A192243 -id:A192243 - cp's OEIS frontend

A192232 Constant term of the reduction of n-th Fibonacci polynomial by x^2 -> x+1. (See Comments.)

1, 0, 2, 1, 6, 7, 22, 36, 89, 168, 377, 756, 1630, 3353, 7110, 14783, 31130, 65016, 136513, 285648, 599041, 1254456, 2629418, 5508097, 11542854, 24183271, 50674318, 106173180, 222470009, 466131960, 976694489, 2046447180, 4287928678, 8984443769, 18825088134

Offset: 1

Clark Kimberling, Jun 26 2011

Polynomial reduction: an introduction
...
We begin with an example.  Suppose that p(x) is a polynomial, so that p(x)=(x^2)t(x)+r(x) for some polynomials t(x) and r(x), where r(x) has degree 0 or 1.  Replace x^2 by x+1 to get (x+1)t(x)+r(x), which is (x^2)u(x)+v(x) for some u(x) and v(x), where v(x) has degree 0 or 1.  Continuing in this manner results in a fixed polynomial w(x) of degree 0 or 1.  If p(x)=x^n, then w(x)=x*F(n)+F(n-1), where F=A000045, the sequence of Fibonacci numbers.
In order to generalize, write d(g) for the degree of an arbitrary polynomial g(x), and suppose that p, q, s are polynomials satisfying d(s)s in this manner until reaching w such that d(w)s.
The coefficients of (reduction of p by q->s) comprise a vector of length d(q)-1, so that a sequence p(n,x) of polynomials begets a sequence of vectors, such as (F(n), F(n-1)) in the above example.  We are interested in the component sequences (e.g., F(n-1) and F(n)) for various choices of p(n,x).
Following are examples of reduction by x^2->x+1:
n-th Fibonacci p(x) -> A192232+x*A112576
n-th cyclotomic p(x) -> A192233+x*A051258
n-th 1st-kind Chebyshev p(x) -> A192234+x*A071101
n-th 2nd-kind Chebyshev p(x) -> A192235+x*A192236
x(x+1)(x+2)...(x+n-1) -> A192238+x*A192239
(x+1)^n -> A001519+x*A001906
(x^2+x+1)^n -> A154626+x*A087635
(x+2)^n -> A020876+x*A030191
(x+3)^n -> A192240+x*A099453
...
Suppose that b=(b(0), b(1),...) is a sequence, and let p(n,x)=b(0)+b(1)x+b(2)x^2+...+b(n)x^n.  We define (reduction of sequence b by q->s) to be the vector given by (reduction of p(n,x) by q->s), with components in the order of powers, from 0 up to d(q)-1.  For k=0,1,...,d(q)-1, we then have the "k-sequence of (reduction of sequence b by q->s)".  Continuing the example, if b is the sequence given by b(k)=1 if k=n and b(k)=0 otherwise, then the 0-sequence of (reduction of b by x^2->x+1) is (F(n-1)), and the 1-sequence is (F(n)).
...
For selected sequences b, here are the 0-sequences and 1-sequences of (reduction of b by x^2->x+1):
b=A000045, Fibonacci sequence (1,1,2,3,5,8,...) yields
   0-sequence A166536 and 1-sequence A064831.
b=(1,A000045)=(1,1,1,2,3,5,8,...) yields
   0-sequence A166516 and 1-sequence A001654.
b=A000027, natural number sequence (1,2,3,4,...) yields
  0-sequence A190062 and 1-sequence A122491.
b=A000032, Lucas sequence (1,3,4,7,11,...) yields
  0-sequence A192243 and 1-sequence A192068.
b=A000217, triangular sequence (1,3,6,10,...) yields
  0-sequence A192244 and 1-sequence A192245.
b=A000290, squares sequence (1,4,9,16,...) yields
  0-sequence A192254 and 1-sequence A192255.
More examples:  A192245-A192257.
...
More comments:
(1) If s(n,x)=(reduction of x^n by q->s) and
    p(x)=p(0)x^n+p(1)x^(n-1)+...+p(n)x^0, then
    (reduction of p by q->s)=p(0)s(n,x)+p(1)s(n-1,x)
    +...+p(n-1)s(1,x)+p(n)s(0,x).  See A192744.
(2) For any polynomial p(x), let P(x)=(reduction of p(x)
    by q->s).  Then P(r)=p(r) for each zero r of
    q(x)-s(x).  In particular, if q(x)=x^2 and s(x)=x+1,
    then P(r)=p(r) if r=(1+sqrt(5))/2 (golden ratio) or
    r=(1-sqrt(5))/2.

