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A101463 Expansion of g.f. (x^3+x^2+2*x+1)/(x^4+5*x^2+1).

%I A101463 #25 Mar 14 2024 15:20:39
%S A101463 1,2,-4,-9,19,43,-91,-206,436,987,-2089,-4729,10009,22658,-47956,
%T A101463 -108561,229771,520147,-1100899,-2492174,5274724,11940723,-25272721,
%U A101463 -57211441,121088881,274116482,-580171684,-1313370969,2779769539,6292738363,-13318676011
%N A101463 Expansion of g.f. (x^3+x^2+2*x+1)/(x^4+5*x^2+1).
%C A101463 A floretion-generated sequence relating to Pythagoras' theorem generalized.
%C A101463 Floretion Algebra Multiplication Program. FAMP code: em[J* ]sigcycseq[ + .75'i + .5'k + .25i' + .5j' + .5k' - .25'ii' + .25'jj' - .25'kk' - .75'jk' + .5'ki' - .25'kj' + .25e]
%D A101463 F. A. Haight, On a generalization of Pythagoras' theorem, pp. 73-77 of J. C. Butcher, editor, A Spectrum of Mathematics. Auckland University Press, 1971.
%H A101463 James A. Sellers, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL5/Sellers/sellers4.html">Domino Tilings and Products of Fibonacci and Pell Numbers</a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.2.
%H A101463 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,-5,0,-1)
%F A101463 Let b(1)=1, b(2)=2, b(3)=4 and b(n)=(b(n-1)*b(n-2)+(3+(-1)^n)/2)/b(n-3) then b(n)=abs(a(n)) - _Benoit Cloitre_, Mar 03 2007
%F A101463 a(n) = -5*a(n-2)-a(n-4), n>3. [_Harvey P. Dale_, Apr 15 2012]
%F A101463 G.f.: ( 1+2*x+x^2+x^3 ) / ( 1+5*x^2+x^4 ). - _R. J. Mathar_, Jun 18 2014
%F A101463 a(n) = -3a(n-1)+2a(n-2) if n even. a(n) = (5*a(n-1)+a(n-2))/2 if n odd. - _R. J. Mathar_, Jun 18 2014
%t A101463 CoefficientList[Series[(x^3+x^2+2x+1)/(x^4+5x^2+1),{x,0,30}],x] (* or *) LinearRecurrence[{0,-5,0,-1},{1,2,-4,-9},31] (* _Harvey P. Dale_, Apr 15 2012 *)
%Y A101463 Elements of even index in the sequence gives A004253. Elements of odd index in the sequence gives A002310.
%Y A101463 Cf. A004253, A002310.
%K A101463 easy,sign
%O A101463 0,2
%A A101463 _Creighton Dement_, Jan 20 2005