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A082840 a(n) = 4*a(n-1) - a(n-2) + 3, with a(0) = -1, a(1) = 1. 5
-1, 1, 8, 34, 131, 493, 1844, 6886, 25703, 95929, 358016, 1336138, 4986539, 18610021, 69453548, 259204174, 967363151, 3610248433, 13473630584, 50284273906, 187663465043, 700369586269, 2613814880036, 9754889933878, 36405744855479, 135868089488041 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Apart from the initial -1, these are the numbers k such that the triangular number k*(k + 1)/2 is the sum of three consecutive triangular numbers - see A129803. - Brian Nowell, Nov 03 2009

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (5,-5,1).

FORMULA

a(n) = A001571(n) - 1. - N. J. A. Sloane, Nov 03 2009

G.f.: -(1 -6*x +2*x^2)/((1 - x)*(1 - 4*x + x^2)).

a(n) = -3/2 + (1/12)*( (a -2*b +5)*a^n + (b -2*a +5)*b^n ), with a = 2 + sqrt(3), b = 2 - sqrt(3):.

a(n) = -3/2 + (3/4)*A003500(n) - (1/4)*A003500(n-1).

a(n) = (1/2)*(A001834(n) - 3).

E.g.f.: ((1 + sqrt(3))*exp((2 + sqrt(3))*x) + (1 - sqrt(3))*exp((2 - sqrt(3))*x) - 6*exp(x))/4. - Franck Maminirina Ramaharo, Nov 12 2018

MATHEMATICA

CoefficientList[Series[(-1+6x-2x^2)/((1-x)(1-4x+x^2)), {x, 0, 30}], x] (* Vincenzo Librandi, Apr 15 2014 *)

LinearRecurrence[{5, -5, 1}, {-1, 1, 8}, 30] (* G. C. Greubel, Feb 25 2019 *)

PROG

(PARI) is(n)=ispolygonal(3/2*n*(n+1)+4, 3) || n==-1 \\ Charles R Greathouse IV, Apr 14 2014

(PARI) my(x='x+O('x^30)); Vec(-(1-6*x+2*x^2)/((1-x)*(1-4*x+x^2))) \\ G. C. Greubel, Feb 25 2019

(MAGMA) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( -(1-6*x+2*x^2)/((1-x)*(1-4*x+x^2)) )); // G. C. Greubel, Feb 25 2019

(Sage) (-(1-6*x+2*x^2)/((1-x)*(1-4*x+x^2))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 25 2019

(GAP) a:=[-1, 1, 8];; for n in [4..30] do a[n]:=5*a[n-1]-5*a[n-2]+a[n-3]; od; a; # G. C. Greubel, Feb 25 2019

CROSSREFS

Cf. A001571, A071954.

Sequence in context: A294844 A302083 A124843 * A224240 A101644 A126395

Adjacent sequences:  A082837 A082838 A082839 * A082841 A082842 A082843

KEYWORD

easy,sign

AUTHOR

Mario Catalani (mario.catalani(AT)unito.it), Apr 14 2003

STATUS

approved

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Last modified July 22 07:23 EDT 2019. Contains 325216 sequences. (Running on oeis4.)