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 1-4 of 4 results.

A186679 First differences of A116697.

Original entry on oeis.org

0, -3, 4, -4, 7, -14, 22, -33, 54, -90, 145, -232, 376, -611, 988, -1596, 2583, -4182, 6766, -10945, 17710, -28658, 46369, -75024, 121392, -196419, 317812, -514228, 832039, -1346270, 2178310, -3524577, 5702886, -9227466, 14930353, -24157816, 39088168, -63245987, 102334156, -165580140
Offset: 0

Views

Author

Reinhard Zumkeller, Feb 25 2011

Keywords

Links

Crossrefs

Cf. A000045, A075193, A081714, A116697, A128533.

Programs

  • Magma
    A186679:= func< n | (-1)^n*Fibonacci(n+2) - (-1)^Floor(n/2) >;
[A186679(n): n in [0..40]]; // G. C. Greubel, Aug 24 2025
  • Mathematica
    Table[(-1)^n*Fibonacci[n+2] -(-1)^Floor[n/2], {n,0,40}] (* G. C. Greubel, Aug 24 2025 *)
  • SageMath
    def A186679(n): return (-1)**n*fibonacci(n+2) -(-1)**(n//2)
print([A186679(n) for n in range(41)]) # G. C. Greubel, Aug 24 2025

Formula

a(n) = A116697(n+1) - A116697(n).
a(2*n) = A128533(n).
a(2*n+1) = A081714(n+1).
a(n+2) = A075193(n+2) - a(n).
G.f.: x*(-3+x)/((1+x-x^2)*(1+x^2)). - Colin Barker, Sep 08 2012
From G. C. Greubel, Aug 24 2025: (Start)
a(n) = (-1)^n*Fibonacci(n+2) - (-1)^floor(n/2).
E.g.f.: exp(-x/2)*(cosh(p*x) - (3/sqrt(5))*sinh(p*x)) - cos(x) - sin(x), where 2*p = sqrt(5). (End)

A128535 a(n) = F(n)*L(n-2) where F = Fibonacci and L = Lucas numbers.

Original entry on oeis.org

0, -1, 2, 2, 9, 20, 56, 143, 378, 986, 2585, 6764, 17712, 46367, 121394, 317810, 832041, 2178308, 5702888, 14930351, 39088170, 102334154, 267914297, 701408732, 1836311904, 4807526975, 12586269026, 32951280098, 86267571273, 225851433716, 591286729880
Offset: 0

Views

Author

Axel Harvey, Mar 09 2007

Keywords

Comments

Generally, F(n)*L(n+k) = F(2*n + k) + F(k)*(-1)^(n+1):
if k=0 the sequence is A001906, if k=1 it is A081714.
For n>2, a(n) is twice the area of the triangle with vertices at (F(n-3), F(n-2)), (F(n-1), F(n)), and (L(n), L(n-1)), where F(n)=A000045(n) and L(n)=A000032(n). - J. M. Bergot, May 22 2014
a(n) is the maximum area of a quadrilateral with lengths of sides in order L(n-2), L(n-2), F(n), F(n) for n>2. - J. M. Bergot, Jan 28 2016

Examples

			a(7) = 143 because F(7)*L(5) = 13*11.
G.f. = -x + 2*x^2 + 2*x^3 + 9*x^4 + 20*x^5 + 56*x^6 + 143*x^7 + ...

References

  • Mohammad K. Azarian, Identities Involving Lucas or Fibonacci and Lucas Numbers as Binomial Sums, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 45, 2012, pp. 2221-2227.

Links

Crossrefs

Cf. A001906, A081714, A128533, A128534.

Programs

  • Magma
    [Fibonacci(n)*Lucas(n-2): n in [0..30]]; // Vincenzo Librandi, Feb 20 2013
  • Maple
    a:= n-> (<<0|1|0>, <0|0|1>, <-1|2|2>>^n. <<0,-1,2>>)[1, 1]:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 28 2016
  • Mathematica
    Table[Fibonacci[i]LucasL[i-2], {i,0,30}] (* Harvey P. Dale, Feb 16 2011 *)
LinearRecurrence[{2, 2, -1}, {0, -1, 2}, 40] (* Vincenzo Librandi, Feb 20 2013 *)
a[ n_] := Fibonacci[2 n - 2] + (-1)^n; (* Michael Somos, May 26 2014 *)
  • PARI
    a(n)=([0,1,0; 0,0,1; -1,2,2]^n*[0;-1;2])[1,1] \\ Charles R Greathouse IV, Feb 01 2016
  • PARI
    a(n) = round(((-1)^n+(2^(-1-n)*(-(3-sqrt(5))^n*(3+sqrt(5))-(-3+sqrt(5))*(3+sqrt(5))^n))/sqrt(5))) \\ Colin Barker, Sep 28 2016

Formula

a(n) = F(2*(n-1)) - (-1)^(n+1), assuming F(0)=0 and L(0)=2.
From R. J. Mathar, Apr 16 2009: (Start)
a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3).
G.f.: x*(-1+4*x)/((1+x)*(x^2-3*x+1)). (End)
a(n+1) = - A116697(2*n). - Reinhard Zumkeller, Feb 25 2011
a(-n) = - A128533(n). - Michael Somos, May 26 2014
0 = a(n)*(+4*a(n) + a(n+1) - 17*a(n+2)) + a(n+1)*(-14*a(n+1) + a(n+2)) + a(n+2)*(+4*a(n+2)) for all n in Z. - Michael Somos, May 26 2014
a(n) = ((-1)^n+(2^(-1-n)*(-(3-sqrt(5))^n*(3+sqrt(5))-(-3+sqrt(5))*(3+sqrt(5))^n)) / sqrt(5)). - Colin Barker, Sep 28 2016

Extensions

More terms from Harvey P. Dale, Feb 16 2011

A115008 a(n) = a(n-1) + a(n-3) + a(n-4).

