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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A135019 a(n) = -a(n-1) + 2a(n-2) - a(n-3), with a(0) = 0, a(1) = 1, a(2) = -3.

Original entry on oeis.org

0, 1, -3, 5, -12, 25, -54, 116, -249, 535, -1149, 2468, -5301, 11386, -24456, 52529, -112827, 242341, -520524, 1118033, -2401422, 5158012, -11078889, 23796335, -51112125, 109783684, -235804269, 506483762, -1087875984, 2336647777
Offset: 0

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Author

Paul Curtz, Feb 10 2008

Keywords

Comments

Sequence is identical to its signed second differences less first 3 terms.-
R. J. Mathar, May 17 2009

Links

Formula

From R. J. Mathar, May 17 2009: (Start)
a(n)*(-1)^(n+1) = A002478(n-1) + 2*A002478(n-2).
G.f.: x*(1 - 2*x)/(1 + x - 2*x^2 + x^3). (End)

Extensions

More terms from R. J. Mathar, May 17 2009

A192805 Constant term in the reduction of the polynomial 1+x+x^2+...+x^n by x^3->x^2+2x+1. See Comments.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 12, 25, 53, 113, 242, 519, 1114, 2392, 5137, 11033, 23697, 50898, 109323, 234814, 504356, 1083305, 2326829, 4997793, 10734754, 23057167, 49524466, 106373552, 228479649, 490751217, 1054084065, 2264066146, 4862985491
Offset: 0

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Author

Clark Kimberling, Jul 10 2011

Keywords

Comments

For discussions of polynomial reduction, see A192232 and A192744.

Examples

			The first five polynomials p(n,x) and their reductions:
p(1,x)=1 -> 1
p(2,x)=x+1 -> x+1
p(3,x)=x^2+x+1 -> x^2+x+1
p(4,x)=x^3+x^2+x+1 -> 2x^2+3x+2
p(5,x)=x^4+x^3+x^2+x+1 -> 5x^2+6*x+3, so that
A192805=(1,1,1,2,3,...), A002478=(0,1,1,3,6,...), A077864=(0,0,1,2,5,...).

Links

Crossrefs

Cf. A192744, A192232, A002478.

Programs

  • Mathematica
    q = x^3; s = x^2 + 2 x + 1; z = 40;
p[0, x_] := 1; p[n_, x_] := x^n + p[n - 1, x];
Table[Expand[p[n, x]], {n, 0, 7}]
reduce[{p1_, q_, s_, x_}] :=
FixedPoint[(s PolynomialQuotient @@ #1 +
       PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]
  (* A192805 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
  (* A002478  *)
u3 = Table[Coefficient[Part[t, n], x, 2], {n, 1, z}]
  (* A077864 *)

Formula

a(n)=2*a(n-1)+a(n-2)-a(n-3)-a(n-4).
G.f.: -(1+x)*(2*x-1) / ( (x-1)*(x^3+2*x^2+x-1) ). - R. J. Mathar, May 06 2014
a(n)-a(n-1) = A002478(n-3). - R. J. Mathar, May 06 2014

A221688 T(n,k)=Number of nXk arrays of occupancy after each element stays put or moves to some king-move neighbor, without consecutive moves in the same direction.

Original entry on oeis.org

1, 3, 3, 6, 35, 6, 13, 396, 396, 13, 28, 4429, 17480, 4429, 28, 60, 49387, 780692, 780692, 49387, 60, 129, 550264, 34704624, 140268025, 34704624, 550264, 129, 277, 6129659, 1542552756
Offset: 1

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Author

R. H. Hardin Jan 22 2013

Keywords

Comments

Table starts
...1......3........6........13.......28.........60.....129.277
...3.....35......396......4429....49387.....550264.6129659
...6....396....17480....780692.34704624.1542552756
..13...4429...780692.140268025
..28..49387.34704624
..60.550264
.129

Examples

			Some solutions for n=3 k=4
..0..1..2..0....0..1..0..1....0..4..0..0....0..4..1..0....0..2..3..0
..0..3..1..0....0..3..0..3....0..1..4..0....0..2..0..2....0..3..3..0
..0..5..0..0....0..0..4..0....0..3..0..0....1..2..0..0....1..0..0..0

Crossrefs

Column 1 is A002478

A378786 G.f. A(x) satisfies A(x) = 1 + x * (1+x)^2 * A(x)^4.

Original entry on oeis.org

1, 1, 6, 39, 296, 2435, 21138, 190603, 1767968, 16761424, 161697576, 1582171216, 15664531716, 156637712953, 1579664567130, 16048129755157, 164085811289360, 1687224436103842, 17436287104620980, 181001686332329224, 1886522317836670988, 19734386503541838083
Offset: 0

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Author

Seiichi Manyama, Dec 07 2024

Keywords

Crossrefs

Cf. A365178, A366216, A366272.
Cf. A002478, A073155, A366221.
Cf. A002293.

Programs

  • PARI
    a(n, r=1, s=2, t=4, u=0) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(s*k, n-k)/(t*k+u*(n-k)+r));

Formula

a(n) = Sum_{k=0..n} binomial(4*k+1,k) * binomial(2*k,n-k)/(4*k+1) = Sum_{k=0..n} binomial(2*k,n-k) * A002293(k).

A105260 Triangle read by rows: T(n,k)=C(2n-2k,k), n>=0, 0<=k<=floor(2n/3).

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 1, 1, 6, 6, 1, 8, 15, 4, 1, 10, 28, 20, 1, 1, 12, 45, 56, 15, 1, 14, 66, 120, 70, 6, 1, 16, 91, 220, 210, 56, 1, 1, 18, 120, 364, 495, 252, 28, 1, 20, 153, 560, 1001, 792, 210, 8, 1, 22, 190, 816, 1820, 2002, 924, 120, 1, 1, 24, 231, 1140, 3060, 4368, 3003
Offset: 0

Views

Author

Emeric Deutsch, Apr 14 2005

Keywords

Examples

			Triangle begins:
1;
1;
1,2;
1,4,1;
1,6,6;
1,8,15,4;
Row n contains 1+floor(2n/3) terms.

References

  • E. Deutsch, Math. Magazine, vol. 75, No. 3, 2002, p. 228, problem 1623.

Crossrefs

Row sums yield A002478.

Programs

  • Maple
    T:=(n,k)->binomial(2*n-2*k,k): for n from 0 to 14 do seq(T(n,k),k=0..floor(2*n/3)) od;# yields sequence in triangular form

Formula

T(n, k)=C(2n-2k, k), n>=0, 0<=k<=floor(2n/3). G.f.=1/[1-z(1+tz)^2].
T(n,k) = A102547(2*n,k). - R. J. Mathar, Aug 21 2016
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