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A141516 The main diagonal of the array of A141425 and its higher order differences.

%I A141516 #18 Nov 23 2022 19:07:41
%S A141516 1,2,1,-7,-23,-1,7,-103,-251,-133,-149,-1387,-3143,-3001,-4913,-19663,
%T A141516 -42611,-55693,-101549,-291667,-612863,-960001,-1831433,-4460023,
%U A141516 -9185771,-15980053,-31162949,-69500347,-141392183,-261261001
%N A141516 The main diagonal of the array of A141425 and its higher order differences.
%C A141516 The sequence A141425 and higher order differences in subsequent rows starts (see A141533):
%C A141516 1, 2, 4,  5,  7, 8, 1, 2, 4,  5,  7, 8, 1, 2, 4,...
%C A141516 1, 2, 1,  2,  1,-7, 1, 2, 1,  2,  1,-7, 1, 2, 1, 2,...
%C A141516 1,-1, 1, -1, -8, 8, 1,-1, 1, -1, -8, 8, 1, -1,..
%C A141516 -2, 2,-2, -7, 16,-7,-2, 2,-2, -7, 16,-7, -2,..
%C A141516 4,-4,-5, 23,-23, 5, 4,-4,-5, 23,-23, 5, 4,..
%C A141516 -8,-1,28,-46, 28,-1,-8,-1,28,-46, 28,-1,..
%C A141516 Reading downwards the main diagonal of this array defines the sequence.
%H A141516 Vincenzo Librandi, <a href="/A141516/b141516.txt">Table of n, a(n) for n = 0..200</a>
%H A141516 <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (1,-1,3,6)
%F A141516 a(n) = ( -3*(-1)^n -2^n +3*(-1)^(floor((n-1)/2))*A108411(n) )/2, n>0. - _R. J. Mathar_, Mar 08 2011
%F A141516 a(2n)+a(2n+1)= -A002023(n-1) = -3*A081294(n), n>0.
%F A141516 a(4n)+a(4n+1)+a(4n+2)+a(4n+3) = -120*16^(n-1), n>0.
%F A141516 a(4n+2)+a(4n+3)+a(4n+4)+a(4n+5) = -30*A001025(n).
%F A141516 G.f. x*(-2+x+6*x^2+21*x^3) / ( (2*x-1)*(1+x)*(3*x^2+1) ). - _R. J. Mathar_, Mar 08 2011
%p A141516 A108411 := proc(n) 3^floor(n/2) ; end proc:
%p A141516 A141516 := proc(n) if n = 0 then 1; else (-3*(-1)^n-2^n+3*(-1)^(floor((n-1)/2))*A108411(n))/2 ; end if; end proc: # _R. J. Mathar_, Mar 08 2011
%t A141516 LinearRecurrence[{1,-1,3,6},{1,2,1,-7,-23},30] (* _Harvey P. Dale_, Nov 23 2022 *)
%K A141516 sign,easy
%O A141516 0,2
%A A141516 _Paul Curtz_, Aug 11 2008