Original entry on oeis.org

1, 0, 2, 4, 5, 7, 13, 22, 34, 54, 89, 145, 233, 376, 610, 988, 1597, 2583, 4181, 6766, 10946, 17710, 28657, 46369, 75025, 121392, 196418, 317812, 514229, 832039, 1346269, 2178310, 3524578, 5702886, 9227465, 14930353, 24157817, 39088168
Offset: 0

Views

Author

Creighton Dement, Feb 23 2006

Keywords

Comments

a(n+2) - a(n+1) - a(n) gives match to A000034, apart from signs.

Links

Crossrefs

Cf. A000034, A000045, A001519, A001906, A006498, A116697, A116698, A116699.

Programs

  • Magma
    A115008:= func< n | Fibonacci(n+1) - (n mod 2) + 2*0^((n+1) mod 4) >;
[A115008(n): n in [0..50]]; // G. C. Greubel, Aug 24 2025
  • Mathematica
    Table[Fibonacci[n+1] -I^(n-1)*Mod[n,2], {n,0,50}] (* G. C. Greubel, Aug 24 2025 *)
  • SageMath
    def A115008(n): return fibonacci(n+1) -i**(n-1)*(n%2)
print([A115008(n) for n in range(51)]) # G. C. Greubel, Aug 24 2025

Formula

a(2*n) = A000045(2*n+1) = A001519(n).
G.f.: (1-x+2*x^2+x^3)/((1+x^2)*(1-x-x^2)).
a(2*n+1) = (-1)^(n+1) + A001906(n+1) (compare with a similar property for A116697) - Creighton Dement, Mar 31 2006
From G. C. Greubel, Aug 24 2025: (Start)
a(n) = A000045(n+1) - i^(n-1)*(n mod 2).
E.g.f.: exp(x/2)*(cosh(p*x) + (1/(2*p))*sinh(p*x)) - sin(x), where 2*p = sqrt(5). (End)

A116698 Expansion of (1-x+3*x^2+x^3) / ((1-x-x^2)*(1+2*x^2)).

Original entry on oeis.org

1, 0, 2, 5, 5, 4, 13, 29, 34, 39, 89, 176, 233, 313, 610, 1115, 1597, 2328, 4181, 7277, 10946, 16687, 28657, 48416, 75025, 117297, 196418, 326003, 514229, 815656, 1346269, 2211077, 3524578, 5637351, 9227465
Offset: 0

Views

Author

Creighton Dement, Feb 23 2006

Keywords

Links

Crossrefs

Cf. A000045, A001519, A006498, A115008, A116697, A116699.

Programs

  • Magma
    A116698:= func< n | Fibonacci(n+1) -((n mod 2) -2*0^((n+1) mod 4))*2^Floor(n/2) >;
[A116898(n): n in [0..50]]; // G. C. Greubel, Aug 24 2025
  • Mathematica
    CoefficientList[Series[(1-x+3x^2+x^3)/((1-x-x^2)(1+2x^2)),{x,0,100}],x] (* or *) LinearRecurrence[{1,-1,2,2},{1,0,2,5},100] (* Harvey P. Dale, May 14 2022 *)
Table[Fibonacci[n+1] -I^(n-1)*Mod[n,2]*2^Floor[n/2], {n,0,50}] (* G. C. Greubel, Aug 24 2025 *)
  • PARI
    Vec((1-x +3*x^2 +x^3)/((1-x-x^2)*(1+2*x^2)) + O(x^40)) \\ Colin Barker, May 18 2019
  • SageMath
    def A116898(n): return fibonacci(n+1) - (-1)**((n-1)//2)*(n%2)*2**(n//2)
print([A116898(n) for n in range(51)]) # G. C. Greubel, Aug 24 2025

Formula

a(2*n) = A000045(2*n+1) = A001519(n).
a(n) = a(n-1) - a(n-2) + 2*a(n-3) + 2*a(n-4) for n > 3. - Colin Barker, May 18 2019
From G. C. Greubel, Aug 24 2025: (Start)
a(n) = A000045(n+1) - (-1)^floor((n-1)/2) * (n mod 2) * 2^floor(n/2).
E.g.f.: exp(x/2)*(cosh(sqrt(5)*x/2) + (1/sqrt(5))*sinh(sqrt(5)*x/2)) - sin(sqrt(2)*x)/sqrt(2). (End)
Showing 1-4 of 4 results.

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