A171516 -id:A171516 - cp's OEIS frontend

A192951 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

0, 1, 3, 9, 20, 40, 74, 131, 225, 379, 630, 1038, 1700, 2773, 4511, 7325, 11880, 19252, 31182, 50487, 81725, 132271, 214058, 346394, 560520, 906985, 1467579, 2374641, 3842300, 6217024, 10059410, 16276523, 26336025, 42612643, 68948766

Offset: 0

Clark Kimberling, Jul 13 2011

The titular polynomials are defined recursively:  p(n,x) = x*p(n-1,x) + 3n - 1, with p(0,x)=1.  For an introduction to reductions of polynomials by substitutions such as x^2 -> x+1, see A192232 and A192744.
...
The list of examples at A192744 is extended here; the recurrence is given by p(n,x) = x*p(n-1,x) + v(n), with p(0,x)=1, and the reduction of p(n,x) by x^2 -> x+1 is represented by u1 + u2*x:
...
If v(n)=        n, then u1=A001595, u2=A104161.
If v(n)=      n-1, then u1=A001610, u2=A066982.
If v(n)=     3n-1, then u1=A171516, u2=A192951.
If v(n)=     3n-2, then u1=A192746, u2=A192952.
If v(n)=     2n-1, then u1=A111314, u2=A192953.
If v(n)=      n^2, then u1=A192954, u2=A192955.
If v(n)=   -1+n^2, then u1=A192956, u2=A192957.
If v(n)=    1+n^2, then u1=A192953, u2=A192389.
If v(n)=   -2+n^2, then u1=A192958, u2=A192959.
If v(n)=    2+n^2, then u1=A192960, u2=A192961.
If v(n)=    n+n^2, then u1=A192962, u2=A192963.
If v(n)=   -n+n^2, then u1=A192964, u2=A192965.
If v(n)= n(n+1)/2, then u1=A030119, u2=A192966.
If v(n)= n(n-1)/2, then u1=A192967, u2=A192968.
If v(n)= n(n+3)/2, then u1=A192969, u2=A192970.
If v(n)=     2n^2, then u1=A192971, u2=A192972.
If v(n)=   1+2n^2, then u1=A192973, u2=A192974.
If v(n)=  -1+2n^2, then u1=A192975, u2=A192976.
If v(n)=  1+n+n^2, then u1=A027181, u2=A192978.
If v(n)=  1-n+n^2, then u1=A192979, u2=A192980.
If v(n)=  (n+1)^2, then u1=A001891, u2=A053808.
If v(n)=  (n-1)^2, then u1=A192981, u2=A192982.

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (3, -2, -1, 1).

Cf. A000045, A192232, A192744.

F:=Fibonacci;; List([0..40], n-> F(n+4)+2*F(n+2)-(3*n+5)); # G. C. Greubel, Jul 12 2019

I:=[0, 1, 3, 9]; [n le 4 select I[n] else 3*Self(n-1)-2*Self(n-2)-1*Self(n-3)+Self(n-4): n in [1..40]]; // Vincenzo Librandi, Nov 16 2011

F:=Fibonacci; [F(n+4)+2*F(n+2)-(3*n+5): n in [0..40]]; // G. C. Greubel, Jul 12 2019

(* First program *)
q = x^2; s = x + 1; z = 40;
p[0, x]:= 1;
p[n_, x_]:= x*p[n-1, x] + 3n - 1;
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}] (* A171516 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192951 *)
(* Additional programs *)
LinearRecurrence[{3,-2,-1,1},{0,1,3,9},40] (* Vincenzo Librandi, Nov 16 2011 *)
With[{F=Fibonacci}, Table[F[n+4]+2*F[n+2]-(3*n+5), {n,0,40}]] (* G. C. Greubel, Jul 12 2019 *)

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; 1,-1,-2,3]^n*[0;1;3;9])[1,1] \\ Charles R Greathouse IV, Mar 22 2016

vector(40, n, n--; f=fibonacci; f(n+4)+2*f(n+2)-(3*n+5)) \\ G. C. Greubel, Jul 12 2019

f=fibonacci; [f(n+4)+2*f(n+2)-(3*n+5) for n in (0..40)] # G. C. Greubel, Jul 12 2019

a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4).
From Bruno Berselli, Nov 16 2011: (Start)
G.f.: x*(1+2*x^2)/((1-x)^2*(1 - x - x^2)).
a(n) = ((25+13*t)*(1+t)^n + (25-13*t)*(1-t)^n)/(10*2^n) - 3*n - 5 = A000285(n+2) - 3*n - 5  where t=sqrt(5). (End)
a(n) = Fibonacci(n+4) + 2*Fibonacci(n+2) - (3*n+5). - G. C. Greubel, Jul 12 2019

A014739 Expansion of (1+x^2)/(1-2*x+x^3).

Original entry on oeis.org

1, 2, 5, 9, 16, 27, 45, 74, 121, 197, 320, 519, 841, 1362, 2205, 3569, 5776, 9347, 15125, 24474, 39601, 64077, 103680, 167759, 271441, 439202, 710645, 1149849, 1860496, 3010347, 4870845, 7881194, 12752041, 20633237, 33385280, 54018519, 87403801

Offset: 0

N. J. A. Sloane

Number of wedged n-spheres in the homotopy type of the Boolean complex of the affine Coxeter group A~ n. - _Bridget Tenner, Jun 04 2008

In an infinite set of sequences such that a(n) = a(n-1) + a(n-2) + k; with a(0) = 1, a(1) = 2, and in A014739, k = 2. Cf. A171516 for a(0) = 1, a(1) = 2, k = 3. - Gary W. Adamson, Dec 10 2009

			The Boolean complex of the affine Coxeter group \widetilde{A}_3 is homotopy equivalent to the wedge of 5 3-spheres.

