login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 56th year, we are closing in on 350,000 sequences, and we’ve crossed 9,700 citations (which often say “discovered thanks to the OEIS”).

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A192356 Coefficients of x in the reduction of the polynomial p(n,x) = ((x+2)^n + (x-2)^n)/2 by x^2->x+2. 2
0, 1, 1, 15, 29, 211, 561, 3095, 9829, 46971, 164921, 728575, 2707629, 11450531, 43942081, 181348455, 708653429, 2884834891, 11388676041, 46006694735, 182670807229, 734751144051, 2926800830801, 11743814559415, 46865424529029, 187791199242011, 750176293590361 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

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

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (2,11,-12).

FORMULA

Empirical g.f.: x^2*(1-x+2*x^2)/((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 * J_{n-2*k}, where J_{n} = A001045(n) are the Jacobsthal numbers.

a(n) = (4^n - (-3)^n - 1 + 2^n*delta(n,0))/6, with delta(n,0) = 1 if n=0, 0 otherwise. (End)

MATHEMATICA

(See A192355.)

Join[{0}, Table[(4^n - (-3)^n - 1)/6, {n, 1, 50}]] (* G. C. Greubel, Oct 20 2018 *)

PROG

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

(MAGMA) [0] cat [(4^n - (-3)^n - 1)/6: n in [1..50]]; // G. C. Greubel, Oct 20 2018

CROSSREFS

Cf. A192232, A192355.

Sequence in context: A146427 A202512 A014095 * A196184 A201136 A200896

Adjacent sequences:  A192353 A192354 A192355 * A192357 A192358 A192359

KEYWORD

nonn

AUTHOR

Clark Kimberling, Jun 29 2011

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 7 20:15 EST 2021. Contains 349587 sequences. (Running on oeis4.)