%I #22 Sep 08 2022 08:45:32

%S 1,4,8,20,36,84,148,340,596,1364,2388,5460,9556,21844,38228,87380,

%T 152916,349524,611668,1398100,2446676,5592404,9786708,22369620,

%U 39146836,89478484,156587348,357913940,626349396,1431655764,2505397588

%N 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.

%C This is essentially a duplicate of A097164. - _R. J. Mathar_, Jun 08 2008

%C Partial sums of A084221.

%H Vincenzo Librandi, <a href="/A133628/b133628.txt">Table of n, a(n) for n = 1..3000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1, 4, -4).

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

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

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

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

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

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

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

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

%p 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

%t 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 *)

%o (Magma) [4^Floor(n/2)+4^Floor((n+1)/2)/3-4/3: n in [1..40]]; // _Vincenzo Librandi_, Aug 17 2011

%o (PARI) vector(40, n, (3*4^floor(n/2) + 4^floor((n+1)/2) - 4)/3) \\ _G. C. Greubel_, Nov 08 2018

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

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

%Y Related sequences: A132666, A132667, A132668, A132669.

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

%K nonn

%O 1,2

%A _Hieronymus Fischer_, Sep 19 2007