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

Author

Cino Hilliard, Feb 15 2003

Keywords

• easy
• nonn

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).

Crossrefs

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

Programs

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

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

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 *)

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)

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

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