A182228 a(n) = 3*a(n-2) - a(n-1) for n > 1, a(0) = 0, a(1) = 1.

0, 1, -1, 4, -7, 19, -40, 97, -217, 508, -1159, 2683, -6160, 14209, -32689, 75316, -173383, 399331, -919480, 2117473, -4875913, 11228332, -25856071, 59541067, -137109280, 315732481, -727060321, 1674257764, -3855438727, 8878212019, -20444528200, 47079164257, -108412748857

Offset: 0

Author

Alex Ratushnyak, Apr 19 2012

Keywords

• sign
• easy

Comments

This is A006130 with minus signs on every other term. - T. D. Noe, Apr 23 2012

Links

• G. C. Greubel, Table of n, a(n) for n = 0..200
• Index entries for linear recurrences with constant coefficients, signature (-1,3).

Crossrefs

Cf. A140167.

Programs

Magma

[n le 2 select n-1 else -Self(n-1) + 3*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Aug 31 2015

Mathematica

RecurrenceTable[{a[n]== - a[n-1] + 3*a[n-2], a[0]== 0, a[1]== 1}, a, {n,0,200}] (* G. C. Greubel, Aug 30 2015 *)
LinearRecurrence[{-1,3},{0,1},40] (* Harvey P. Dale, Oct 23 2016 *)

Python

prpr = 0
prev = 1
for i in range(2,55):
    current = prpr*3-prev
    print (current, end=',')
    prpr = prev
    prev = current

Formula

From R. J. Mathar, Apr 22 2013: (Start)

a(n) = -A140167(n).

G.f.: x/(1 + x - 3*x^2). (End)

G.f.: 1 - Q(0), where Q(k) = 1 + 3*x^2 - (k+2)*x + x*(k+1 - 3*x)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 06 2013

E.g.f.: (-1/sqrt(13))*(exp(-(1+sqrt(13))*x/2) - exp(-(1-sqrt(13))*x/2)). - G. C. Greubel, Aug 30 2015

a(n) = (-(-1)^n/sqrt(13))*(((1 + sqrt(13))/2)^n - ((1 - sqrt(13))/2)^n). - Taras Goy, Jul 17 2018