A051258 -id:A051258 - 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

A054433 Numbers formed by interpreting the reduced residue set of every even number as a Zeckendorf Expansion.

Original entry on oeis.org

1, 4, 9, 33, 80, 174, 588, 1596, 3135, 9950, 28512, 56268, 196040, 496496, 888300, 3524577, 9224880, 18118362, 63239220, 150527400, 310190454, 1129200138, 2971168704, 5834056536, 18513646430, 53213956640, 104687896833

Offset: 1

Antti Karttunen

nonn

Amiram Eldar, Table of n, a(n) for n = 1..2392

Cf. A054432, A048757, A051258, A063683.

with(combinat,fibonacci); # one_or_zero given at A054431.
A054433_as_sum := proc(n) local i; RETURN(add((one_or_zero(igcd(n,i))*fibonacci(i+1)),i=1..(n-1))); end;

r[n_] := Sum[If[GCD[n, k] == 1, Fibonacci[n + 1 - k], 0], {k, 1, n}]; r /@ (2*Range[27]) (* Amiram Eldar, Oct 19 2019 *)

a(n) = A054433_as_sum(2*n).

A063704 Cyclotomic polynomials Phi_n at x=phi divided by sqrt(5) and floored down (where phi = tau = (sqrt(5)+1)/2).

Original entry on oeis.org

0, 0, 1, 2, 1, 7, 0, 20, 3, 10, 2, 143, 2, 376, 5, 11, 21, 2583, 6, 6764, 15, 74, 34, 46367, 18, 7435, 89, 2618, 104, 832039, 25, 2178308, 987, 3399, 610, 20160, 136, 39088168, 1597, 23228, 861, 267914295, 182, 701408732, 4895, 35920, 10946, 4807526975

Offset: 0

Antti Karttunen, Aug 03 2001

nonn

Paolo Xausa, Table of n, a(n) for n = 0..1000

Cf. A063703, A051258, A063706, A063708.

with(numtheory); Phi_at_x := (n,y) -> subs(x=y,cyclotomic(n,x)); [seq(floor(evalf(simplify(Phi_at_x(j,(sqrt(5)+1)/2))/(sqrt(5)))),j=0..120)];

Floor[Simplify[Cyclotomic[Range[0, 50], GoldenRatio]]/Sqrt[5]] (* Paolo Xausa, Feb 27 2024 *)

a(47) corrected by Sean A. Irvine, May 08 2023

A063706 Cyclotomic polynomials Phi_n at x=phi, divided by sqrt(5) and rounded to nearest integer (where phi = tau = (sqrt(5)+1)/2).

Original entry on oeis.org

1, 0, 1, 2, 2, 7, 1, 20, 4, 10, 2, 143, 2, 376, 5, 12, 21, 2583, 7, 6764, 15, 75, 34, 46367, 18, 7435, 89, 2618, 104, 832039, 25, 2178308, 987, 3400, 610, 20161, 136, 39088168, 1597, 23229, 861, 267914295, 182, 701408732, 4895, 35921, 10946, 4807526975

Offset: 0

Antti Karttunen, Aug 03 2001

nonn

Paolo Xausa, Table of n, a(n) for n = 0..1000

Cf. A063705, A051258, A063704, A063708.

with(numtheory); Phi_at_x := (n,y) -> subs(x=y,cyclotomic(n,x)); [seq(round(evalf(simplify(Phi_at_x(j,(sqrt(5)+1)/2))/(sqrt(5)))),j=0..120)];

Join[{1}, Round[Simplify[Cyclotomic[Range[50], GoldenRatio]]/Sqrt[5]]] (* Paolo Xausa, Feb 27 2024 *)

a(43) and a(47) corrected by Sean A. Irvine, May 08 2023

A063708 Cyclotomic polynomials Phi_n at x=phi divided by sqrt(5) and ceiled up (where phi = tau = (sqrt(5)+1)/2).

