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A127546 a(n) = F(n)^2 + F(n+1)^2 + F(n+2)^2, where F(n) denotes the n-th Fibonacci number.

%I A127546 #31 Oct 11 2023 04:36:07
%S A127546 2,6,14,38,98,258,674,1766,4622,12102,31682,82946,217154,568518,
%T A127546 1488398,3896678,10201634,26708226,69923042,183060902,479259662,
%U A127546 1254718086,3284894594,8599965698,22515002498,58945041798,154320122894,404015326886,1057725857762
%N A127546 a(n) = F(n)^2 + F(n+1)^2 + F(n+2)^2, where F(n) denotes the n-th Fibonacci number.
%C A127546 The following conjecture, if not already well-known, is probably easy to prove: a(n) = 3a(n-1)-a(n-2)-2(-1)^n, for n=4,5,6,... . (This has been verified up to n=1000.)
%H A127546 Harvey P. Dale, <a href="/A127546/b127546.txt">Table of n, a(n) for n = 0..1000</a>
%H A127546 Shalosh B. Ekhad and Doron Zeilberger, <a href="http://arxiv.org/abs/1206.4864">Automatic Counting of Tilings of Skinny Plane Regions</a>, arXiv preprint arXiv:1206.4864 [math.CO], 2012.
%H A127546 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-1).
%F A127546 a(n) = 2*A061646(n+1) = 4*F(n+1)^2-2*(-1)^(n+1). - _Emeric Deutsch_, Apr 04 2007; _Gary Detlefs_, Nov 27 2010
%F A127546 a(n) = 2*(F(n)^2+F(n+1)^2+F(n)*F(n+1)). - _Emeric Deutsch_, Apr 04 2007
%F A127546 G.f.: 2(1+x-x^2)/((1+x)(1-3x+x^2)). - _R. J. Mathar_, Nov 25 2008
%e A127546 a(2)=14 because F(2)^2+F(3)^2+F(4)^2=1+4+9=14.
%p A127546 with(combinat): a:=n->fibonacci(n)^2+fibonacci(n+1)^2+fibonacci(n+2)^2: seq(a(n),n=0..32); # _Emeric Deutsch_, Apr 04 2007
%p A127546 A000045 := proc(n) combinat[fibonacci](n) ; end: A127546 := proc(n) add( A000045(i+1)^2,i=n..n+2) ; end: for n from 1 to 33 do printf("%d, ",A127546(n)) ; od ; # _R. J. Mathar_, Apr 03 2007
%p A127546 with(combinat): seq(4*fibonacci(n+1)^2-2*(-1)^n, n=0..29)
%t A127546 Total/@(Partition[Fibonacci[Range[0,30]],3,1]^2) (* _Harvey P. Dale_, Oct 20 2011 *)
%o A127546 (PARI) for(n=0,10,print1(4*fibonacci(n+1)^2-2*(-1)^n,", "))
%Y A127546 Cf. A061646.
%K A127546 nonn
%O A127546 0,1
%A A127546 _Simone Severini_, Apr 01 2007
%E A127546 Edited and extended by _R. J. Mathar_, _Emeric Deutsch_ and _John W. Layman_, Apr 09 2007