OFFSET
1,2
LINKS
Colin Barker, Table of n, a(n) for n = 1..700
Index entries for linear recurrences with constant coefficients, signature (22,-1).
FORMULA
a(n+2) = 22*a(n+1) - a(n); a(n+1) = 11*a(n) + 2*sqrt(30*a(n)^2 + 6).
a(n) = (sqrt(30)/10 - 1/2)*(11 + 2*sqrt(30))^n - (sqrt(30)/10 + 1/2) * (11 - 2*sqrt(30))^n. - Emeric Deutsch, Oct 24 2007
G.f.: x*(1+x)/(1-22*x+x^2). - R. J. Mathar, Nov 14 2007
a(n) = Chebyshev(n-1, 11) + Chebyshev(n-2, 11). - G. C. Greubel, Jan 13 2020
MAPLE
a[1]:=1: a[2]:=23: for n to 14 do a[n+2]:=22*a[n+1]-a[n] end do: seq(a[n], n= 1..16); # Emeric Deutsch, Oct 24 2007
MATHEMATICA
Table[n /. {ToRules[Reduce[n > 0 && k >= 0 && 30*n^2+6 == k^2, n, Integers] /. C[1] -> c]} // Simplify, {c, 1, 20}] // Flatten // Union (* Jean-François Alcover, Dec 19 2013 *)
Rest@ CoefficientList[Series[x(1+x)/(1-22x+x^2), {x, 0, 20}], x] (* Michael De Vlieger, Jul 14 2016 *)
LinearRecurrence[{22, -1}, {1, 23}, 20] (* Harvey P. Dale, Sep 22 2017 *)
Table[ChebyshevU[n-1, 11] + ChebyshevU[n-2, 11], {n, 20}] (* G. C. Greubel, Jan 13 2020 *)
PROG
(PARI) Vec(x*(1+x)/(1-22*x+x^2) + O(x^20)) \\ Colin Barker, Jul 14 2016
(PARI) vector(20, n, polchebyshev(n-1, 2, 11) + polchebyshev(n-2, 2, 11) ) \\ G. C. Greubel, Jan 13 2020
(Magma) I:=[1, 23]; [n le 2 select I[n] else 22*Self(n-1) - Self(n-2): n in [1..20]]; // G. C. Greubel, Jan 13 2020
(Sage) [chebyshev_U(n-1, 11) + chebyshev_U(n-2, 11) for n in (1..20)] # G. C. Greubel, Jan 13 2020
(GAP) a:=[1, 23];; for n in [3..20] do a[n]:=22*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Jan 13 2020
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Richard Choulet, Oct 16 2007
EXTENSIONS
More terms from Emeric Deutsch, Oct 24 2007
STATUS
approved