A192352 -id:A192352 - cp's OEIS frontend

A049602 a(n) = (Fibonacci(2n)-(-1)^nFibonacci(n))/2.

0, 1, 1, 5, 9, 30, 68, 195, 483, 1309, 3355, 8900, 23112, 60813, 158717, 416325, 1088661, 2852242, 7463884, 19546175, 51163695, 133962621, 350695511, 918170280, 2403740304, 6293172025, 16475579353, 43133883845, 112925557953

Offset: 0

Clark Kimberling

A049602 gives the coefficients of x in the reduction of the polynomial p(n,x)=(1/2)((x+1)^n+(x-1)^n) by x^2->x+1. For the constant terms, see A192352. For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232. - Clark Kimberling, Jun 29 2011

Index entries for linear recurrences with constant coefficients, signature (2,3,-4,1).

Cf. A049601.

LinearRecurrence[{2,3,-4,1},{0,1,1,5},30] (* Harvey P. Dale, Jul 07 2017 *)

a(n)=(fibonacci(2*n)-(-1)^n*fibonacci(n))/2 \\ Charles R Greathouse IV, Oct 07 2015

a(n)=Sum{T(2i+1, n-2i-1): i=0, 1, ..., [ (n+1)/2 ]}, array T as in A049600.
Cosh transform of Fibonacci numbers A000045 (or mean of binomial and inverse binomial transforms of A000045). E.g.f.: cosh(x)(2/sqrt(5))*exp(x/2)*sinh(sqrt(5)*x/2). - Paul Barry, May 10 2003
a(n)=sum{k=0..floor(n/2), C(n, 2k)Fib(n-2k)}; - Paul Barry, May 01 2005
a(n)=2a(n-1)+3a(n-2)-4a(n-3)+a(n-4). - Paul Curtz, Jun 16 2008
G.f.: x(1-x)/((1+x-x^2)(1-3x+x^2)); a(n)=sum{k=0..n-1, (-1)^(n-k+1)*F(2k+2)*F(n-k+1)}; - Paul Barry, Jul 11 2008

Simpler description from Vladeta Jovovic and Thomas Baruchel, Aug 24 2004

More terms from Paul Curtz, Jun 16 2008

A192373 Constant term in the reduction of the polynomial p(n,x) defined at A162517 and below in Comments.

Original entry on oeis.org

1, 0, 7, 8, 77, 192, 1043, 3472, 15529, 57792, 240655, 934808, 3789653, 14963328, 60048443, 238578976, 953755537, 3798340224, 15162325975, 60438310184, 241126038941, 961476161856, 3835121918243, 15294304429744, 61000836720313, 243280700771904

Offset: 1

Clark Kimberling, Jun 29 2011

nonn

The polynomials are given by p(n,x)=((x+d)^n-(x-d)^n)/(2d), where d=sqrt(x+4).

For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.

			The first five polynomials p(n,x) and their reductions are as follows:
p(0,x)=1 -> 1
p(1,x)=2x -> 2x
p(2,x)=4+x+3x^2 -> 7+4x
p(3,x)=16x+4x^2+4x^3 -> 8+28x
p(4,x)=16+8x+41x^2+10x^3+5x^4 -> 77+84x.
From these, read A192352=(1,0,7,8,77,...) and A049602=(0,2,4,28,84,...).

Cf. A192232, A192374, A192375, A162517.

q[x_] := x + 1; d = Sqrt[x + 4];
p[n_, x_] := ((x + d)^n - (x - d)^n )/(2 d) (* A162517 *)
Table[Expand[p[n, x]], {n, 1, 6}]
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, 1, 30}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]  (* A192373 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]  (* A192374 *)
Table[Coefficient[Part[t, n]/2, x, 1], {n, 1, 30}] (* A192375 *)

Conjecture: a(n) = 2*a(n-1)+10*a(n-2)-6*a(n-3)-9*a(n-4). G.f.: -x*(x+1)*(3*x-1) / (9*x^4+6*x^3-10*x^2-2*x+1). - Colin Barker, May 09 2014

A192353 Constant term of the reduction of the polynomial p(n,x)=(1/2)((x+2)^n+(x-2)^n) by x^2->x+1.

Original entry on oeis.org

1, 0, 5, 1, 42, 43, 429, 820, 4861, 12597, 58598, 177859, 732825, 2417416, 9358677, 32256553, 120902914, 426440955, 1571649221, 5610955132, 20497829133, 73645557469, 267803779710, 965384509651, 3502058316337, 12646311635088, 45818284122149

Offset: 1

Clark Kimberling, Jun 29 2011

nonn

For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.

