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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A091870 A trinomial transform of Fibonacci(3n).

Original entry on oeis.org

0, 1, 8, 53, 336, 2105, 13144, 81997, 511392, 3189169, 19888040, 124023461, 773419248, 4823095913, 30077155576, 187563189565, 1169656805184, 7294059356257, 45486249993032, 283655347025429, 1768894026280080
Offset: 0

Views

Author

Paul Barry, Feb 06 2004

Keywords

Comments

Binomial transform of A084326.
Second binomial transform of A001076(n) = Fibonacci(3n)/2.

Links

Crossrefs

Cf. A084326.

Programs

  • GAP
    a:=[0,1];; for n in [3..30] do a[n]:=8*a[n-1]-11*a[n-2]; od; a; # G. C. Greubel, May 21 2019
  • Magma
    [n le 2 select n-1 else 8*Self(n-1) -11*Self(n-2): n in [1..30]]; // G. C. Greubel, May 21 2019
  • Mathematica
    LinearRecurrence[{8, -11}, {0, 1}, 30] (* G. C. Greubel, May 21 2019 *)
CoefficientList[Series[x/(1 - 8 x + 11 x^2), {x, 0, 30}], x] (* Michael De Vlieger, Sep 22 2017 *)
  • PARI
    my(x='x+O('x^30)); concat([0], Vec(x/(1 -8*x +11*x^2))) \\ G. C. Greubel, May 21 2019
  • Sage
    [lucas_number1(n,8,11) for n in range(0, 30)] # Zerinvary Lajos, Apr 23 2009

Formula

G.f.: x/(1 - 8*x + 11*x^2).
a(n) = sqrt(5) * ((4+sqrt(5))^n - (4-sqrt(5))^n) / 10.
a(n) = Sum_{i=0..n} Sum_{j=0..n} (n!/(i!*j!*(n-i-j)!)) * Fibonacci(3*i) / 2.

A087635 a(n) = S(n,3) where S(n,m) = Sum_{k=0..n} binomial(n,k)*Fibonacci(m*k).

Original entry on oeis.org

0, 2, 12, 64, 336, 1760, 9216, 48256, 252672, 1323008, 6927360, 36272128, 189923328, 994451456, 5207015424, 27264286720, 142757658624, 747488804864, 3913902194688, 20493457948672, 107305138913280, 561857001684992
Offset: 0

Views

Author

Benoit Cloitre, Oct 23 2003

Keywords

Links

Crossrefs

Cf. A000045, A001906 (S(n, 1)), A030191 (S(n, 2)).
Cf. A084326.

Programs

  • Mathematica
    LinearRecurrence[{6,-4}, {0, 2}, 22] (* Amiram Eldar, Apr 29 2025 *)

Formula

a(n) = 6*a(n-1)-4*a(n-2) = 2*A084326(n).
a(n) = Sum_{0<=j<=i<=n} C(i,j)*C(n,i)*Fibonacci(i+j). - Benoit Cloitre, May 21 2005
a(n) = 2^n*Fibonacci(2*n). - Benoit Cloitre, Sep 13 2005
a(n) = Sum_{k=0..n} C(n,k)*Fibonacci(k)*Lucas(n-k). - Ross La Haye, Aug 14 2006
G.f.: 2*x/(1-6*x+4*x^2). - Colin Barker, Jun 19 2012

A099843 A transform of the Fibonacci numbers.

Original entry on oeis.org

1, -5, 21, -89, 377, -1597, 6765, -28657, 121393, -514229, 2178309, -9227465, 39088169, -165580141, 701408733, -2971215073, 12586269025, -53316291173, 225851433717, -956722026041, 4052739537881, -17167680177565, 72723460248141, -308061521170129, 1304969544928657
Offset: 0

Views

Author

Paul Barry, Oct 27 2004

Keywords

Comments

The g.f. is the transform of the g.f. of A000045 under the mapping G(x) -> (-1/(1+x))*G((x-1)/(x+1)). In general this mapping transforms x/(1-k*x-k*x^2) into (1-x)/(1 + 2(k+1)*x - (2*k-1)*x^2).
Pisano period lengths: 1, 1, 8, 2, 20, 8, 16, 4, 8, 20, 10, 8, 28, 16, 40, 8, 12, 8, 6, 20, ... - R. J. Mathar, Aug 10 2012

Links

Crossrefs

Cf. A000045, A001076, A015448, A099842.
Cf. A084326 (shifted unsigned inverse binomial transform), A152174 (binomial transform).

