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

1, 3, 8, 20, 50, 125, 313, 784, 1964, 4920, 12325, 30875, 77344, 193752, 485362, 1215865, 3045825, 7630000, 19113672, 47881056, 119945321, 300471235, 752701000, 1885567500, 4723475586, 11832629493, 29641546393, 74254101600

Offset: 1

Author

Roger L. Bagula and Gary W. Adamson, Sep 19 2006

Keywords

• nonn
• easy

Examples

			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 + ...
		

References

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.
• Index entries for linear recurrences with constant coefficients, signature (3,-1,-1,1).

Crossrefs

Cf. A066170.

Programs

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

Mathematica

nn = 30; CoefficientList[Series[x/(1 - 3*x + x^2 + x^3 - x^4), {x, 0, nn}], x]
LinearRecurrence[{3,-1,-1,1},{1,3,8,20},40] (* Harvey P. Dale, Nov 09 2020 *)

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

Formula

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

a(n) ~ 0.50556... * 2.505068...^n. - Charles R Greathouse IV, Aug 06 2012

a(n) = 3*a(n-1) - a(n-2) - a(n-3) + a(n-4) for n>4. - Wesley Ivan Hurt, Sep 18 2015

Extensions

Edited by N. J. A. Sloane, Sep 21 2006; definition corrected Aug 06 2012