A079362 Sequence of sums of alternating powers of 3.

1, 4, 5, 14, 17, 44, 53, 134, 161, 404, 485, 1214, 1457, 3644, 4373, 10934, 13121, 32804, 39365, 98414, 118097, 295244, 354293, 885734, 1062881, 2657204, 3188645, 7971614, 9565937, 23914844, 28697813, 71744534, 86093441, 215233604

Offset

1,2

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (1,3,-3).

Formula

G.f.: x*(1+3*x-2*x^2)/((1-x)*(1-3*x^2)). - Michael Somos, Feb 18 2003

For n >= 1, a(2n-1) = (2/3)*3^n - 1, a(2n) = (5/3)*3^n - 1. - Benoit Cloitre, Feb 16 2003

Maple

a[1]:=1:a[2]:=4:for n from 3 to 100 do a[n]:=3*a[n-2]+2 od: seq(a[n], n=1..33); # Zerinvary Lajos, Mar 17 2008

Mathematica

LinearRecurrence[{1, 3, -3}, {1, 4, 5}, 40] (* Harvey P. Dale, Oct 18 2016 *)

Prog

(PARI) a(n)=if(n<1, 0, 1+sum(k=2, n, 3^((k\2)-(k%2))))
(PARI) a(n)=if(n<0, 0, (5/3-3*n%2)*2^ceil(n/2)-1)
(Magma) I:=[1, 4, 5]; [n le 3 select I[n] else Self(n-1) +3*Self(n-2) -3*Self(n-3): n in [1..40]]; // G. C. Greubel, Aug 07 2019
(Sage)
@CachedFunction
def a(n):
    if (n==0): return 1
    elif (1<=n<=2): return n+3
    else: return a(n-1) + 3*a(n-2) - 3*a(n-3)
[a(n) for n in (0..40)] # G. C. Greubel, Aug 07 2019
(GAP) a:=[1, 4, 5];; for n in [4..30] do a[n]:=a[n-1]+3*a[n-2]-3*a[n-3]; od; a; # G. C. Greubel, Aug 07 2019

Crossrefs

Cf. A079360, A079363, A028242, A048473 (bisection).

Sequence in context: A191142 A049770 A262903 * A222364 A222372 A302348

Adjacent sequences: A079359 A079360 A079361 * A079363 A079364 A079365

Keyword

easy,nonn

Author

Cino Hilliard, Feb 15 2003

Status

approved