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A038342 G.f.: 1/(1 - 3 x - 3 x^2 + 4 x^3 + x^4 - x^5). 6
1, 3, 12, 41, 146, 511, 1798, 6314, 22187, 77946, 273856, 962142, 3380337, 11876254, 41725295, 146595013, 515037713, 1809501081, 6357387289, 22335644540, 78472648463, 275700866485, 968630080476, 3403123989780 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Middle line of 5-wave sequence A038201.

Let M denotes the 5 X 5 matrix = row by row (1,1,1,1,1)(1,1,1,1,0)(1,1,1,0,0)(1,1,0,0,0)(1,0,0,0,0) and A(n) the vector (x(n),y(n),z(n),t(n),u(n))=M^n*A where A is the vector (1,1,1,1,1) then a(n)=z(n). - Benoit Cloitre, Apr 02 2002

a(n) appears in the formula for 1/rho(11)^n, with rho(11) := 2*cos(Pi/11) (length ratio (smallest diagonal/side) in the regular 11-gom) when written in the power basis of the degree 5 number field Q(rho(11)): 1/rho(11)^n = a(n)*1 + A230080(n)*rho(11) - A230081(n)*rho(11)^2 - A069006(n-1)* rho(11)^3 + a(n-1)*rho(11)^4, n >= 0, with A069006(-1) = 0 = a(-1). See A230080 with the example for n=4. - Wolfdieter Lang, Nov 04 2013

From Wolfdieter Lang, Nov 20 2013: (Start)

The limit a(n+1)/a(n) for n -> infinity is omega(11) := S(4, x) = 1 - 3*x^2 + x^4 with x = rho(11). omega(11) = 1/(2*cos(Pi*5/11)), approx. 3.51333709. For the Chebyshev S-polynomial see A049310. For rho(11) see the preceding comment. The decimal expansion of omega(11) is given in A231186. omega(11) is an integer in Q(rho(11)) with power basis coefficients [1,0,-3,0,1]. It is known to be the length ratio (longest diagonal)/side in the regular 11-gon.

This limit follows from the a(n)-recurrence and the solutions of X^5 - 3*X^4 - 3*X^3 + 4*X^2 + X - 1 = 0, which are given by the inverse of the known solutions of the minimal polynomial C(11, x) of rho(11) (see A187360). The other four X solutions are 1/rho(11), with coefficients [3,3,-4,-1,1] in the power basis of Q(rho(11)), approx. 0.52110856, 1/(2*cos(Pi*3/11)) with coefficients [-1,-1,1,0,0], approx. 0.763521119, 1/(2*cos(Pi*7/11)) with coefficients [0,-3,3,1,-1], approx. -1.20361562, and 1/(2*cos(Pi*9/11)) with coefficients [0,1,3,0,-1], approx. -0.59435114. These solutions for X are therefore irrelevant for this sequence.

The same limit omega(11) is therefore obtained for the sequences A069006, A230080 and A230081. See the Nov 04 2013 comment.

(End)

REFERENCES

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

LINKS

Table of n, a(n) for n=0..23.

F. v. Lamoen, Wave sequences

P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), p. 22-31 (formula 5).

Index entries for linear recurrences with constant coefficients, signature (3,3,-4,-1,1).

FORMULA

a(n) = 3a(n-1)+3a(n-2)-4a(n-3)-a(n-4)+a(n-5). Also a(n) = b(4n+2) with b(n) as in 5-wave sequence A038201.

G.f.: 1/(1 - 3 x - 3 x^2 + 4 x^3 + x^4 - x^5) = -1/C(11, x), with C(11, x) the minimal polynomial of 2*cos(Pi/11) (see the name and A187360 for C). - Wolfdieter Lang, Nov 07 2013

MATHEMATICA

b = {-1, 3, 3, -4, -1, 1}; p[x_] := Sum[x^(n - 1)*b[[7 - n]], {n, 1, 6}] q[x_] := ExpandAll[x^5*p[1/x]] Table[ SeriesCoefficient[ Series[x/q[x], {x, 0, 30}], n], {n, 0, 30}] (* Roger L. Bagula and Gary W. Adamson, Sep 19 2006 *)

LinearRecurrence[{3, 3, -4, -1, 1}, {1, 3, 12, 41, 146}, 30] (* Harvey P. Dale, Aug 27 2012 *)

CROSSREFS

Cf. A006358, A069006, A230080, A230081: same recurrence formula.

Cf. A066170.

Sequence in context: A038345 A127120 A017940 * A260153 A135264 A084529

Adjacent sequences:  A038339 A038340 A038341 * A038343 A038344 A038345

KEYWORD

nonn

AUTHOR

Floor van Lamoen

EXTENSIONS

More terms from Benoit Cloitre, Apr 02 2002

Edited by N. J. A. Sloane at the suggestion of Andrew Plewe, Jun 08 2007

STATUS

approved

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Last modified August 19 23:35 EDT 2017. Contains 290821 sequences.