G. C. Greubel, Table of n, a(n) for n = 0..1000
K. Ragnarsson and B. E. Tenner, Homotopy type of the Boolean complex of a Coxeter system, Advances in Mathematics, Volume 222, Issue 2, 1 October 2009, Pages 409-430.
Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

Cf. A000032, A000045, A171516.

List([0..40], n-> Lucas(1,-1,n+2)[2] -2); # G. C. Greubel, Jul 22 2019

[Lucas(n+2)-2: n in [0..40]]; // G. C. Greubel, Jul 22 2019

with(combinat): seq(fibonacci(n)+fibonacci(n+2)-2, n=1..37); # Zerinvary Lajos, Jan 31 2008
g:=(1+z^2)/(1-z-z^2): gser:=series(g, z=0, 43): seq(coeff(gser, z, n)-2, n=2..38); # Zerinvary Lajos, Jan 09 2009

CoefficientList[Series[(1+x^2)/(1-2*x+x^3), {x,0,40}], x] (* Robert G. Wilson v, Feb 25 2005 *)
a[0]=1; a[1]=2; a[2]=5; a[n_]:= a[n] = 2a[n-1]-a[n-3]; Array[a, 40, 0]
LinearRecurrence[{2,0,-1},{1,2,5},40] (* Harvey P. Dale, Jun 26 2011 *)
LucasL[Range[0,40]+2]-2 (* G. C. Greubel, Jul 22 2019 *)

Vec((1+x^2)/(1-2*x+x^3)+O(x^40)) \\ Charles R Greathouse IV, Sep 23 2012

vector(40, n, n--; f=fibonacci; f(n+3)+f(n+1)-2) \\ G. C. Greubel, Jul 22 2019

[lucas_number2(n+2,1,-1)-2 for n in (0..40)] # G. C. Greubel, Jul 22 2019

Partial sums of Lucas numbers A000032 less 1.
From Paul Barry, Sep 03 2003: (Start)
G.f.: (1+x^2)/((1-x)*(1-x-x^2)).
a(n) = ((3+sqrt(5))((1+sqrt(5))/2)^n+(3-sqrt(5))((1-sqrt(5))/2)^n)/2-2. (End)
From Zerinvary Lajos, Jan 31 2008: (Start)
a(n) = A001610(n+1)-1.
a(n) = F(n+1) + F(n+3) - 2 = A000071(n+1) + A000071(n+3), where F(n) is the n-th Fibonacci number. [corrected by R. J. Mathar, Mar 14 2011] (End)
a(n) = A000032(n+2) - 2. - Matthew Vandermast, Nov 05 2009
a(n) = 2*a(n-1) - a(n-3). - Vincenzo Librandi, Dec 31 2010

More terms from Robert G. Wilson v, Feb 25 2005

A182415 a(0) = 1, a(1) = 2; for n>1, a(n) = a(n-1) + a(n-2) + 4.

Original entry on oeis.org

1, 2, 7, 13, 24, 41, 69, 114, 187, 305, 496, 805, 1305, 2114, 3423, 5541, 8968, 14513, 23485, 38002, 61491, 99497, 160992, 260493, 421489, 681986, 1103479, 1785469, 2888952, 4674425, 7563381, 12237810, 19801195, 32039009, 51840208, 83879221, 135719433, 219598658

Offset: 0

Alex Ratushnyak, Apr 28 2012

			a(3) = 7 + 2 + 4 = 13.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

Cf. A000045, A014739, A022095 (first differences), A171516.

F:=Fibonacci;; List([0..40], n-> F(n+3)+3*F(n+1)-4); # G. C. Greubel, Jul 22 2019

F:=Fibonacci; [F(n+3)+3*F(n+1)-4: n in [0..40]]; // G. C. Greubel, Jul 22 2019

With[{f=Fibonacci}, Table[F[n+3]+3*F[n+1]-4, {n,0,40}]] (* G. C. Greubel, Jul 22 2019 *)
RecurrenceTable[{a[0]==1,a[1]==2,a[n]==a[n-1]+a[n-2]+4},a,{n,40}] (* or *) LinearRecurrence[{2,0,-1},{1,2,7},40] (* Harvey P. Dale, Nov 24 2020 *)

vector(40, n, n--; f=fibonacci; f(n+3) +3*f(n+1) -4 ) \\ G. C. Greubel, Jul 22 2019

f=fibonacci; [f(n+3)+3*f(n+1)-4 for n in (0..40)] # G. C. Greubel, Jul 22 2019

From Colin Barker, May 07 2012: (Start)
a(n) = 2*a(n-1) - a(n-3).
G.f.: (1+3*x^2)/((1-x)*(1-x-x^2)). (End)
a(n) = Fibonacci(n+3) + 3*Fibonacci(n+1) - 4. - G. C. Greubel, Jul 22 2019

cp's OEIS Frontend

A192951 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Magma

Mathematica

PARI

PARI

Sage

Formula

A014739 Expansion of (1+x^2)/(1-2*x+x^3).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

PARI

Sage

Formula

Extensions

A182415 a(0) = 1, a(1) = 2; for n>1, a(n) = a(n-1) + a(n-2) + 4.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

Sage

Formula