Programs

  • Magma
    [(-1)^n*Fibonacci(3*n+2): n in [0..40]]; // G. C. Greubel, Apr 20 2023
  • Maple
    a:= n-> (<<0|1>, <1|-4>>^n.<<1, -5>>)[1,1]:
seq(a(n), n=0..24);  # Alois P. Heinz, Apr 21 2023
  • Mathematica
    CoefficientList[Series[(1-x)/(1+4*x-x^2), {x,0,30}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 10 2011 *)
LinearRecurrence[{-4,1},{1,-5},30] (* Harvey P. Dale, Aug 13 2015 *)
  • SageMath
    [(-1)^n*fibonacci(3*n+2) for n in range(41)] # G. C. Greubel, Apr 20 2023

Formula

G.f.: (1-x)/(1+4*x-x^2).
a(n) = (sqrt(5)-2)^n * (1/2 - 3*sqrt(5)/10) + (-sqrt(5)-2)^n * (1/2 + 3*sqrt(5)/10).
a(n) = (-1)^n*Fibonacci(3*n+2).
a(n) = -4*a(n-1) + a(n-2), a(0)=1, a(1)=-5. - Philippe Deléham, Nov 03 2008
a(n) = (-1)^n*(A001076(n) + A001076(n+1)). - R. J. Mathar, Aug 10 2012
a(n) = (-1)^n*A015448(n+1). - R. J. Mathar, May 07 2019

A206800 Riordan array (1/(1-3*x+x^2), x*(1-x)/(1-3*x+x^2)).

Original entry on oeis.org

1, 3, 1, 8, 5, 1, 21, 19, 7, 1, 55, 65, 34, 9, 1, 144, 210, 141, 53, 11, 1, 377, 654, 534, 257, 76, 13, 1, 987, 1985, 1905, 1111, 421, 103, 15, 1, 2584, 5911, 6512, 4447, 2041, 641, 134, 17, 1, 6765, 17345, 21557, 16837, 9038, 3440, 925, 169, 19, 1
Offset: 0

Views

Author

Philippe Deléham, Feb 12 2012

Keywords

Examples

			Triangle begins :
1
3, 1
8, 5, 1
21, 19, 7, 1
55, 65, 34, 9, 1
144, 210, 141, 53, 11, 1
377, 654, 534, 257, 76, 13, 1
987, 1985, 1905, 1111, 421, 103, 15, 1
2584, 5911, 6512, 4447, 2041, 641, 134, 17, 1
6765, 17345, 21557, 16837, 9038, 3440, 925, 169, 19, 1
Triangle (0,3,-1/3,1/3,0,0,0,0,0,...) DELTA (1,0,-1/3,1/3,0,0,0,0,...) begins :
1
0, 1
0, 3, 1
0, 8, 5, 1
0, 21, 19, 7, 1
0, 55, 65, 34, 9, 1...

References

  • Subtriangle of the triangle given by (0, 3, -1/3, 1/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, -1/3, 1/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
  • Antidiagonal sums are A072264(n).

Crossrefs

Cf. A000045, A001519, A001906, A001870,

Formula

T(n,k) = 3*T(n-1,k) + T(n-1,k-1) - T(n-2,k) - T(n-2,k-1).
G.f.: 1/(1-(y+3)*x+(y+1)*x^2).
Sum_{k, 0<=k<=n} T(n,k)*x^k = (-1)^n* A015587(n+1), (-1)^n*A190953(n+1), (-1)^n*A015566(n+1), (-1)*A189800(n+1), (-1)^n*A015541(n+1), (-1)^n*A085939(n+1), (-1)^n*A015523(n+1), (-1)^n*A063727(n), (-1)^n*A006130(n), A077957(n), A000045(n+1), A000079(n), A001906(n+1), A007070(n), A116415(n), A084326(n+1), A190974(n+1), A190978(n+1), A190984(n+1), A190990(n+1), A190872(n) for x = -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8 respectively.
Previous Showing 11-14 of 14 results.

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