			(See A192352 for a related example.)

Cf. A192232, A192354, A192352.

q[x_] := x + 1; d = 2;
p[n_, x_] := ((x + d)^n + (x - d)^n )/2 (* similar to polynomials defined at A161516 *)
Table[Expand[p[n, x]], {n, 0, 6}]
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}](* A192353 *)
Table[Coefficient[Part[t,n],x,1], {n,1,30}]  (* A192354 *)

Empirical G.f.: x*(x^3-4*x^2-2*x+1)/((x^2+3*x+1)*(5*x^2-5*x+1)). [Colin Barker, Sep 11 2012]

A192355 Constant term of the reduction of the polynomial p(n,x)=(1/2)((x+2)^n+(x-2)^n) by x^2->x+2.

Original entry on oeis.org

1, 0, 6, 2, 70, 90, 926, 2002, 13110, 37130, 194446, 640002, 2973350, 10653370, 46333566, 174174002, 730176790, 2820264810, 11582386286, 45425564002, 184414199430, 729520967450, 2942491360606, 11696742970002, 47006639297270, 187367554937290

Offset: 1

Clark Kimberling, Jun 29 2011

nonn

For an introduction to reductions of polynomials by substitutions such as x^2->x+2, see A192232.

Direct sums can be obtained for A192355 and A192356 in the following way. The polynomials p_{n}(x) can be given in series form by p_{n}(x) = Sum_{k=0..floor(n/2)} binomial(n, 2*k)*4*k*x^(n-2*k). For the reduction x^2 -> x+2 then the general form can be seen as x^n -> J_{n}*x + phi_{n}, where J_{n} = A001045(n) are the Jacobsthal numbers and phi_{n} = A078008. The reduction of p_{n}(x) now takes the form p_{n}(x) = x * Sum_{k=0..floor(n/2)} binomial(n,2*k)*4^k*J_{n-2*k} + Sum_{k=0..floor(n/2)} binomial(n,2*k)*4^k*phi_{n-2*k}. Evaluating the series leads to p_{n}(x) = x * (4^n - (-3)^n - 1 + 2^n*delta(n,0))/6 + (4^n + 2*(-3)^n + 2 + 2^n*delta(n,0))/6, where delta(n,k) is the Kronecker delta. - G. C. Greubel, Oct 29 2018

			(See A192352 for a related example.)

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,11,-12).

Cf. A192232, A192356.

[1] cat [(4^n + 2*(-3)^n + 2)/6: n in [1..50]]; // G. C. Greubel, Oct 20 2018

q[x_] := x + 2; d = 2;
p[n_, x_] := ((x + d)^n + (x - d)^n )/2 (* similar to polynomials defined at A161516 *)
Table[Expand[p[n, x]], {n, 0, 6}]
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}]
  (* A192355 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]
  (* A192356 *)
Join[{1}, Table[(4^n + 2*(-3)^n + 2)/6, {n, 1, 50}]] (* G. C. Greubel, Oct 20 2018 *)

for(n=0, 50, print1(if(n==0, 1, (4^n + 2*(-3)^n + 2)/6), ", ")) \\ G. C. Greubel, Oct 20 2018

Empirical G.f.: x*(2*x^3-5*x^2-2*x+1)/((x-1)*(3*x+1)*(4*x-1)). - Colin Barker, Sep 12 2012
From G. C. Greubel, Oct 28 2018: (Start)
a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2*k) * 4^k * phi_{n-2*k}, where phi_{n} = A078008(n).a(n) = (4^n + 2*(-3)^n + 2 + 2^n*delta(n,0))/6, with delta(n,0) = 1 if n=0, 0 else. (End)

cp's OEIS Frontend

A049602 a(n) = (Fibonacci(2*n)-(-1)^n*Fibonacci(n))/2.

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A192373 Constant term in the reduction of the polynomial p(n,x) defined at A162517 and below in Comments.

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A192353 Constant term of the reduction of the polynomial p(n,x)=(1/2)((x+2)^n+(x-2)^n) by x^2->x+1.

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A192355 Constant term of the reduction of the polynomial p(n,x)=(1/2)((x+2)^n+(x-2)^n) by x^2->x+2.

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A049602 a(n) = (Fibonacci(2n)-(-1)^nFibonacci(n))/2.