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.

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A133038 Cubes of A000931.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 1, 8, 8, 27, 64, 125, 343, 729, 1728, 4096, 9261, 21952, 50653, 117649, 274625, 636056, 1481544, 3442951, 8000000, 18609625, 43243551, 100544625, 233744896, 543338496, 1263214441, 2936493568, 6826561273, 15870019697, 36892780289
Offset: 0

Views

Author

Omar E. Pol, Nov 02 2007

Keywords

Examples

			a(10)=27 because Padovan(10)=3 and 3^3=27.

Links

Crossrefs

Cf. A000578, A030078, A056570. Padovan sequence: A000931.

Programs

  • Mathematica
    CoefficientList[Series[(1 - x^2)/(1 - x^2 - x^3), {x, 0, 50}], x]^3

Formula

a(n) = A000931(n)^3.
a(n) = a(n-1)+2*a(n-2)+3*a(n-3)-2*a(n-4)+4*a(n-5)-4*a(n-6)-a(n-7)-a(n-8)-a(n-10).
G.f.: (x^9-x^8+x^7+x^6-5*x^5+x^4-2*x^3-2*x^2-x+1) / ((x-1) * (x^3-2*x^2+3*x-1) * (x^6+3*x^5+5*x^4+5*x^3+5*x^2+3*x+1)). - Colin Barker, Sep 18 2013

A226958 a(n) = Fibonacci(n-2)*Fibonacci(n)*Fibonacci(n+2).

Original entry on oeis.org

2, 0, 10, 24, 130, 504, 2210, 9240, 39338, 166320, 705058, 2985840, 12649570, 53582256, 226981610, 961503816, 4073004770, 17253510120, 73087065922, 309601740360, 1311494081482, 5555577978720, 23533806138050, 99690802301664, 422297015715650, 1788878864564064, 7577812474943050
Offset: 1

Views

Author

Ron Knott, Jun 27 2013

Keywords

Examples

			a(3) = F(1)*F(3)*F(5) = 1*2*5 = 10.

Links

Crossrefs

Cf. A000045 (Fibonacci numbers), A000032 (Lucas numbers).
Products of 3 Fibonaccis: A065563, A056570, A220362, A110224.

Programs

  • Mathematica
    Table[Fibonacci[n - 2] Fibonacci[n] Fibonacci[n + 2], {n, 1, 20}]
LinearRecurrence[{3,6,-3,-1},{2,0,10,24},30] (* Harvey P. Dale, Apr 10 2022 *)
Join[{2},#[[1]]#[[3]]#[[5]]&/@Partition[Fibonacci[Range[0,40]],5,1]] (* Harvey P. Dale, May 20 2025 *)
  • PARI
    a(n)=fibonacci(n-2)*fibonacci(n)*fibonacci(n+2); \\ Joerg Arndt, Jul 07 2013

Formula

a(n) = 3*a(n-1) + 6*a(n-2) - 3*a(n-3) - a(n-4).
G.f.: 2*(1-3*x-x^2)/(1-3*x-6*x^2+3*x^3+x^4).
a(n) = Lucas(n-1)*Fibonacci(n+2) = Fibonacci(n-2)*Lucas(n+1).
a(n) = (1/5)*(Fibonacci(3*n)-8*(-1)^n*Fibonacci(n)). - Ehren Metcalfe, Mar 26 2016
For n >= 3, a(n) is the numerator of the continued fraction [1,..,1, 3 ,1,..,1, 3 ,1,..,1] with three runs of 1's each of length n-3 and each separated by a single 3. For example, a(5)=130 which is the numerator of the continued fraction [1,1, 3 ,1,1, 3 ,1,1]. - Greg Dresden, Jan 01 2022

Extensions

More terms from Joerg Arndt, Jul 07 2013

A226976 Fibonacci(n)^3 + Fibonacci(n+2)^3.

Original entry on oeis.org

1, 9, 28, 133, 539, 2322, 9773, 41501, 175636, 744273, 3152359, 13354306, 56568617, 239630337, 1015087436, 4299984173, 18215017507, 77160064914, 326855259829, 1384581132277, 5865179743556, 24845300179929, 105246380344463, 445830821750018, 1888569667033489
Offset: 0

Views

Author

Ron Knott, Jun 27 2013

Keywords

Examples

			a(2) = Fibonacci(2)^3 + Fibonacci(4)^3 = 1^3 + 2^3 = 9

Links

Crossrefs

Cf. A000045 (Fibonacci), A056570 (Fibonacci^3).
Cf. A110224 (Fib(n)^3 + Fib(n+1)^3).

Programs

  • Mathematica
    Table[Fibonacci[n]^3 + Fibonacci[n+2]^3,{n,0,50}]
#[[1]]+#[[3]]&/@Partition[Fibonacci[Range[0,30]]^3,3,1] (* or *) LinearRecurrence[{3,6,-3,-1},{1,9,28,133},30] (* Harvey P. Dale, Jun 30 2025 *)
  • PARI
    a(n) = fibonacci(n)^3+fibonacci(n+2)^3; \\ Joerg Arndt, Jul 07 2013

Formula

a(n) = 3a(n-1)+6a(n-2)-3a(n-3)-a(n-4)
G.f.: (1+6x-5x^2-2x^3)/(1-3x-6x^2+3x^3+x^4)= (2x^2+7x+1)(1-x)/((x^2-x-1)(x^2+4x-1))
Previous Showing 31-33 of 33 results.

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