Original entry on oeis.org

1, 1, 2, 3, 2, 8, 1, 21, 4, 11, 3, 144, 3, 377, 6, 12, 22, 2584, 7, 6765, 16, 75, 35, 46368, 19, 7436, 90, 2619, 105, 832040, 26, 2178309, 988, 3400, 611, 20161, 137, 39088169, 1598, 23229, 862, 267914296, 183, 701408733, 4896, 35921, 10947, 4807526976

Offset: 0

Antti Karttunen, Aug 03 2001

nonn

Paolo Xausa, Table of n, a(n) for n = 0..1000

Cf. A063707, A051258, A063704, A063706.

with(numtheory); Phi_at_x := (n,y) -> subs(x=y,cyclotomic(n,x)); [seq(ceil(evalf(simplify(Phi_at_x(j,(sqrt(5)+1)/2))/(sqrt(5)))),j=0..120)];

Ceiling[Simplify[Cyclotomic[Range[0, 50], GoldenRatio]]/Sqrt[5]] (* Paolo Xausa, Feb 27 2024 *)

A192233 Constant term of the reduction of the n-th cyclotomic polynomial by x^2->x+1.

Original entry on oeis.org

-1, 1, 2, 2, 5, 2, 13, 3, 7, 3, 89, 2, 233, 5, 8, 14, 1597, 5, 4181, 10, 47, 23, 28657, 12, 4596, 57, 1619, 65, 514229, 16, 1346269, 611, 2102, 379, 12461, 85, 24157817, 989, 14357, 533, 165580141, 113, 433494437, 3026, 22201, 6767, 2971215073, 598, 171486453, 3836

Offset: 1

Clark Kimberling, Jun 26 2011

sign

See A192232.

Cf. A192232.

q[x_] := x + 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] &,
      Cyclotomic[n, x]]]], {n, 1, 50}];
Table[Coefficient[Part[t, n], x, 0], {n, 1, 50}]  (* A192233 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1,
  50}]  (* A051258 Fibocyclotomic numbers *)
(* by Peter J. C. Moses, Jun 25 2011 *)

A318884 a(n) is the sum of absolute values of the coefficients in the n-th cyclotomic polynomial.

Original entry on oeis.org

2, 2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 3, 13, 7, 7, 2, 17, 3, 19, 5, 9, 11, 23, 3, 5, 13, 3, 7, 29, 7, 31, 2, 15, 17, 17, 3, 37, 19, 17, 5, 41, 9, 43, 11, 7, 23, 47, 3, 7, 5, 23, 13, 53, 3, 17, 7, 25, 29, 59, 7, 61, 31, 9, 2, 31, 15, 67, 17, 31, 17, 71, 3, 73, 37, 7, 19, 31, 17, 79, 5, 3, 41, 83, 9, 41, 43, 39, 11, 89

Offset: 1

Antti Karttunen, Sep 10 2018

nonn

Differs from A051664 in the positions given by A013590, thus for the first time at n=105, where a(105) = 35, while A051664(105) = 33 as the 105th cyclotomic polynomial is the first one that has a coefficient other than 1, 0, or -1.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Eric Weisstein's World of Mathematics, Cyclotomic Polynomial
Wikipedia, Cyclotomic polynomial

Cf. A013590, A013595, A051258, A051664, A318886.

Array[Total@ Abs@ CoefficientList[Cyclotomic[#, x], x] &, 89] (* Michael De Vlieger, Sep 10 2018 *)

A318884(n) = vecsum(apply(abs,Vec(polcyclo(n)))); \\ Antti Karttunen, Sep 10 2018

A051259 Every 25th Fibocyclotomic number.

Original entry on oeis.org

7435, 6205, 93175555, 101508825, 354226959676926237175, 111479225, 4875225599199811387356000, 23415180442039155, 5354430843336297973830800, 354222736706770131025, 255273607757358738643290991867871250745195

Offset: 1

Antti Karttunen, Oct 24 1999

nonn

All divisible by fib(5) = 5.

Cf. A051258.

a(n) = A051258(25n)

A051260 Every 25th Fibocyclotomic number divided by 5.

