqrt(5))^n + (2-sqrt(5))(5-sqrt(5))^n)/2.

2, 15, 110, 800, 5800, 42000, 304000, 2200000, 15920000, 115200000, 833600000, 6032000000, 43648000000, 315840000000, 2285440000000, 16537600000000, 119667200000000, 865920000000000, 6265856000000000, 45340160000000000

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Jul 13 2009

nonn

Binomial transform of A162772. Fifth binomial transform of A162963.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10, -20).

Cf. A162772, A162963.

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

LinearRecurrence[{10,-20},{2,15},30] (* Harvey P. Dale, Dec 24 2012 *)

a(n) = 10*a(n-1) - 20*a(n-2) for n > 1; a(0) = 2, a(1) = 15.

G.f.: (2-5*x)/(1-10*x+20*x^2).

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 19 2009

Formulae corrected and clarified by Harvey P. Dale, Dec 24 2012

A162771 a(n) = ((2+sqrt(5))(3+sqrt(5))^n + (2-sqrt(5))(3-sqrt(5))^n)/2.

Original entry on oeis.org

2, 11, 58, 304, 1592, 8336, 43648, 228544, 1196672, 6265856, 32808448, 171787264, 899489792, 4709789696, 24660779008, 129125515264, 676109975552, 3540157792256, 18536506851328, 97058409938944, 508204432228352

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Jul 13 2009

nonn

Binomial transform of A001077 without initial 1. Third binomial transform of A162963. Inverse binomial transform of A162772.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (6,-4).

Cf. A001077, A002878, A162963, A162772.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-5); S:=[ ((2+r)*(3+r)^n+(2-r)*(3-r)^n)/2: n in [0..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 19 2009

LinearRecurrence[{6,-4},{2,11},30] (* Harvey P. Dale, Aug 15 2013 *)
CoefficientList[Series[(2 - x) / (1 - 6 x + 4 x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 16 2013 *)

a(n) = 6*a(n-1) - 4*a(n-2) for n > 1; a(0) = 2, a(1) = 11. [corrected by Harvey P. Dale, Aug 15 2013]
G.f.: (2-x)/(1-6*x+4*x^2).
a(n) = 2^(n-1) * A002878(n+1). - Diego Rattaggi, Jun 16 2020
a(n) = Sum_{k>=1} binomial(k+n-1,n) * A000032(k) / 2^(k+1). - Diego Rattaggi, Aug 02 2020

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 19 2009

A162772 a(n) = ((2+sqrt(5))(4+sqrt(5))^n + (2-sqrt(5))(4-sqrt(5))^n)/2.

Original entry on oeis.org

2, 13, 82, 513, 3202, 19973, 124562, 776793, 4844162, 30208573, 188382802, 1174768113, 7325934082, 45685023413, 284894912402, 1776624041673, 11079148296962, 69090321917293, 430851944071762, 2686822011483873

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Jul 13 2009

nonn

Binomial transform of A162771. Fourth binomial transform of A162963. Inverse binomial transform of A162773.

Index entries for linear recurrences with constant coefficients, signature (8, -11).

Cf. A162771, A162963, A162773.

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

a(n) = 8*a(n-1) - 11*a(n-1) for n > 1; a(0) = 2, a(1) = 13.

G.f.: (2-3*x)/(1-8*x+11*x^2).

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 19 2009

A162770 a(n) = ((2+sqrt(5))(1+sqrt(5))^n + (2-sqrt(5))(1-sqrt(5))^n)/2.

Original entry on oeis.org

2, 7, 22, 72, 232, 752, 2432, 7872, 25472, 82432, 266752, 863232, 2793472, 9039872, 29253632, 94666752, 306348032, 991363072, 3208118272, 10381688832, 33595850752, 108718456832, 351820316672, 1138514460672, 3684310188032

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Jul 13 2009

nonn

Binomial transform of A162963. Inverse binomial transform of A001077 without initial 1.

Michael De Vlieger, Table of n, a(n) for n = 0..1960
Silvana Ramaj, New Results on Cyclic Compositions and Multicompositions, Master's Thesis, Georgia Southern Univ., 2021. See p. 35.
Index entries for linear recurrences with constant coefficients, signature (2,4).

Cf. A000032, A001077, A162963.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-5); S:=[ ((2+r)*(1+r)^n+(2-r)*(1-r)^n)/2: n in [0..24] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 19 2009

LinearRecurrence[{2,4},{2,7},30] (* Harvey P. Dale, Jan 13 2015 *)

a(n) = 2*a(n-1) + 4*a(n-2) for n > 1; a(0) = 2, a(1) = 7.
G.f.: (2+3*x)/(1-2*x-4*x^2).
a(n) = 2^(n-1) * A000032(n+3). - Diego Rattaggi, Jun 24 2020

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 19 2009

cp's OEIS Frontend

A162773 a(n) = ((2+sqrt(5))*(5+sqrt(5))^n + (2-sqrt(5))*(5-sqrt(5))^n)/2.

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A162771 a(n) = ((2+sqrt(5))*(3+sqrt(5))^n + (2-sqrt(5))*(3-sqrt(5))^n)/2.

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Magma

Mathematica

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A162772 a(n) = ((2+sqrt(5))*(4+sqrt(5))^n + (2-sqrt(5))*(4-sqrt(5))^n)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Formula

Extensions

A162770 a(n) = ((2+sqrt(5))*(1+sqrt(5))^n + (2-sqrt(5))*(1-sqrt(5))^n)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

Extensions

A162773 a(n) = ((2+sqrt(5))(5+sqrt(5))^n + (2-sqrt(5))(5-sqrt(5))^n)/2.

A162771 a(n) = ((2+sqrt(5))(3+sqrt(5))^n + (2-sqrt(5))(3-sqrt(5))^n)/2.

A162772 a(n) = ((2+sqrt(5))(4+sqrt(5))^n + (2-sqrt(5))(4-sqrt(5))^n)/2.

A162770 a(n) = ((2+sqrt(5))(1+sqrt(5))^n + (2-sqrt(5))(1-sqrt(5))^n)/2.