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A122595 Expansion of x/(1 - 3*x + x^2 + x^3 - x^4). 2

%I

%S 1,3,8,20,50,125,313,784,1964,4920,12325,30875,77344,193752,485362,

%T 1215865,3045825,7630000,19113672,47881056,119945321,300471235,

%U 752701000,1885567500,4723475586,11832629493,29641546393,74254101600

%N Expansion of x/(1 - 3*x + x^2 + x^3 - x^4).

%D Jay Kappraff, Beyond Measure, A Guided Tour Through Nature, Myth and Number, World Scientific, 2002.

%H Vincenzo Librandi, <a href="/A122595/b122595.txt">Table of n, a(n) for n = 1..1000</a>

%H P. Steinbach, <a href="http://www.jstor.org/stable/2691048">Golden fields: a case for the heptagon</a>, Math. Mag. 70 (1997), no. 1, 22-31.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-1,1).

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

%F a(n) ~ 0.50556... * 2.505068...^n. - _Charles R Greathouse IV_, Aug 06 2012

%F a(n) = 3*a(n-1) - a(n-2) - a(n-3) + a(n-4) for n>4. - _Wesley Ivan Hurt_, Sep 18 2015

%e x + 3*x^2 + 8*x^3 + 20*x^4 + 50*x^5 + 125*x^6 + 313*x^7 + 784*x^8 + 1964*x^9 + ...

%t nn = 30; CoefficientList[Series[x/(1 - 3*x + x^2 + x^3 - x^4), {x, 0, nn}], x]

%t LinearRecurrence[{3,-1,-1,1},{1,3,8,20},40] (* _Harvey P. Dale_, Nov 09 2020 *)

%o (PARI) a(n)=polcoeff(x/(1-3*x+x^2+x^3-x^4)+O(x^(n+1)),n) \\ _Charles R Greathouse IV_, Aug 06 2012

%o (MAGMA) I:=[1, 3, 8, 20]; [n le 4 select I[n] else 3*Self(n-1)-Self(n-2)-Self(n-3)+Self(n-4): n in [1..40]] // _Vincenzo Librandi_, Aug 07 2012

%Y Cf. A066170.

%K nonn,easy

%O 1,2

%A _Roger L. Bagula_ and _Gary W. Adamson_, Sep 19 2006

%E Edited by _N. J. A. Sloane_, Sep 21 2006; definition corrected Aug 06 2012

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Last modified June 13 00:04 EDT 2021. Contains 344980 sequences. (Running on oeis4.)