A164038 Expansion of (5-19*x)/(1-10*x+23*x^2).

5, 31, 195, 1237, 7885, 50399, 322635, 2067173, 13251125, 84966271, 544886835, 3494644117, 22414043965, 143763624959, 922113238395, 5914569009893, 37937085615845, 243335768930911, 1560804720144675, 10011324516035797

Offset: 0

Author

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

Keywords

• nonn
• easy

Comments

Binomial transform of A161731 without initial 1. Fifth binomial transform of A164095. Inverse binomial transform of A164110.

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 (10,-23).

Crossrefs

Cf. A161731, A164095, A164110.

Programs

Magma

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((5+3*r)*(5+r)^n+(5-3*r)*(5-r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 10 2009

Mathematica

LinearRecurrence[{10,-23},{5,31},50] (* or *) CoefficientList[Series[(5 - 19*x)/(1 - 10*x + 23*x^2), {x,0,50}], x] (* G. C. Greubel, Sep 08 2017 *)

PARI

Vec((5-19*x)/(1-10*x+23*x^2)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

Formula

a(n) = 10*a(n-1) - 23*a(n-2) for n > 1; a(0) = 5, a(1) = 31.

G.f.: (5-19*x)/(1-10*x+23*x^2).

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

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

Extensions

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