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Search: a055845 -id:a055845
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a(n) = ((6 + sqrt(3))^n - (6 - sqrt(3))^n)/(2*sqrt(3)).
+10
2
1, 12, 111, 936, 7569, 59940, 469503, 3656016, 28378593, 219894588, 1702241487, 13170376440, 101870548209, 787824155988, 6092161780959, 47107744223904, 364251591915201, 2816463543593580, 21777259989921327, 168383822940467784
OFFSET
1,2
COMMENTS
Fourth binomial transform of A055845.
lim_{n -> infinity} a(n)/a(n-1) = 6 + sqrt(3) = 7.73205080756887729....
FORMULA
G.f.: x/(1 - 12*x + 33*x^2). - Klaus Brockhaus, Dec 31 2008, (corrected Oct 11 2009)
a(n) = 12*a(n-1) - 33*a(n-2) for n>1; a(0)=0, a(1)=1. - Philippe Deléham, Jan 01 2009
E.g.f.: sinh(sqrt(3)*x)*exp(6*x)/sqrt(3). - Ilya Gutkovskiy, Aug 23 2016
MATHEMATICA
LinearRecurrence[{12, -33}, {1, 12}, 25] (* G. C. Greubel, Aug 22 2016 *)
PROG
(Magma) Z<x>:= PolynomialRing(Integers()); N<r>:=NumberField(x^2-3); S:=[ ((6+r)^n-(6-r)^n)/(2*r): n in [1..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Dec 31 2008
(Sage) [lucas_number1(n, 12, 33) for n in range(1, 21)] # Zerinvary Lajos, Apr 27 2009
(Magma) I:=[1, 12]; [n le 2 select I[n] else 12*Self(n-1)-33*Self(n-2): n in [1..25]]; // Vincenzo Librandi, Aug 23 2016
CROSSREFS
Cf. A002194 (decimal expansion of sqrt(3)), A055845.
KEYWORD
nonn
AUTHOR
Al Hakanson (hawkuu(AT)gmail.com), Dec 29 2008
EXTENSIONS
Extended beyond a(7) by Klaus Brockhaus, Dec 31 2008
Edited by Klaus Brockhaus, Oct 11 2009
STATUS
approved
a(n) = 3*a(n-2) for n > 2; a(1) = 1, a(2) = 6.
+10
1
1, 6, 3, 18, 9, 54, 27, 162, 81, 486, 243, 1458, 729, 4374, 2187, 13122, 6561, 39366, 19683, 118098, 59049, 354294, 177147, 1062882, 531441, 3188646, 1594323, 9565938, 4782969, 28697814, 14348907, 86093442, 43046721, 258280326
OFFSET
1,2
COMMENTS
Interleaving of A000244 and 6*A000244.
Second binomial transform is A055845. Sixth binomial transform is A153597.
FORMULA
a(n) = (3+(-1)^n)*3^(1/4*(2*n-1+(-1)^n))/2.
G.f.: x*(1+6*x)/(1-3*x^2).
MATHEMATICA
LinearRecurrence[{0, 3}, {1, 6}, 50] (* G. C. Greubel, May 14 2016 *)
PROG
(Magma) [ n le 2 select 5*n-4 else 3*Self(n-2): n in [1..35] ];
CROSSREFS
Cf. A000244 (powers of 3), A055845, A153597.
KEYWORD
nonn,easy
AUTHOR
Klaus Brockhaus, Oct 13 2009
STATUS
approved
Expansion of ( -4+15*x-8*x^2 ) / ( (x-1)*(x^2-4*x+1) ).
+10
1
4, 5, 13, 44, 160, 593, 2209, 8240, 30748, 114749, 428245, 1598228, 5964664, 22260425, 83077033, 310047704, 1157113780, 4318407413, 16116515869, 60147656060, 224474108368, 837748777409, 3126521001265, 11668335227648, 43546819909324
OFFSET
0,1
LINKS
FORMULA
a(n) = (5*A001353(n+1)-13*A001353(n)+3)/2. - R. J. Mathar, May 26 2016
MAPLE
A257487 := proc(n)
(5+sqrt(3))/4*(2-sqrt(3))^n+(5-sqrt(3))/4*(2+sqrt(3))^n+3/2 ;
expand(%) ;
end proc:
seq(A257487(n), n=0..30) ;
PROG
(PARI) Vec(( -4+15*x-8*x^2 ) / ( (x-1)*(x^2-4*x+1) ) + O(x^50)) \\ Michel Marcus, Apr 26 2015
CROSSREFS
Cf. A055845 (first differences).
KEYWORD
nonn,easy
AUTHOR
R. J. Mathar, Apr 26 2015
STATUS
approved

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