Original entry on oeis.org

1487, 1241, 18635111, 20301765, 70845391935385247435, 22295845, 975045119839962277471200, 4683036088407831, 1070886168667259594766160, 70844547341354026205, 51054721551471747728658198373574250149039

Offset: 1

Antti Karttunen, Oct 24 1999

nonn

Cf. A051258, A051259.

a(n) = A051258(n)/5

A278158 Least number with the prime signature of the n-th Fibocyclotomic number, with a(6) = 0.

Original entry on oeis.org

1, 1, 2, 1, 2, 0, 12, 2, 6, 1, 6, 2, 24, 4, 2, 6, 60, 6, 60, 6, 6, 6, 6, 12, 6, 24, 210, 24, 30, 4, 420, 30, 30, 30, 20160, 24, 9240, 420, 12, 30, 60060, 30, 420, 30, 240, 30, 420, 210, 27720, 30, 60, 720, 420, 420, 6, 720, 2310, 30, 210, 30, 2042040, 4620, 24, 210, 7680, 60, 60060, 210, 6, 30240, 510510, 2160, 6486480, 840, 2310, 9240, 9240, 420, 60060, 210

Offset: 1

Antti Karttunen, Nov 18 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..550

Cf. A046523, A051258.

ps[n_] := Sort[Last /@ FactorInteger[n]]; g[n_] := Module[{cy = CoefficientList[ Cyclotomic[n, x], x]}, Total[Times @@@ Thread[{Fibonacci[Range[0, Length[cy] - 1]], cy}]]]; f[n_] := Block[{c = ps[g[n]]}, lng = Length@ c; Times @@ (Reverse[ Prime[ Range[ lng]]]^c)]; f[6] = 0; f[1] = f[2] = f[4] = f[10] = 1; Array[f, 70] (* Robert G. Wilson v, Nov 19 2016 *)
Table[Times @@ MapIndexed[(Prime@ First@ #2)^#1 &, #] &@ If[Length@ # == 1 && #[[1, 1]] == 1, {0}, Reverse@ Sort@ #[[All, -1]]] &@ FactorInteger[
Total[Times @@@ Thread[{Fibonacci[Range[0, Length@ # - 1]], #}]] &@ CoefficientList[Cyclotomic[n, x], x] + Boole[n == 0]], {n, 120}] (* Michael De Vlieger, Nov 21 2016, after Harvey P. Dale at A051258 *)

A051258(n) = my(P=polcyclo(n)); sum(i=1, poldegree(P), polcoeff(P, i)*fibonacci(i)); \\ From Charles R Greathouse IV, Jan 05 2013
A046523(n) = my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]) \\ From Charles R Greathouse IV, Aug 17 2011
A278158(n) = if(6==n,0,A046523(A051258(n)));
for(n=1, 550, write("b278158.txt", n, " ", A278158(n)));

(define (A278158 n) (let ((k (A051258 n))) (if (zero? k) k (A046523 k))))

a(n) = A046523(A051258(n)), except for n=6, a(6) = 0.

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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A054433 Numbers formed by interpreting the reduced residue set of every even number as a Zeckendorf Expansion.

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A063704 Cyclotomic polynomials Phi_n at x=phi divided by sqrt(5) and floored down (where phi = tau = (sqrt(5)+1)/2).

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A063706 Cyclotomic polynomials Phi_n at x=phi, divided by sqrt(5) and rounded to nearest integer (where phi = tau = (sqrt(5)+1)/2).

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A063708 Cyclotomic polynomials Phi_n at x=phi divided by sqrt(5) and ceiled up (where phi = tau = (sqrt(5)+1)/2).

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A192233 Constant term of the reduction of the n-th cyclotomic polynomial by x^2->x+1.

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A318884 a(n) is the sum of absolute values of the coefficients in the n-th cyclotomic polynomial.

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A051259 Every 25th Fibocyclotomic number.

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A051260 Every 25th Fibocyclotomic number divided by 5.

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