			The first four Fibonacci polynomials and their reductions by x^2->x+1 are shown here:
F1(x)=1 -> 1 + 0x
F2(x)=x -> 0 + 1x
F3(x)=x^2+1 -> 2+1x
F4(x)=x^3+2x -> 1+4x
F5(x)=x^4+3x^2+1 -> (x+1)^2+3(x+1)+1 -> 6+6x.
From these, read A192232=(1,0,1,1,6,...) and A112576=(0,1,1,4,6,...).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,3,-1,-1).

Cf. A168561, A192233 - A192240, A192744.

Cf. A206296, A265398, A265399, A265752, A265753.

q[x_] := x + 1;
reductionRules = {x^y_?EvenQ -> q[x]^(y/2),  x^y_?OddQ -> x q[x]^((y - 1)/2)};
t = Table[FixedPoint[Expand[#1 /. reductionRules] &, Fibonacci[n, x]], {n, 1, 40}];
Table[Coefficient[Part[t, n], x, 0], {n, 1, 40}]
  (* A192232 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 40}]
(* A112576 *)
(* Peter J. C. Moses, Jun 25 2011 *)
LinearRecurrence[{1, 3, -1, -1}, {1, 0, 2, 1}, 60] (* Vladimir Joseph Stephan Orlovsky, Feb 08 2012 *)

Vec((1-x-x^2)/(1-x-3*x^2+x^3+x^4)+O(x^99)) \\ Charles R Greathouse IV, Jan 08 2013

Empirical G.f.: -x*(x^2+x-1)/(x^4+x^3-3*x^2-x+1). - Colin Barker, Sep 11 2012
The above formula is correct. - Charles R Greathouse IV, Jan 08 2013
a(n) = A265752(A206296(n)). - Antti Karttunen, Dec 15 2015
a(n) = A112576(n) -A112576(n-1) -A112576(n-2). - R. J. Mathar, Dec 16 2015

Example corrected by Clark Kimberling, Dec 18 2017

A192068 a(n) = Fibonacci(2*n) - (n mod 2).

Original entry on oeis.org

0, 3, 7, 21, 54, 144, 376, 987, 2583, 6765, 17710, 46368, 121392, 317811, 832039, 2178309, 5702886, 14930352, 39088168, 102334155, 267914295, 701408733, 1836311902, 4807526976, 12586269024, 32951280099, 86267571271, 225851433717, 591286729878

Offset: 1

Clark Kimberling, Jun 26 2011

nonn

Previous name was: 1-sequence of reduction of Lucas sequence by x^2 -> x+1.

See A192232 for definition of "k-sequence of reduction of [sequence] by [substitution]".

			(See A192243.)

Cf. A000032, A000045, A192243, A192068.

Partial sums of A081714.

a := n -> combinat[fibonacci](2*n)-(n mod 2):
seq(a(n), n=1..29); # Peter Luschny, Mar 10 2015

c[n_] := LucasL[n];
Table[c[n], {n, 1, 15}]
q[x_] := x + 1; p[0, x_] := 1;
p[n_, x_] :=  p[n - 1, x] + (x^n)*c[n + 1] reductionRules = {x^y_?EvenQ -> q[x]^(y/2),
   x^y_?OddQ -> x q[x]^((y - 1)/2)};
t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0,
   30}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]  (* A192243 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]  (* A192068 *)
(* Peter J. C. Moses, Jun 26 2011 *)
Table[Fibonacci[2n]-Mod[n,2],{n,30}] (* Harvey P. Dale, Jul 11 2020 *)

