A133628 a(1)=1, a(n) = a(n-1) + (p-1)*p^(n/2-1) if n is even, else a(n) = a(n-1) + p^((n-1)/2), where p=4.

1, 4, 8, 20, 36, 84, 148, 340, 596, 1364, 2388, 5460, 9556, 21844, 38228, 87380, 152916, 349524, 611668, 1398100, 2446676, 5592404, 9786708, 22369620, 39146836, 89478484, 156587348, 357913940, 626349396, 1431655764, 2505397588

Offset: 1

Author

Hieronymus Fischer, Sep 19 2007

Keywords

• nonn

Comments

This is essentially a duplicate of A097164. - R. J. Mathar, Jun 08 2008

Partial sums of A084221.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..3000
• Index entries for linear recurrences with constant coefficients, signature (1, 4, -4).

Crossrefs

Sequences with similar recurrence rules: A027383(p=2), A087503(p=3), A133629(p=5).

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

Related sequences: A132666, A132667, A132668, A132669.

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

Programs

Magma

[4^Floor(n/2)+4^Floor((n+1)/2)/3-4/3: n in [1..40]]; // Vincenzo Librandi, Aug 17 2011

Maple

a[0]:=0:a[1]:=1:for n from 2 to 100 do a[n]:=4*a[n-2]+4 od: seq(a[n], n=1..31); # Zerinvary Lajos, Mar 17 2008

Mathematica

nxt[{n_,a_}]:={n+1,If[OddQ[n],a+3*4^((n+1)/2-1),a+4^(n/2)]}; Transpose[ NestList[ nxt,{1,1},30]][[2]] (* Harvey P. Dale, Mar 31 2013 *)

PARI

vector(40, n, (3*4^floor(n/2) + 4^floor((n+1)/2) - 4)/3) \\ G. C. Greubel, Nov 08 2018

Formula

a(n) = Sum_{k=1..n} A084221(k).

G.f.: x*(1+3*x)/((1-4*x^2)*(1-x)).

a(n) = (4/3)*(4^(n/2)-1) if n is even, otherwise a(n) = (4/3)*(7*4^((n-3)/2)-1).

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

a(n) = 4^floor(n/2) + 4^floor((n+1)/2)/3 - 4/3.

a(n) = A132668(a(n+1)) - 1.

a(n) = A132668(a(n-1) + 1) for n > 0.

A132668(a(n)) = a(n-1) + 1 for n > 0.