A087503 a(n) = 3(a(n-2) + 1), with a(0) = 1, a(1) = 3.

1, 3, 6, 12, 21, 39, 66, 120, 201, 363, 606, 1092, 1821, 3279, 5466, 9840, 16401, 29523, 49206, 88572, 147621, 265719, 442866, 797160, 1328601, 2391483, 3985806, 7174452, 11957421, 21523359, 35872266, 64570080, 107616801, 193710243

Offset: 0

Author

Reinhard Zumkeller, Sep 11 2003

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..2000
• Index entries for linear recurrences with constant coefficients, signature (1, 3, -3).

Crossrefs

Sequences with similar recurrence rules: A027383 (p=2), A133628 (p=4), A133629 (p=5).

Other related sequences for different p: A016116 (p=2), A038754 (p=3), A084221 (p=4), A133632 (p=5).

See A133629 for general formulas with respect to the recurrence rule parameter p.

Related sequences: A132666, A132667, A132668, A132669.

Programs

Magma

[(3/2)*(3^Floor((n+1)/2)+3^Floor(n/2)-3^Floor((n-1)/2)-1): n in [0..40]]; // Vincenzo Librandi, Aug 16 2011

Maple

A087503 := proc(n)
    option remember;
    if n <=1 then
        op(n+1,[1,3]) ;
    else
        3*procname(n-2)+3 ;
    end if;
end proc:
seq(A087503(n),n=0..20) ; # R. J. Mathar, Sep 10 2021

Mathematica

RecurrenceTable[{a[0]==1,a[1]==3,a[n]==3(a[n-2]+1)},a,{n,40}] (* or *) LinearRecurrence[{1,3,-3},{1,3,6},40] (* Harvey P. Dale, Jan 01 2015 *)

Formula

a(n) = a(n-1) + A038754(n). (i.e., partial sums of A038754).

From Hieronymus Fischer, Sep 19 2007, formulas adjusted to offset, Dec 29 2012: (Start)

G.f.: g(x) = (1+2x)/((1-3x^2)(1-x)).

a(n) = (3/2)*(3^((n+1)/2)-1) if n is odd, else a(n) = (3/2)*(5*3^((n-2)/2)-1).

a(n) = (3/2)*(3^floor((n+1)/2) + 3^floor(n/2) - 3^floor((n-1)/2)-1).

a(n) = 3^floor((n+1)/2) + 3^floor((n+2)/2)/2 - 3/2.

a(n) = A132667(a(n+1)) - 1.

a(n) = A132667(a(n-1) + 1) for n > 0.

A132667(a(n)) = a(n-1) + 1 for n > 0.

Also numbers such that: a(0)=1, a(n) = a(n-1) + (p-1)*p^((n+1)/2 - 1) if n is odd, else a(n) = a(n-1) + p^(n/2), where p=3.

(End)

a(n) = A052993(n)+2*A052993(n-1). - R. J. Mathar, Sep 10 2021

Extensions

Additional comments from Hieronymus Fischer, Sep 19 2007

Edited by N. J. A. Sloane, May 04 2010. I merged two essentially identical entries with different offsets, so some of the formulas may need to be adjusted.

Formulas and MAGMA prog adjusted to offset 0 by Hieronymus Fischer, Dec 29 2012