cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

2, 15, 110, 800, 5800, 42000, 304000, 2200000, 15920000, 115200000, 833600000, 6032000000, 43648000000, 315840000000, 2285440000000, 16537600000000, 119667200000000, 865920000000000, 6265856000000000, 45340160000000000
Offset: 0

Views

Author

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

Keywords

Comments

Binomial transform of A162772. Fifth binomial transform of A162963.

Links

Crossrefs

Cf. A162772, A162963.

Programs

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

Formula

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).

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 19 2009
Formulae corrected and clarified by Harvey P. Dale, Dec 24 2012

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

Original entry on oeis.org

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

Views

Author

Klaus Brockhaus, Jul 19 2009

Keywords

Comments

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.

Links

Crossrefs

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

Programs

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

Formula

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.

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

Views

Author

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

Keywords

Comments

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

Links

Crossrefs

Cf. A001077, A002878, A162963, A162772.

Programs

  • Magma
    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
  • Mathematica
    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 *)

Formula

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

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 19 2009
Showing 1-3 of 3 results.

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