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A143375 Expansion of x / ( 1 -x^2 -2*x^5 -x^8 -x^10 -x^12 ). 1
1, 0, 1, 0, 1, 2, 1, 4, 2, 6, 8, 8, 19, 14, 34, 36, 54, 86, 93, 172, 194, 308, 427, 552, 878, 1076, 1675, 2224, 3120, 4546, 5986, 8928, 11933, 17104, 24005, 32928, 47534, 64640, 92523, 128348, 179418, 253994, 350622, 498000, 690790, 971508, 1362840 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

A new 4 symbol polynomial of the Weaver telegraphic type ( Prime like powers) : dot:x^2; dash:x^5; Letter space: x^3 ; Word space: x^7 ; p(x)=-1 - x^2 - x^4 - 2 x^7 - x^10 + x^12.

At C=-Log[0.7139184783743413]=0.336986 this has a lower channel capacity

than the Weaver C=0.539.

REFERENCES

Claude Shannon and Warren Weaver, A Mathematical Theory of Communication, University of Illinois Press, Chicago, 1963, p37 - 38

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (0, 1, 0, 0, 2, 0, 0, 1, 0, 1, 0, 1).

FORMULA

a(n) = +a(n-2) +2*a(n-5) +a(n-8) +a(n-10) +a(n-12).

EXAMPLE

Weaver determinant:

A0 = x^2;

B0 = x^5;

C0 = x^3;

D0 = x^7;

Expand[FullSimplify[ExpandAll[x^12*Det[{{-1, (1/B0 + 1/A0)}, {(1/D0 + 1/C0),

1/A0 + 1/B0 - 1}}]]]]

MATHEMATICA

p[x_] = -1 - x^2 - x^4 - 2 x^7 - x^10 + x^12; q[x_] = ExpandAll[x^12*p[1/x]]; a = Table[SeriesCoefficient[Series[1/q[x], {x, 0, 30}], n], {n, 0, 30}]

CoefficientList[Series[x/(1-x^2-2x^5-x^8-x^10-x^12), {x, 0, 60}], x] (* or *) LinearRecurrence[{0, 1, 0, 0, 2, 0, 0, 1, 0, 1, 0, 1}, {0, 1, 0, 1, 0, 1, 2, 1, 4, 2, 6, 8}, 60] (* Harvey P. Dale, Oct 01 2012 *)

PROG

(PARI) x='x+O('x^50); Vec(x/(1-x^2-2*x^5-x^8-x^10-x^12)) \\ G. C. Greubel, Sep 27 2017

CROSSREFS

Cf. A122762.

Sequence in context: A239960 A292402 A205685 * A074364 A256610 A276055

Adjacent sequences:  A143372 A143373 A143374 * A143376 A143377 A143378

KEYWORD

nonn,uned

AUTHOR

Roger L. Bagula and Gary W. Adamson, Oct 22 2008

EXTENSIONS

More terms from Harvey P. Dale, Oct 01 2012

STATUS

approved

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Last modified July 22 10:45 EDT 2018. Contains 312891 sequences. (Running on oeis4.)