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.

Showing 1-4 of 4 results.

A192744 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.

Original entry on oeis.org

1, 1, 3, 8, 29, 133, 762, 5215, 41257, 369032, 3676209, 40333241, 483094250, 6271446691, 87705811341, 1314473334832, 21017294666173, 357096406209005, 6424799978507178, 122024623087820183, 2439706330834135361, 51219771117454755544
Offset: 0

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Author

Clark Kimberling, Jul 09 2011

Keywords

Comments

The titular polynomial is defined recursively by p(n,x)=x*p(n-1,x)+n! for n>0, where p(0,x)=1; see the Example. For an introduction to polynomial reduction, see A192232. The discussion at A192232 Comments continues here:
...
Let R(p,q,s) denote the "reduction of polynomial p by q->s" as defined at A192232. Suppose that q(x)=x^k for some k>0 and that s(x)=s(k,0)*x^(k-1)+s(k,1)*x^(k-2)+...+s(k,k-2)x+s(k,k-1).
...
First, we shall take p(x)=x^n, where n>=0; the results will be used to formulate R(p,q,s) for general p. Represent R(x^n,q,s) by
...
R(x^n)=s(n,0)*x^(k-1)+s(n,1)*x^(k-2)+...+s(n,k-2)*x+s(n,k-1).
...
Then each of the sequences u(n)=s(n,h), for h=0,1,...,k-1, satisfies this linear recurrence relation:
...
u(n)=s(k,0)*u(n-1)+s(k,1)*u(n-2)+...+s(k,k-2)*u(n-k-1)+s(k,k-1)*u(n-k), with initial values tabulated here:
...
n: ..s(n,0)...s(n,1)..s(n,2).......s(n,k-2)..s(n,k-1)
0: ....0........0.......0..............0.......1
1: ....0........0.......0..............1.......0
...
k-2: ..0........1.......0..............0.......0
k-1: ..0........0.......0..............0.......0
k: ..s(k,0)...s(k,1)..s(k,2).......s(k,k-2)..s(k,k-1)
...
That completes the formulation for p(x)=x^n. Turning to the general case, suppose that
...
p(n,x)=p(n,0)*x^n+p(n,1)*x^(n-1)+...+p(n,n-1)*x+p(n,n)
...
is a polynomial of degree n>=0. Then the reduction denoted by (R(p(n,x) by x^k -> s(x)) is the polynomial of degree k-1 given by the matrix product P*S*X, where P=(p(n,0)...p(n,1).........p(n-k)...p(n,n-k+1); X has all 0's except for main diagonal (x^(k-1), x^(k-2)...x,1); and S has
...
row 1: ... s(n,0) ... s(n,1) ...... s(n,k-2) . s(n,k-1)
row 2: ... s(n-1,0) . s(n-1,1) .... s(n-1,k-2) s(n-1,k-1)
...
row n-k+1: s(k,0).... s(k,1) ...... s(k,k-2) ..s(k,k-1)
row n-k+2: p(n,n-k+1) p(n,n-k+2) .. p(n,n-1) ..p(n,n)
*****
As a class of examples, suppose that (v(n)), for n>=0, is a sequence, that p(0,x)=1, and p(n,x)=v(n)+p(n-1,x) for n>0. If q(x)=x^2 and s(x)=x+1, and we write the reduction R(p(n,x)) as u1(n)*x+u2(n), then the sequences u1 and u2 are convolutions with the Fibonacci sequence, viz., let F=(0,1,1,2,3,5,8,...)=A000045 and let G=(1,0,1,1,2,3,5,8...); then u1=G**v and u2=F**v, where ** denotes convolution. Examples (with a few exceptions for initial terms):
...
If v(n)=n! then u1=A192744, u2=A192745.
If v(n)=n+1 then u1=A000071, u2=A001924.
If v(n)=2n then u1=A014739, u2=A027181.
If v(n)=2n+1 then u1=A001911, u2=A001891.
If v(n)=3n+1 then u1=A027961, u2=A023537.
If v(n)=3n+2 then u1=A192746, u2=A192747.
If v(n)=3n then u1=A154691, u2=A192748.
If v(n)=4n+1 then u1=A053311, u2=A192749.
If v(n)=4n+2 then u1=A192750, u2=A192751.
If v(n)=4n+3 then u1=A192752, u2=A192753.
If v(n)=4n then u1=A147728, u2=A023654.
If v(n)=5n+1 then u1=A192754, u2=A192755.
If v(n)=5n then u1=A166863, u2=A192756.
If v(n)=floor((n+1)tau) then u1=A192457, u2=A023611.
If v(n)=floor((n+2)/2) then u1=A052952, u2=A129696.
If v(n)=floor((n+3)/3) then u1=A004695, u2=A178982.
If v(n)=floor((n+4)/4) then u1=A080239, u2=A192758.
If v(n)=floor((n+5)/5) then u1=A124502, u2=A192759.
If v(n)=n+2 then u1=A001594, u2=A192760.
If v(n)=n+3 then u1=A022318, u2=A192761.
If v(n)=n+4 then u1=A022319, u2=A192762.
If v(n)=2^n then u1=A027934, u2=A008766.
If v(n)=3^n then u1=A106517, u2=A094688.

