A162773 -id:A162773 - cp's OEIS frontend

A162963 a(n) = 5*a(n-2) for n > 2; a(1) = 2, a(2) = 5.

2, 5, 10, 25, 50, 125, 250, 625, 1250, 3125, 6250, 15625, 31250, 78125, 156250, 390625, 781250, 1953125, 3906250, 9765625, 19531250, 48828125, 97656250, 244140625, 488281250, 1220703125, 2441406250, 6103515625, 12207031250

Offset: 1

Klaus Brockhaus, Jul 19 2009

Binomial transform is A162770, second binomial transform is A001077 without initial 1, third binomial transform is A162771, fourth binomial transform is A162772, fifth binomial transform is A162773.

Index entries for linear recurrences with constant coefficients, signature (0,5).

Cf. A000351 (powers of 5), A026383, A001077, A162770, A162771, A162772, A162773.

[ n le 2 select 3*n-1 else 5*Self(n-2): n in [1..29] ];

a(n) = (3-(-1)^n)*5^(1/4*(2*n-1+(-1)^n))/2.
G.f.: x*(2+5*x)/(1-5*x^2).
a(n) = A026383(n) for n >= 1.

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

A163066 a(n) = 12a(n-1) - 31a(n-2) for n > 1; a(0) = 2, a(1) = 17.

Original entry on oeis.org

2, 17, 142, 1177, 9722, 80177, 660742, 5443417, 44838002, 369310097, 3041743102, 25052304217, 206333614442, 1699381942577, 13996241263222, 115274054938777, 949405180105442, 7819366458163217, 64400836914689902, 530409682773219097

Offset: 0

Klaus Brockhaus, Jul 20 2009

nonn

Binomial transform of A162773. Inverse binomial transform of A163067.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-31).

Cf. A162773, A163067.

[ n le 2 select 15*n-13 else 12*Self(n-1)-31*Self(n-2): n in [1..20] ];

CoefficientList[Series[(2-7*x)/(1-12*x+31*x^2), {x,0,50}],x] (* or *) LinearRecurrence[{12,-31}, {2,17}, 30] (* G. C. Greubel, Dec 22 2017 *)

x='x+O('x^30); Vec((2-7*x)/(1-12*x+31*x^2)) \\ G. C. Greubel, Dec 22 2017

a(n) = ((2+sqrt(5))*(6+sqrt(5))^n + (2-sqrt(5))*(6-sqrt(5))^n)/2.

G.f.: (2-7*x)/(1-12*x+31*x^2).

cp's OEIS Frontend

A162963 a(n) = 5*a(n-2) for n > 2; a(1) = 2, a(2) = 5.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Formula

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

A163066 a(n) = 12a(n-1) - 31a(n-2) for n > 1; a(0) = 2, a(1) = 17.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

cp's OEIS Frontend

A162963 a(n) = 5*a(n-2) for n > 2; a(1) = 2, a(2) = 5.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Formula

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

A163066 a(n) = 12*a(n-1) - 31*a(n-2) for n > 1; a(0) = 2, a(1) = 17.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

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

A163066 a(n) = 12a(n-1) - 31a(n-2) for n > 1; a(0) = 2, a(1) = 17.