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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A099561 Sum C(n-3k,k-1), k=0..floor(n/4).

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 3, 4, 5, 6, 10, 14, 19, 25, 36, 50, 69, 94, 131, 181, 250, 344, 476, 657, 907, 1251, 1728, 2385, 3292, 4543, 6272, 8657, 11949, 16492, 22765, 31422, 43371, 59863, 82629, 114051, 157422, 217285, 299915, 413966, 571388, 788673
Offset: 0

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Author

Paul Barry, Oct 22 2004

Keywords

Links

Crossrefs

Cf. A074331, A099560, A099562.

Formula

G.f.: x^4/((1-x^4)(1-x-x^4)); a(n)=a(n-1)+2a(n-4)-a(n-5)-a(n-8).

A074585 a(n)= Sum_{j=0..floor(n/2)} A073145(2*j + q), where q = 2*(n/2 - floor(n/2)).

Original entry on oeis.org

3, -1, 2, 4, -3, 3, 8, -12, 11, 11, -30, 32, 13, -73, 96, -8, -157, 263, -110, -308, 685, -485, -504, 1676, -1653, -525, 3858, -4984, 605, 8239, -13824, 6192, 15875, -35889, 26210, 25556, -87651, 88307, 24904, -200860, 264267, -38501, -426622
Offset: 0

Views

Author

Mario Catalani (mario.catalani(AT)unito.it), Aug 28 2002

Keywords

Comments

a(n) is the convolution of A073145(n) with the sequence (1,0,1,0,1,0, ...).
a(n) is also the sum of the reflected (see A074058) sequence of the generalized tribonacci sequence (A001644).

Links

Crossrefs

Cf. A073145, A074058, A001644, A074331, A074392, A074475.

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (3+2*x+x^2)/(1+x-2*x^3-x^4+x^5) )); // G. C. Greubel, Apr 13 2019
  • Mathematica
    CoefficientList[ Series[(3+2*x+x^2)/(1+x-2*x^3-x^4+x^5), {x, 0, 50}], x]
  • PARI
    my(x='x+O('x^50)); Vec((3+2*x+x^2)/(1+x-2*x^3-x^4+x^5)) \\ G. C. Greubel, Apr 13 2019
  • Sage
    ((3+2*x+x^2)/(1+x-2*x^3-x^4+x^5)).series(x, 50).coefficients(x, sparse=False) # G. C. Greubel, Apr 13 2019

Formula

a(n) = -a(n-1) + 2*a(n-3) + a(n-4) - a(n-5), a(0) = 3, a(1) = -1, a(2) = 2, a(3) = 4, a(4) = -3.
G.f.: (3 + 2*x + x^2)/(1 + x - 2*x^3 - x^4 + x^5).

Extensions

More terms from Robert G. Wilson v, Aug 29 2002

A159864 Difference array of Fibonacci numbers A000045 read by antidiagonals.

Original entry on oeis.org

0, 1, 1, 1, 0, -1, 2, 1, 1, 2, 3, 1, 0, -1, -3, 5, 2, 1, 1, 2, 5, 8, 3, 1, 0, -1, -3, -8, 13, 5, 2, 1, 1, 2, 5, 13, 21, 8, 3, 1, 0, -1, -3, -8, -21, 34, 13, 5, 2, 1, 1, 2, 5, 13, 34, 55, 21, 8, 3, 1, 0, -1, -3, -8, -21, -55, 89, 34, 13, 5, 2, 1, 1, 2, 5, 13, 34, 89
Offset: 0

Views

Author

Philippe Deléham, Apr 24 2009

Keywords

Examples

			Triangle begins:
  0;
  1,  1;
  1,  0, -1;
  2,  1,  1,  2;
  3,  1,  0, -1, -3;
  ...

Links

Crossrefs

Cf. A000045, A074331.
Main diagonal gives A039834.

Programs

  • Maple
    A159864Q := proc(n,k) option remember; if n = 0 then combinat[fibonacci](k) ; else procname(n-1,k+1) -procname(n-1,k) ; fi; end: A159864 := proc(n,k) A159864Q(k,n-k) ; end: for n from 0 to 5 do for k from 0 to n do printf("%d,",A159864(n,k)) ; od: od: # R. J. Mathar, May 29 2009
# second Maple program:
T:= (n, k)-> (<<0|1>, <1|1>>^(n-2*k))[1, 2]:
seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, Oct 27 2022
  • Mathematica
    nmax = 10; f = Table[Fibonacci[n], {n, 0, nmax}]; t = Table[Differences[f, n], {n, 0, nmax}]; Table[t[[n-k+1, k+1]], {n, 0, nmax}, {k, n, 0, -1}]  // Flatten (* Jean-François Alcover, Apr 14 2015 *)
T[ n_, k_] := If[ k<0 || k>n, 0, Fibonacci[n - 2*k]]; Join@@Table[T[n, k], {n, 0, 10}, {k, 0, n}] (* Michael Somos, Oct 27 2022 *)
  • PARI
    {T(n, k) = If(k<0 || k>n, 0, fibonacci(n - 2*k))}; /* Michael Somos, Oct 27 2022 */

