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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A173495 a(n) = Lucas(n) - floor(Lucas(n)/2).

Original entry on oeis.org

1, 1, 2, 2, 4, 6, 9, 15, 24, 38, 62, 100, 161, 261, 422, 682, 1104, 1786, 2889, 4675, 7564, 12238, 19802, 32040, 51841, 83881, 135722, 219602, 355324, 574926, 930249, 1505175, 2435424, 3940598, 6376022, 10316620, 16692641, 27009261, 43701902, 70711162
Offset: 0

Views

Author

Roger L. Bagula, Nov 23 2010

Keywords

Links

Crossrefs

Cf. A000032, A001077.

Programs

  • Magma
    [Ceiling(Lucas(n)/2): n in [0..40]]; // Vincenzo Librandi, Feb 19 2014
  • Maple
    a:= n-> (<<0|1|0|0|0>, <0|0|1|0|0>, <0|0|0|1|0>, <0|0|0|0|1>,
          <-1|-1|1|1|1>>^n. <<1, 1, 2, 2, 4>>)[1, 1]:
seq(a(n), n=0..50);  # Alois P. Heinz, Feb 17 2013
  • Mathematica
    l[0] = 2; l[1] = 1; l[n_] := l[n] = l[n - 1] + l[n - 2]; Table[l[n] - Floor[l[n]/2], {n, 0, 30}]
Table[Ceiling[LucasL[n]/2], {n, 0, 39}] (* Jean-François Alcover, Feb 17 2014 *)
CoefficientList[Series[(1 - 2 x^3)/((1 - x) (1 + x + x^2) (1 - x - x^2)), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 19 2014 *)

Formula

G.f.: (1-2*x^3)/[(1-x)*(1+x+x^2)*(1-x-x^2)].
a(n) = ceiling(Lucas(n)/2).
a(3n) = A001077(n). - Christopher Hohl, Aug 19 2021

A226447 Expansion of (1-x+x^3)/(1-x^2+2*x^3-x^4).

Original entry on oeis.org

1, -1, 1, -2, 4, -5, 9, -15, 23, -38, 62, -99, 161, -261, 421, -682, 1104, -1785, 2889, -4675, 7563, -12238, 19802, -32039, 51841, -83881, 135721, -219602, 355324, -574925, 930249, -1505175, 2435423, -3940598, 6376022, -10316619, 16692641, -27009261, 43701901, -70711162, 114413064, -185124225
Offset: 0

Views

Author

Paul Curtz, Jun 28 2013

Keywords

Comments

a(n) and its differences:
. 1, -1, 1, -2, 4, -5, 9, -15, 23, -38, ...
. -2, 2, -3, 6, -9, 14, -24, 38, -61, 100, ...
. 4, -5, 9, -15, 23, -38, 62, -99, 161, -261, ...
. -9, 14, -24, 38, -61, 100, -161, 260, -422, 682, ...
. 23, -38, 62, -99, 161, -261, 421, -682, 1104, -1785, ...
. -61, 100, -161, 260, -422, 682, -1103, 1786, -2889, 4674, ...
. 161, -261, 421, -682, 1104, -1785, 2889, -4675, 7563, -12238, ...
The third row is the first shifted .
The main diagonal is A001077(n). The fourth is -A001077(n+1). By "shifted" antidiagonals there are one 1, two 2's (-2 of the first row and 2), generally A001651(n) (-1)^n *A001077(n).
a(n+1)/a(n) tends to A001622 (the golden ratio) as n -> infinity.

Links

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x+x^3)/(1-x^2+2*x^3-x^4))); // Bruno Berselli, Jul 04 2013
  • Mathematica
    a[0] = 1; a[1] = -1; a[n_] := a[n] = a[n-2] - a[n-1] - {-1, 0, 1, 1, 0, -1}[[Mod[n+1, 6] + 1]]; Table[a[n], {n, 0, 41}] (* Jean-François Alcover, Jul 04 2013 *)

Formula

a(0)=1, a(1)=-1; for n>1, a(n) = a(n-2) - a(n-1) + A010892(n+2).
a(n) = a(n-2) -2*a(n-3) +a(n-4).
a(n) = A226956(-n).
a(n+1) = A039834(n) - (-1)^n*A094686(n).
a(n+6) - a(n) = 2*(-1)^n* A000032(n+3).
a(2n+1) = -A226956(2n+1).
G.f. ( -1+x-x^3 ) / ( (x^2-x-1)*(1-x+x^2) ). - R. J. Mathar, Jun 29 2013
2*a(n) = A010892(n+2)+A061084(n+1). - R. J. Mathar, Jun 29 2013

A165251 Variation on Delannoy array/triangle; based on a triangular sum with the base multiplied by 2.

Original entry on oeis.org

1, 2, 2, 4, 9, 4, 8, 28, 28, 8, 16, 76, 121, 76, 16, 32, 192, 422, 422, 192, 32, 64, 464, 1304, 1809, 1304, 464, 64, 64, 464, 1304, 1809, 1304, 464, 64, 128, 1088, 3728, 6648, 6648, 3728, 1088, 128, 256, 2496, 10096, 22056, 28401, 22056, 10096, 2496, 256, 512
Offset: 1

Views

Author

Mark Dols, Sep 10 2009

Keywords

Comments

If a(1,2,3)=1 sums of rows give A001077.

Examples

			Triangle begins:
1
2,2
4,9,4
8,28,28,8
16,76,121,76,16
32,192,422,422,192,32

Crossrefs

Cf. A001077, A001076 (row sums), A008288
Previous Showing 51-53 of 53 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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