Examples

			The first five polynomials and their reductions:
1 -> 1
1+x -> 1+x
2+x+x^2 -> 3+2x
6+2x+x^2+x^3 -> 8+5x
24+6x+2x^2+x^3+x^4 -> 29+13x, so that
A192744=(1,1,3,8,29,...) and A192745=(0,1,2,5,13,...).

Crossrefs

Cf. A192232.

Programs

  • Maple
    A192744p := proc(n,x)
    option remember;
    if n = 0 then
        1;
    else
        x*procname(n-1,x)+n! ;
        expand(%) ;
    end if;
end proc:
A192744 := proc(n)
    local p;
    p := A192744p(n,x) ;
    while degree(p,x) > 1 do
        p := algsubs(x^2=x+1,p) ;
        p := expand(p) ;
    end do:
    coeftayl(p,x=0,0) ;
end proc: # R. J. Mathar, Dec 16 2015
  • Mathematica
    q = x^2; s = x + 1; z = 40;
p[0, n_] := 1; p[n_, x_] := x*p[n - 1, x] + n!;
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}]
  (* A192744 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
  (* A192745 *)

Formula

G.f.: (1-x)/(1-x-x^2)/Q(0), where Q(k)= 1 - x*(k+1)/(1 - x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, May 20 2013
Conjecture: a(n) +(-n-2)*a(n-1) +2*(n-1)*a(n-2) +3*a(n-3) +(-n+2)*a(n-4)=0. - R. J. Mathar, May 04 2014
Conjecture: (-n+2)*a(n) +(n^2-n-1)*a(n-1) +(-n^2+3*n-3)*a(n-2) -(n-1)^2*a(n-3)
=0. - R. J. Mathar, Dec 16 2015

A250729 T(n,k)=Number of (n+1)X(k+1) 0..1 arrays with nondecreasing x(i,j)+x(i,j-1) in the i direction and nondecreasing min(x(i,j),x(i-1,j)) in the j direction.

Original entry on oeis.org

9, 22, 18, 50, 46, 33, 114, 110, 85, 58, 257, 257, 208, 144, 99, 579, 596, 496, 365, 230, 166, 1302, 1376, 1158, 885, 600, 350, 275, 2927, 3173, 2699, 2092, 1500, 942, 513, 452, 6578, 7310, 6257, 4889, 3605, 2434, 1418, 728, 739, 14782, 16838, 14520, 11377, 8514
Offset: 1

Views

Author

R. H. Hardin, Nov 27 2014

Keywords

Comments

Table starts
....9...22...50...114...257...579...1302...2927....6578...14782...33216
...18...46..110...257...596..1376...3173...7310...16838...38777...89300
...33...85..208...496..1158..2699...6257..14520...33640...77999..180744
...58..144..365...885..2092..4889..11377..26419...61330..142336..330417
...99..230..600..1500..3605..8514..19887..46315..107565..249853..579962
..166..350..942..2434..6016.14437..34069..79704..185684..431691.1002869
..275..513.1418..3807..9728.23941..57397.135645..317769..741367.1725118
..452..728.2065..5760.15297.38821..95231.228455..540546.1268605.2963321
..739.1006.2918..8465.23407.61554.155263.380220..912438.2161980.5081193
.1204.1358.4022.12119.34943.95438.248537.623913.1525255.3661515.8684030

Examples

			Some solutions for n=4 k=4
..0..0..0..0..1....0..0..0..0..0....0..0..0..0..0....0..0..0..0..0
..1..0..1..0..1....0..0..0..0..0....0..0..0..0..0....0..0..0..0..0
..0..1..0..1..0....0..1..0..0..0....0..0..0..0..0....0..0..0..0..0
..1..0..1..0..1....1..0..1..0..1....0..0..0..0..0....0..0..0..0..1
..0..1..0..1..0....0..1..0..1..0....0..1..0..1..0....0..1..1..0..1

Links

Crossrefs

Column 1 is A192760(n+2)

