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A157456 Expansion of x*(1-x) / ( 1 - 16*x + x^2 ). 21
1, 15, 239, 3809, 60705, 967471, 15418831, 245733825, 3916322369, 62415424079, 994730462895, 15853271982241, 252657621252961, 4026668668065135, 64174041067789199, 1022757988416562049, 16299953773597203585, 259776502389138695311, 4140124084452621921391 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Previous name was: The general form of the recurrences are the a(j), b(j) and n(j) solutions of the 2 equations problem: C*n(j)+1 = a(j)A^2; (C+2)*n(j)+1 = b(j)A^2; with C, n(j), a(j), b(j) positive integer elements. The example above is the a(j) recurrence for C=7. The general form above, with C=7, is 7*n(j)+1=a(j)A^2 and 9*n(j)+1=b(j)A^2, leading to the single equation 9*a(j)A^2 - 7b(j)A^2 = 2. The single equation in general form is (C+2)*a(t)A^2 - C*b(t)A^2 = 2.

.C a-seque b-seque n-seque

01 A001835 A001834 A045899

02 A001653 A002315 A098602

03 A070997 A057080 -------

04 A072256 A054320 -------

05 A077417 A077416 -------

06 A001570 A028230 -------

07 ------- ------- -------

08 A007805 A049629 -------

09 A075839 A083043 -------

10 A157014 A133283 -------

11 ------- ------- -------

12 A153111 ------- -------

For all higher values of C, no sequences could be found in the list.

Positive values of x (or y) satisfying x^2 - 16xy + y^2 + 14 = 0. - Colin Barker, Feb 11 2014

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..200

J.-C. Novelli, J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014.

Index entries for linear recurrences with constant coefficients, signature (16,-1).

FORMULA

The a(j) recurrence is a(1)=1, a(2)=2*C+1, a(t+2)=(2*C+2)*a(t+1)-a(t), resulting in a(j) terms 1, 2*C+1, 4*CA^2 +6*C+1;

the b(j) recurrence is b(1)=1, b(2)=2*C+3, b(t+2)=(2*C+2)*b(t+1)-b(t), resulting in b(j) terms 1, 2*C+3, 4*CA^2 +10*C+5;

the n(j) recurrence is n(1)=0, n(2)=4*C+4, n(3)=(4*CA^2 +8*C+3)*n(2), n(t+3)=(4*CA^2 +8*C+3)*(n(t+2)-n(t+1)) + n(t), resulting in n(j) terms 0, 4*C+4, 16*CA^3 +48*CA^2 +44*C+12.

Other forms of the recurrence are

n(t+3) = (4*CA^2 + 8*C+2)*n(t+2) - n(t+1) + n(2)

or

n(t)=(b(t)A^2 - a(t)A^2 )/2.

G.f.: -x*(-1+x) / ( 1-16*x+x^2 ). - R. J. Mathar, Oct 31 2011

a(n) = 16*a(n-1)-a(n-2). - Colin Barker, Feb 11 2014

a(n) = (1/18)*(9-sqrt(63))*(1+(8+sqrt(63))^(2*n-1))/(8+sqrt(63))^(n-1). [Bruno Berselli, Feb 25 2014]

a(n) = sqrt(2+(8-3*sqrt(7))^(1+2*n)+(8+3*sqrt(7))^(1+2*n))/(3*sqrt(2)). - Gerry Martens, Jun 06 2015

a(n) = A077412(n-1) - A077412(n-2). - R. J. Mathar, Feb 05 2020

MAPLE

f:= gfun:-rectoproc({a(n)=16*a(n-1)-a(n-2), a(1)=1, a(2)=15}, a(n), remember):

map(f, [$1..30]); # Robert Israel, Jul 07 2015

MATHEMATICA

CoefficientList[Series[(1 - x)/(1 - 16 x + x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 12 2014 *)

LinearRecurrence[{16, -1}, {1, 15}, 20] (* Harvey P. Dale, Sep 17 2019 *)

PROG

(MAGMA) I:=[1, 15]; [n le 2 select I[n] else 16*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Feb 12 2014

CROSSREFS

Cf. A159678.

Cf. similar sequences listed in A238379.

Sequence in context: A209118 A093745 A071811 * A343527 A097262 A158557

Adjacent sequences:  A157453 A157454 A157455 * A157457 A157458 A157459

KEYWORD

nonn,easy,uned

AUTHOR

Paul Weisenhorn, Mar 01 2009

EXTENSIONS

New name (using the g.f. by R. J. Mathar) from Joerg Arndt, Jun 06 2015

STATUS

approved

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Last modified December 6 15:57 EST 2021. Contains 349565 sequences. (Running on oeis4.)