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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.

A226956 a(0)=a(1)=1, a(n+2) = a(n+1) + a(n) - A128834(n).

Original entry on oeis.org

1, 1, 2, 2, 3, 5, 9, 15, 24, 38, 61, 99, 161, 261, 422, 682, 1103, 1785, 2889, 4675, 7564, 12238, 19801, 32039, 51841, 83881, 135722, 219602, 355323, 574925, 930249, 1505175, 2435424, 3940598, 6376021, 10316619, 16692641, 27009261, 43701902, 70711162, 114413063, 185124225, 299537289
Offset: 0

Views

Author

Paul Curtz, Jun 24 2013

Keywords

Comments

a(n) and differences:
1, 1, 2, 2, 3, 5, 9, 15, 24, 38, ... a(n)
0, 1, 0, 1, 2, 4, 6, 9, 14, 23, 38, ... b(n)
1, -1, 1, 1, 2, 2, 3, 5, 9, 15, 24, 38, ... a(n-2)
-2, 2, 0, 1, 0, 1, 2, 4, 6, 9, 14, 23, 38, ... b(n-2)
4, -2, 1,-1, 1, 1, 2, 2, 3, 5, 9, ... a(n-4)
-6, 3,-2, 2, 0, 1, 0, 1, 2, 4, 6, 9, ... b(n-4)
9, -5, 4,-2, 1,-1, 1, 1, 2, 2, 3, 5, 9, ... a(n-6)
-14, 9,-6, 3,-2, 2, 0, 1 0, 1, 2, ... b(n-6)
23,-15, 9,-5, 4,-2, 1, -1, 1, 1, 2, 2, ... a(n-8)
a(n)-b(n+1) = period 6: repeat 0, 1, 1, 0, -1, -1 = A128834(n).
Diagonals with the same number give 1, 2, 9, 38, ... = A001077(n).
Second column: the (n+2)-th term is identical to a(n+1) signed.
a(n+1) is identical to its twice shifted inverse binomial transform signed.
a(n+1)/a(n) tends to A001622 (the golden ratio) as n -> infinity.

Examples

			a(0) = a(1) = 1.
a(2) = a(3) = 2.
a(4) = 2*a(3) - a(2) + a(0) = 4-2+1 = 3.
a(5) = 6-2+1 = 5.

Links

Crossrefs

Cf. Diagonals in A024490.

Programs

  • Magma
    I:=[1, 1, 2, 2]; [n le 4 select I[n] else 2*Self(n-1) - Self(n-2) + Self(n-4): n in [1..30]]; // G. C. Greubel, Jan 15 2018
  • Mathematica
    a[n_] := (LucasL[n] + {0, 1, 1, 0, -1, -1}[[Mod[n, 6] + 1]])/2; Table[a[n], {n, 0, 42}] (* Jean-François Alcover, Jun 28 2013, after R. J. Mathar *)
LinearRecurrence[{2,-1,0,1}, {1,1,2,2}, 30] (* G. C. Greubel, Jan 15 2018 *)
  • PARI
    x='x+O('x^30); Vec((x-1)*(1+x^2)/((x^2+x-1)*(x^2-x+1))) \\ G. C. Greubel, Jan 15 2018

Formula

a(n+6) - a(n-6) = 20*A000045(n).
a(n) = 2*a(n-1) - a(n-2) + a(n-4).
a(n) = 3*a(n-3) + 5*a(n-6) + a(n-9) (plus many similar by telescoping the fundamental recurrence).
a(n+3) - a(n-3) = 2*A000032(n).
G.f.: (x-1)*(1+x^2) / ( (x^2+x-1)*(x^2-x+1) ). - R. J. Mathar, Jun 26 2013
2*a(n) = A000032(n) + A010892(n-1). - R. J. Mathar, Jun 26 2013
a(n+5) = a(n+4) + a(n+2) + A108014(n).
a(2n+1) + A226447(2n+2) = 2*A182895(n).
a(n) - a(n-2) = 0,2,1,1,1,3,6,... = abs(A111734(n-2)).