Formula

Empirical for column k:
k=1: a(n) = 3*a(n-1) -2*a(n-2) -a(n-3) +a(n-4)
k=2: a(n) = 4*a(n-1) -5*a(n-2) +5*a(n-4) -4*a(n-5) +a(n-6); also a polynomial of degree 4 plus a quasipolynomial of degree 0 with period 2
k=3: a(n) = 4*a(n-1) -5*a(n-2) +5*a(n-4) -4*a(n-5) +a(n-6); also a polynomial of degree 4 plus a quasipolynomial of degree 0 with period 2
k=4: a(n) = 5*a(n-1) -9*a(n-2) +5*a(n-3) +5*a(n-4) -9*a(n-5) +5*a(n-6) -a(n-7) for n>8; also a polynomial of degree 5 plus a quasipolynomial of degree 0 with period 2 for n>1
k=5: [order 8; also a polynomial of degree 6 plus a quasipolynomial of degree 0 with period 2] for n>10
k=6: [order 9; also a polynomial of degree 7 plus a quasipolynomial of degree 0 with period 2] for n>14
k=7: [order 10; also a polynomial of degree 8 plus a quasipolynomial of degree 0 with period 2] for n>17
Empirical for row n:
n=1: a(n) = 3*a(n-1) -a(n-2) -2*a(n-3) +a(n-4)
n=2: a(n) = 2*a(n-1) +2*a(n-2) -3*a(n-3) for n>4
n=3: a(n) = 4*a(n-1) -2*a(n-2) -9*a(n-3) +12*a(n-4) -2*a(n-5) -3*a(n-6) +a(n-7) for n>8
n=4: [order 7] for n>9
n=5: [order 9] for n>12
n=6: [order 11] for n>15
n=7: [order 14] for n>19

A210673 a(n) = a(n-1)+a(n-2)+n-4, a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, -1, -1, -2, -2, -2, -1, 1, 5, 12, 24, 44, 77, 131, 219, 362, 594, 970, 1579, 2565, 4161, 6744, 10924, 17688, 28633, 46343, 74999, 121366, 196390, 317782, 514199, 832009, 1346237, 2178276, 3524544, 5702852, 9227429, 14930315, 24157779, 39088130, 63245946
Offset: 0

Views

Author

Alex Ratushnyak, May 09 2012

Keywords

Comments

Second differences are Fibonacci numbers A000045 with offset -4. - Olivier Gérard, Aug 21 2016

Links

Crossrefs

Cf. A066982: a(n)=a(n-1)+a(n-2)+n-2, a(0)=0, a(1)=1 (except the first term).
Cf. A104161: a(n)=a(n-1)+a(n-2)+n-1, a(0)=0, a(1)=1.
Cf. A001924: a(n)=a(n-1)+a(n-2)+n, a(0)=0, a(1)=1.
Cf. A192760: a(n)=a(n-1)+a(n-2)+n+1, a(0)=0, a(1)=1.
Cf. A192761: a(n)=a(n-1)+a(n-2)+n+2, a(0)=0, a(1)=1.
Cf. A192762: a(n)=a(n-1)+a(n-2)+n+3, a(0)=0, a(1)=1.
Cf. A210675: a(n)=a(n-1)+a(n-2)+n+4, a(0)=0, a(1)=1.

Programs

  • Mathematica
    RecurrenceTable[{a[0]==0,a[1]==1,a[n]==a[n-1]+a[n-2]+n-4},a,{n,50}] (* or *) LinearRecurrence[{3,-2,-1,1},{0,1,-1,-1},50] (* Harvey P. Dale, Oct 03 2012 *)

Formula

a(0)=0, a(1)=1, a(2)=-1, a(3)=-1, a(n) = 3*a(n-1)-2*a(n-2)-a(n-3)+a(n-4). - Harvey P. Dale, Oct 03 2012
G.f.: x/Q(0), where Q(k)= 1 + (k+1)*x/(1 - x - x*(1-x)/(x + (k+1)*(1-x)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Apr 24 2013
G.f.: -x*(2*x-1)^2 / ((x-1)^2*(x^2+x-1)). - Colin Barker, May 31 2013

A210675 a(n)=a(n-1)+a(n-2)+n+4, a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 7, 15, 30, 54, 94, 159, 265, 437, 716, 1168, 1900, 3085, 5003, 8107, 13130, 21258, 34410, 55691, 90125, 145841, 235992, 381860, 617880, 999769, 1617679, 2617479, 4235190, 6852702, 11087926, 17940663, 29028625, 46969325, 75997988, 122967352, 198965380
Offset: 0

Views

Author

Alex Ratushnyak, May 09 2012

Keywords

Links

Crossrefs

Cf. A210673: a(n)=a(n-1)+a(n-2)+n-4, a(0)=0,a(1)=1.
Cf. A066982: a(n)=a(n-1)+a(n-2)+n-2, a(0)=0,a(1)=1 (except the first term).
Cf. A104161: a(n)=a(n-1)+a(n-2)+n-1, a(0)=0,a(1)=1.
Cf. A001924: a(n)=a(n-1)+a(n-2)+n, a(0)=0,a(1)=1.
Cf. A192760: a(n)=a(n-1)+a(n-2)+n+1, a(0)=0,a(1)=1.
Cf. A192761: a(n)=a(n-1)+a(n-2)+n+2, a(0)=0,a(1)=1.
Cf. A192762: a(n)=a(n-1)+a(n-2)+n+3, a(0)=0,a(1)=1.

Programs

Formula

a(n) = 3*a(n-1)-2*a(n-2)-a(n-3)+a(n-4). G.f.: x*(4*x^2-4*x-1) / ((x-1)^2*(x^2+x-1)). - Colin Barker, May 31 2013
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