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A116423
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Binomial transform of A006053.
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5
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0, 1, 3, 9, 26, 74, 209, 588, 1651, 4631, 12983, 36388, 101972, 285741, 800660, 2243445, 6286059, 17613241, 49351342, 138279586, 387451077, 1085614208, 3041824015, 8523002359, 23880923183, 66912861640, 187485674652
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| a(n)/a(n-1) tends to 2.801...= 1 + 2*Cos Pi/7.
A(n):= a(n+1)*(-1)^(n+1) appears in the following formula for the nonpositive powers of rho*sigma, where rho:=2*cos(Pi/7) and sigma:=sin(3*Pi/7)/sin(Pi/7) = rho^2-1 are the ratios of the smaller and larger diagonal length to the side length in a regular 7-gon (heptagon). See the Steinbach reference where the basis <1,rho,sigma> is used in an extension of the rational field. (rho*sigma)^(-n) = C(n) + B(n)*rho + A(n)*sigma,n>=0, with C(n)= A085810(n)*(-1)^n, and B(n)= A181880(n-2)*(-1)^n. For the nonnegative powers see A120757(n), |A122600(n-1)| and A181879(n), respectively. See also a comment under A052547.
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LINKS
| P. Steinbach, Golden Fields: A Case for the Heptagon, Mathematics Magazine, 70,1 (1997) 22-31.
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FORMULA
| Binomial transform of A006053 starting with A006053(1): (0, 1, 1, 3, 4, 9, 14...)
O.g.f.: x^2(1-x)/(1-4x+3x^2+x^3). a(n)=4*a(n-1)-3*a(n-2)-a(n-3). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 02 2008
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EXAMPLE
| a(5) = 26 = 1*0 + 1*4 + 4*1 + 4*3 + 6*1 = 4 + 4 + 12 + 6 = 26
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CROSSREFS
| Cf. A006053.
Sequence in context: A054447 A061667 A127911 * A077845 A171277 A000243
Adjacent sequences: A116420 A116421 A116422 * A116424 A116425 A116426
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KEYWORD
| nonn
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AUTHOR
| Gary W. Adamson. (qntmpkt(AT)yahoo.com), Feb 14 2006
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EXTENSIONS
| More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 02 2008
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