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.

A163141 a(n) = 5*a(n-2) for n > 2; a(1) = 4, a(2) = 5.

Original entry on oeis.org

4, 5, 20, 25, 100, 125, 500, 625, 2500, 3125, 12500, 15625, 62500, 78125, 312500, 390625, 1562500, 1953125, 7812500, 9765625, 39062500, 48828125, 195312500, 244140625, 976562500, 1220703125, 4882812500, 6103515625, 24414062500
Offset: 1

Views

Author

Klaus Brockhaus, Jul 21 2009

Keywords

Comments

Apparently the same as A133632 without initial 1.
Binomial transform is A163069, second binomial transform is A163070, third binomial transform is A163071, fourth binomial transform is A108404 without initial 1, fifth binomial transform is A163072.

Links

Crossrefs

Cf. A133632, A108404, A163069, A163070, A163071, A163072.

Programs

  • Magma
    [ n le 2 select n+3 else 5*Self(n-2): n in [1..29] ];
  • Mathematica
    LinearRecurrence[{0,5},{4,5},30] (* Harvey P. Dale, Dec 20 2021 *)
  • PARI
    Vec(x*(4+5*x)/(1-5*x^2) + O(x^30)) \\ Jinyuan Wang, Mar 23 2020

Formula

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

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

Views

Author

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

Keywords

Comments

Binomial transform A163141. Inverse binomial transform A163070.

Links

Crossrefs

Cf. A163141, A163070.

Programs

  • Magma
    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
  • Mathematica
    LinearRecurrence[{2, 4}, {4, 9}, 30] (* G. C. Greubel, Jan 08 2018 *)
  • PARI
    x='x+O('x^30); Vec((4+x)/(1-2*x-4*x^2)) \\ G. C. Greubel, Jan 08 2018

Formula

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

Extensions

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

Views

Author

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

Keywords

Comments

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

Links

Crossrefs

Cf. A163070, A163141, A108404.

Programs

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

Formula

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

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 21 2009
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.