A164037 Expansion of (5-9*x)/(1-6*x+7*x^2).

5, 21, 91, 399, 1757, 7749, 34195, 150927, 666197, 2940693, 12980779, 57299823, 252933485, 1116502149, 4928478499, 21755355951, 96032786213, 423909225621, 1871225850235, 8259990522063, 36461362180733, 160948239429957

Offset: 0

Author

Al Hakanson (hawkuu(AT)gmail.com), Aug 08 2009

Keywords

• nonn
• easy

Comments

Binomial transform of A161941 without initial 2. Third binomial transform of A164095. Inverse binomial transform of A161731 without initial 1.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..100 from Vincenzo Librandi)
• Index entries for linear recurrences with constant coefficients, signature (6,-7).

Crossrefs

Cf. A161941, A164095 (5, 6, 10, 12, 20, 24, ...), A161731.

Programs

Magma

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((5+3*r)*(3+r)^n+(5-3*r)*(3-r)^n)/2: n in [0..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 10 2009

Mathematica

CoefficientList[Series[(5-9x)/(1-6x+7x^2),{x,0,30}],x] (* or *) LinearRecurrence[{6,-7},{5,21},30] (* Harvey P. Dale, Apr 27 2017 *)

PARI

Vec((5-9*x)/(1-6*x+7*x^2)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

Formula

a(n) = 6*a(n-1)-7*a(n-2) for n > 1; a(0) = 5, a(1) = 21.

G.f.: (5-9*x)/(1-6*x+7*x^2).

a(n) = ((5+3*sqrt(2))*(3+sqrt(2))^n+(5-3*sqrt(2))*(3-sqrt(2))^n)/2.

E.g.f.: (5*cosh(sqrt(2)*x) + 3*sqrt(2)*sinh(sqrt(2)*x))*exp(3*x). - G. C. Greubel, Sep 08 2017

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 10 2009