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, 322850406, 581130732

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..2000

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

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

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}] (* Harvey P. Dale, Jan 01 2015 *)
(* Alternative: *)
LinearRecurrence[{1, 3, -3}, {1, 3, 6}, 40] (* Harvey P. Dale, Jan 01 2015 *)

Prog

(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
(Python)
def A087503(n): return (3+((n+1&1)<<1))*3**(n+1>>1)-3>>1 # Chai Wah Wu, Sep 02 2025

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.

Sequence in context: A396091 A006330 A293636 * A092176 A000991 A345028

Adjacent sequences: A087500 A087501 A087502 * A087504 A087505 A087506

Keyword

nonn,easy,changed

Author

Reinhard Zumkeller, Sep 11 2003

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

Status

approved