A143419 G.f.: 1/p(x), where p(x) = degree 22 Salem polynomial p(x) = x^22 + x^21 - x^19 - 2*x^18 - 3*x^17 - 3*x^16 - 2*x^15 + 2*x^13 + 4*x^12 + 5*x^11 + 4*x^10 + 2*x^9 - 2*x^7 - 3*x^6 - 3*x^5 - 2*x^4 - x^3 + x + 1.

1, -1, 1, 0, 1, 1, 1, 2, 2, 4, 4, 7, 9, 12, 17, 23, 32, 44, 60, 83, 113, 156, 214, 294, 403, 554, 760, 1044, 1433, 1967, 2701, 3708, 5091, 6988, 9596, 13172, 18085, 24828, 34086, 46797, 64246, 88203, 121092, 166246, 228237, 313343, 430185, 590594, 810819

Offset: 0

Author

Roger L. Bagula and Gary W. Adamson, Oct 23 2008

Keywords

• easy
• sign

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Curtis T. McMullen, Dynamics on K3 surfaces: Salem numbers and Siegel disks, 2001
• Index entries for linear recurrences with constant coefficients, signature (-1,0,1,2,3,3,2,0,-2,-4,-5,-4,-2,0,2,3,3,2,1,0,-1,-1).

Crossrefs

Cf. A029826, A117791, A143438, A143472, A143619, A143644, A147663, A173908, A173911, A173924, A173925, A174522, A175740, A175772, A175773, A175782, A181600, A204631, A225391, A225393, A225394, A225482, A225499.

Programs

Magma

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(x^22 +x^21-x^19-2*x^18-3*x^17-3*x^16-2*x^15+2*x^13+4*x^12+5*x^11 + 4*x^10+2*x^9-2*x^7-3*x^6-3*x^5-2*x^4-x^3+x+1))); // G. C. Greubel, Nov 03 2018

Mathematica

f[x_] = x^22 + x^21 - x^19 - 2*x^18 - 3*x^17 - 3*x^16 - 2*x^15 + 2*x^13 + 4*x^12 + 5*x^11 + 4*x^10 + 2*x^9 - 2*x^7 - 3*x^6 - 3*x^5 - 2*x^4 - x^3 + x + 1;
CoefficientList[Series[1/f[x], {x, 0, 50}], x]
LinearRecurrence[{-1,0,1,2,3,3,2,0,-2,-4,-5,-4,-2,0,2,3,3,2,1,0,-1,-1},{1,-1,1,0,1,1,1,2,2,4,4,7,9,12,17,23,32,44,60,83,113,156},50] (* Harvey P. Dale, Aug 18 2024 *)

PARI

p(x)=x^22 + x^21 - x^19 - 2*x^18 - 3*x^17 - 3*x^16 - 2*x^15 + 2*x^13 + 4*x^12 + 5*x^11 + 4*x^10 + 2*x^9 - 2*x^7 - 3*x^6 - 3*x^5 - 2*x^4 - x^3 + x + 1; Vec(1/p(x)+O(x^60)) \\ Charles R Greathouse IV, Feb 13 2011

Formula

a(n) = -a(n-1) + a(n-3) + 2*a(n-4) + 3*a(n-5) + 3*a(n-6) + 2*a(n-7) - 2*a(n-9) - 4*a(n-10) - 5*a(n-11) - 4*a(n-12) - 2*a(n-13) + 2*a(n-15) + 3*a(n-16) + 3*a(n-17) + 2*a(n-18) + a(n-19) - a(n-21) - a(n-22). - Franck Maminirina Ramaharo, Oct 30 2018

Extensions

Edited by N. J. A. Sloane, Dec 12 2008

More terms from Sean A. Irvine, Feb 13 2011

Offset corrected, and more terms from Franck Maminirina Ramaharo, Nov 02 2018