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Whitney transform of 3^n.
1

%I #23 Sep 08 2022 08:45:17

%S 1,4,16,61,232,880,3337,12652,47968,181861,689488,2614048,9910609,

%T 37573972,142453744,540083149,2047610680,7763081488,29432076505,

%U 111585473980,423052651456,1603914376309,6080901083296,23054446378816

%N Whitney transform of 3^n.

%C Partial sums of A030195. The Whitney transform maps the sequence with g.f. g(x) to that with g.f. (1/(1-x))g(x(1+x)).

%H Vincenzo Librandi, <a href="/A103820/b103820.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,0,-3).

%F G.f.: 1/((1-x)(1-3x-3x^2));

%F a(n) = 4a(n-1) - 3a(n-3);

%F a(n) = Sum_{k=0..n} (Sum_{i=0..n} C(k, i-k))*3^k.

%F a(n) = 3(a(n-1) + a(n-2)) + 1, n > 1. [_Gary Detlefs_, Jun 21 2010]

%p a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=3*a[n-1]+3*a[n-2]+1 od: seq(a[n], n=1..33); # _Zerinvary Lajos_, Dec 14 2008

%t Join[{a=0,b=1},Table[c=3*b+3*a+1;a=b;b=c,{n,100}]] (* _Vladimir Joseph Stephan Orlovsky_, Jan 17 2011 *)

%t LinearRecurrence[{4, 0, -3}, {1, 4, 16}, 40] (* _Vincenzo Librandi_, Aug 18 2017 *)

%o (Magma) I:=[1,4,16]; [n le 3 select I[n] else 4*Self(n-1)-3*Self(n-3): n in [1..30]]; // _Vincenzo Librandi_, Aug 18 2017

%Y Cf. A004070, A030195.

%Y Equals (A108306(n+1) - 1)/5.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Feb 16 2005