A182893 Triangle read by rows: T(n,k) is the number of weighted lattice paths in L_n having k (1,0)-steps at level 0. The members of L_n are paths of weight n that start at (0,0) , end on the horizontal axis and whose steps are of the following four kinds: an (1,0)-step with weight 1, an (1,0)-step with weight 2, a (1,1)-step with weight 2, and a (1,-1)-step with weight 1. The weight of a path is the sum of the weights of its steps.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 2, 0, 2, 1, 2, 4, 1, 3, 1, 4, 8, 6, 3, 4, 1, 12, 12, 18, 9, 6, 5, 1, 24, 36, 30, 32, 14, 10, 6, 1, 54, 84, 78, 64, 51, 22, 15, 7, 1, 130, 184, 204, 152, 120, 77, 34, 21, 8, 1, 300, 452, 462, 416, 280, 205, 113, 51, 28, 9, 1, 706, 1084, 1130, 1000, 770, 492, 328, 163, 74, 36, 10, 1
Offset: 0

Views

Author

Emeric Deutsch, Dec 12 2010

Keywords

Comments

Sum of entries in row n is A051286(n).
T(n,0)=A182894(n).
Sum(k*T(n,k), k=0..n)=A182895(n).

Examples

			T(3,2)=2. Indeed, denoting by h (H) the (1,0)-step of weight 1 (2), and u=(1,1), d=(1,-1), the five paths of weight 3 are ud, du, hH, Hh, and hhh; two of them, namely hH and Hh, have exactly two (1,0)-steps at level 0.
Triangle starts:
1;
0,1;
0,1,1;
2,0,2,1;
2,4,1,3,1;
4,8,6,3,4,1.

References

  • M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.
  • E. Munarini, N. Zagaglia Salvi, On the rank polynomial of the lattice of order ideals of fences and crowns, Discrete Mathematics 259 (2002), 163-177.

Crossrefs

Cf. A051286, A182894, A182895.

Programs

  • Maple
    G:=1/((1-t)*z*(1+z)+sqrt((1+z+z^2)*(1-3*z+z^2))): Gser:=simplify(series(G,z=0,15)): for n from 0 to 11 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 11 do seq(coeff(P[n],t,k),k=0..n) od; # yields sequence in triangular form

Formula

G.f.: G(t,z) =1/[(1-t)z(1+z)+sqrt((1+z+z^2)(1-3z+z^2))].

A227047 Expansion of x^2*(1+x^2) / ( (x^2-x+1)*(-x^2-x+1)*(1+x+x^2) ).

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 4, 7, 12, 19, 31, 50, 80, 130, 211, 341, 552, 893, 1444, 2337, 3782, 6119, 9901, 16020, 25920, 41940, 67861, 109801, 177662, 287463, 465124, 752587, 1217712, 1970299, 3188011, 5158310, 8346320, 13504630, 21850951, 35355581, 57206532, 92562113, 149768644
Offset: 0

Views

Author

Paul Curtz, Jun 29 2013

Keywords

Links

Programs

  • Mathematica
    CoefficientList[Series[x^2(1+x^2)/((x^2-x+1)(-x^2-x+1)(1+x+x^2)),{x,0,50}],x] (* or *) LinearRecurrence[{1,0,1,0,1,1},{0,0,1,1,2,3},50] (* Harvey P. Dale, Oct 16 2018 *)

Formula

a(0)=a(1)=0. a(n+2)=a(n+1)+a(n) + A134667(n+1).
a(2n+1) = A182895(n). a(2n+2) = A182895(n+1)-A182895(n).
a(n+1)/a(n) tends to A001622 (the golden ratio) as n->infinity.
a(n) = A079962(n-2) + A079962(n-4). - R. J. Mathar, Jun 30 2013
a(n+6) - a(n-6) = 10*A000045(n).
a(n+3) - a(n-3) = A000032(n).
a(n) = a(n-1) +a(n-3) +a(n-5) +a(n-6). - Joerg Arndt, Jun 30 2013
Showing 1-3 of 3 results.

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