A212336 Expansion of 1/(1 - 23*x + 23*x^2 - x^3).
1, 23, 506, 11110, 243915, 5355021, 117566548, 2581109036, 56666832245, 1244089200355, 27313295575566, 599648413462098, 13164951800590591, 289029291199530905, 6345479454589089320, 139311518709760434136, 3058507932160140461673
Offset: 0
Links
- Bruno Berselli, Table of n, a(n) for n = 0..200
- Robert K. Moniot, A Family of Integer Sequences, 2021.
- Index entries for linear recurrences with constant coefficients, signature (23,-23,1).
Crossrefs
Sequences with g.f. of the type 1/(1-k*x+k*x^2-x^3): A334673 (k=24), A212336 (k=23), A212335 (k=22), A097833 (k=21), A097832 (k=20), A049664 (k=19), A097831-A097829 (k=18,17,16), A076139 (k=15), A097828-A097826 (k=14,13,12), A097784 (k=11), A092420 (k=10), A076765 (k=9), A092521 (k=8), A053142 (k=7), A089817(k=6), A061278 (k=5), A027941 (k=4), A000217 (k=3), A021823 (k=2), A133872 (k=1), A079978 (k=0).
Programs
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Magma
m:=17; R
:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1-23*x+23*x^2-x^3))); -
Magma
I:=[1,23,506]; [n le 3 select I[n] else 23*Self(n-1)-23*Self(n-2)+Self(n-3): n in [1..20]]; // Vincenzo Librandi, Aug 18 2013
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Maple
a:= n-> (<<0|1|0>, <0|0|1>, <1|-23|23>>^n. <<1, 23, 506>>)[1, 1]: seq(a(n), n=0..20); # Alois P. Heinz, Jun 15 2012
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Mathematica
CoefficientList[Series[1/(1 - 23 x + 23 x^2 - x^3), {x, 0, 16}], x] LinearRecurrence[{23, -23, 1}, {1, 23, 506}, 20] (* Vincenzo Librandi, Aug 18 2013 *) -
Maxima
makelist(coeff(taylor(1/(1-23*x+23*x^2-x^3), x, 0, n), x, n), n, 0, 16);
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PARI
Vec(1/(1-23*x+23*x^2-x^3)+O(x^17))
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Sage
[(1/20)*(-1 +21*chebyshev_U(n, 11) -chebyshev_U(n-1, 11)) for n in (0..30)] # G. C. Greubel, Feb 07 2022
Formula
G.f.: 1/((1-x)*(1 - 22*x + x^2)).
a(n) = (((6+sqrt(30))^(2*n+3) + (6-sqrt(30))^(2*n+3))/6^(n+1) - 12)/240.
a(n) = a(-n-3) = 23*a(n-1) - 23*a(n-2) + a(n-3).
a(n)*a(n+2) = a(n+1)*(a(n+1)-1).
a(n+1) - 11*a(n) = A133285(n+2).
11*a(n+1) - a(n) = (1/5)*A157096(n+2).
a(n) = (1/20)*(-1 + 21*ChebyshevU(n, 11) - ChebyshevU(n-1, 11)). - G. C. Greubel, Feb 07 2022
Comments