Formula

Conjecture: row sums are Sum_{k=0..n} T(2n,k)=0. Sum_{k=0..n} T(2n+1,k) = A025169(n). - R. J. Mathar, May 29 2009
(1/2) * Sum_{k=0..n} |T(n,k)| = A074331(n). - Alois P. Heinz, Oct 27 2022

Extensions

Sign of a(65) = -55 corrected by Jean-François Alcover, Apr 14 2015

A279890 Expansion of x*(1 - x + 2*x^3 - x^4)/((1 - x)*(1 + x)*(1 - x + x^2)*(1 - x - x^2)).

Original entry on oeis.org

0, 1, 1, 2, 4, 7, 12, 19, 31, 50, 82, 133, 216, 349, 565, 914, 1480, 2395, 3876, 6271, 10147, 16418, 26566, 42985, 69552, 112537, 182089, 294626, 476716, 771343, 1248060, 2019403, 3267463, 5286866, 8554330, 13841197, 22395528, 36236725, 58632253, 94868978, 153501232, 248370211, 401871444, 650241655, 1052113099, 1702354754
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 22 2016

Keywords

Comments

The integer part of the harmonic mean of Fibonacci(n), Fibonacci(n+1) and Fibonacci(n+2).
The o.g.f. for the numerators of the fractional part of the harmonic mean of Fibonacci(n), Fibonacci(n+1) and Fibonacci(n+2) is 6*x/((1 + x - x^2)*(1 - 4*x - x^2)).
The o.g.f. for the denominators of the fractional part of the harmonic mean of Fibonacci(n), Fibonacci(n+1) and Fibonacci(n+2) is (1 + 3*x - x^2)/((1 + x)*(1 - 3*x + x^2)).
Convolution of Fibonacci numbers and periodic sequence [1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, ...].

Examples

			a(1) = floor(3/(1/F(1)+1/F(2)+1/F(3))) = floor(3/(1/1+1/1+1/2)) = 1;
a(2) = floor(3/(1/F(2)+1/F(3)+1/F(4))) = floor(3/(1/1+1/2+1/3)) = 1;
a(3) = floor(3/(1/F(3)+1/F(4)+1/F(5))) = floor(3/(1/2+1/3+1/5)) = 2, etc.

Links

Crossrefs

Cf. A000045, A001622, A004695, A065563, A236428.
Cf. A062114 (the integer part of the harmonic mean of Fibonacci(n+1) and Fibonacci(n+2) for n>0).
Cf. A074331 (the integer part of the geometric mean of Fibonacci(n), Fibonacci(n+1) and Fibonacci(n+2)).

Programs

  • Mathematica
    LinearRecurrence[{2, 0, -2, 2, 0, -1}, {0, 1, 1, 2, 4, 7}, 46]
Table[Floor[3 Fibonacci[n] Fibonacci[n + 1] Fibonacci[n + 2]/(2 Fibonacci[n + 1] Fibonacci[n + 2] - (-1)^n)], {n, 0, 45}]
  • PARI
    concat(0, Vec((x*(1-x+2*x^3-x^4)/((1-x)*(1+x)*(1-x+x^2))) + O(x^40))) \\ Felix Fröhlich, Dec 22 2016

Formula

G.f.: x*(1 - x + 2*x^3 - x^4)/((1 - x)*(1 + x)*(1 - x + x^2)*(1 - x - x^2)).
a(n) = 2*a(n-1) - 2*a(n-3) + 2*a(n-4) - a(n-6).
a(n) = (9*sqrt(5)*(((1 + sqrt(5))/2)^n - ((1 - sqrt(5))/2)^n) + 5*((-1)^n + 2*cos(Pi*n/3) - 3))/30.
a(n) = floor(3*F(n)*F(n+1)*F(n+2)/(2*F(n+1)*F(n+2)-(-1)^n)), where F(n) is the n-th Fibonacci number (A000045).
a(n) = floor(3*A065563(n)/A236428(n+1)).
a(n) = 3*A000045(n)/2 + ((-1)^n + 2*cos(Pi*n/3) - 3)/6.
a(n) ~ 3*phi^n/(2*sqrt(5)), where phi is the golden ratio (A001622).
Lim_{n->infinity} a(n+1)/a(n) = phi.
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