Empirical g.f. and recurrence: x^2*(3-2*x)/(1-3*x+3*x^3-x^4), a(n) = 3*a(n-1) - 3*a(n-3) + a(n-4). - Colin Barker, Feb 08 2012

a(n) = Fibonacci(2*n) - (n mod 2). - Peter Luschny, Mar 10 2015

New name from Peter Luschny, Mar 10 2015

A246565 Number of rooted ordered trees with n non-root nodes such that the branch lengths are weakly increasing.

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 20, 31, 67, 115, 238, 406, 880, 1494, 3118, 5568, 11408, 20092, 41938, 73687, 151529, 272359, 552611, 987277, 2033173, 3617167, 7371745, 13318869, 26991289, 48496985, 99264686, 177917588, 362349258, 655925568, 1331038563, 2401097768, 4913906801, 8844673793, 18046697901, 32720071992, 66666578597

Offset: 0

Joerg Arndt, Aug 30 2014

nonn

			The a(4) = 6 such trees are:
:
:   1:
:  [ 1 1 1 1 ]
:  [ 0 0 0 0 ]
:
:  O--o
:  .--o
:  .--o
:  .--o
:
:
:   2:
:  [ 1 1 2 ]
:  [ 0 0 0 ]
:
:  O--o
:  .--o
:  .--o--o
:
:
:   3:
:  [ 1 3 ]
:  [ 0 0 ]
:
:  O--o
:  .--o--o--o
:
:
:   4:
:  [ 2 2 ]
:  [ 0 0 ]
:
:  O--o--o
:  .--o--o
:
:
:   5:
:  [ 2 2 ]
:  [ 0 1 ]
:
:  O--o--o
:     .--o--o
:
:
:   6:
:  [ 4 ]
:  [ 0 ]
:
:  O--o--o--o--o
:
See the Arndt link for all trees for 0 <= n <= 7.

Joerg Arndt, All trees for 0 <= n <= 7

Cf. A000108 (all trees), A246566 (descending branch lengths), A193196, A192243.

A246566 Number of rooted ordered trees with n non-root nodes such that the branch lengths are weakly decreasing.

Original entry on oeis.org

1, 1, 2, 4, 10, 24, 62, 155, 396, 992, 2504, 6228, 15512, 38220, 94059, 229498, 558642, 1350384, 3255138, 7801327, 18642118, 44329833, 105111283, 248184255, 584396849, 1371048495, 3208318247, 7483522231, 17413665881, 40405120865, 93543352492

Offset: 0

Joerg Arndt, Aug 30 2014

nonn

			The a(3) = 4 such trees are:
:
:     1:
:    [ 1 1 1 ]
:    [ 0 0 0 ]
:
:  O--o
:  .--o
:  .--o
:
:
:     2:
:    [ 2 1 ]
:    [ 0 0 ]
:
:  O--o--o
:  .--o
:
:
:     3:
:    [ 2 1 ]
:    [ 0 1 ]
:
:  O--o--o
:     .--o
:
:
:     4:
:    [ 3 ]
:    [ 0 ]
:
:  O--o--o--o
:
See the Arndt link for all trees for 0 <= n <= 5.

Joerg Arndt, All trees for 0 <= n <= 5

Cf. A000108 (all trees), A246565 (ascending branch lengths), A193196, A192243, A001519.

cp's OEIS Frontend

A192232 Constant term of the reduction of n-th Fibonacci polynomial by x^2 -> x+1. (See Comments.)

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A192068 a(n) = Fibonacci(2*n) - (n mod 2).

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A246565 Number of rooted ordered trees with n non-root nodes such that the branch lengths are weakly increasing.

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A246566 Number of rooted ordered trees with n non-root nodes such that the branch lengths are weakly decreasing.

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