A215502
a(n) = (1+sqrt(3))^n + (-2)^n + (1-sqrt(3))^n + 1.
Original entry on oeis.org
4, 1, 13, 13, 73, 121, 481, 1009, 3361, 7969, 24193, 61249, 177025, 464257, 1307137, 3493633, 9699841, 26190337, 72173569, 195941377, 537802753, 1464342529, 4010582017, 10937266177, 29920862209, 81665925121, 223274237953, 609678999553, 1666309128193
Offset: 0
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[Round((1+Sqrt(3))^n + (-2)^n + (1-Sqrt(3))^n + 1): n in [0..30]]; // G. C. Greubel, Apr 23 2018
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A215502 := n -> 1+(1+sqrt(3))^n+(-2)^n+(1-sqrt(3))^n;
seq(simplify(A215502(i)),i=0..28);
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Simplify/@Table[(1+Sqrt[3])^n+(1-Sqrt[3])^n+1+(-2)^n,{n,0,30}] (* or *) LinearRecurrence[{1,6,-2,-4},{4,1,13,13},30] (* Harvey P. Dale, Mar 12 2013 *)
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x='x+O('x^30); Vec((4-3*x-12*x^2+2*x^3)/((1-x)*(1+2*x)*(1-2*x-2*x^2))) \\ G. C. Greubel, Apr 23 2018
A215503
a(n) = (u+1)^n + (-s-1)^n + (t+1)^n + (-1)^n + (-t+1)^n + (s-1)^n + (-u+1)^n where s = sqrt(2), t = sqrt(2-s), u = sqrt(2+s).
Original entry on oeis.org
7, 1, 19, 13, 111, 121, 763, 1093, 5575, 9697, 42099, 84173, 324591, 717081, 2538331, 6023173, 20049671, 50079553, 159514963, 413387789, 1275778031, 3394968121, 10242581819, 27780675397, 82461727687, 226743641121, 665232392883, 1847286687181, 5374409263215
Offset: 0
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Wikipedia, Nested radical
- Index entries for linear recurrences with constant coefficients, signature (1,9,-5,-17,1,5,-1).
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I:=[7,1,19,13,111,121,763]; [n le 7 select I[n] else Self(n-1) + 9*Self(n-2) -5*Self(n-3) -17*Self(n-4) + Self(n-5) + 5*Self(n-6) - Self(n-7): n in [1..30]]; // G. C. Greubel, Apr 23 2018
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A215503 := n -> (sqrt(2+sqrt(2))+1)^n+(-sqrt(2)-1)^n+(sqrt(2-sqrt(2))+1)^n+(-1)^n+(-sqrt(2-sqrt(2))+1)^n+(sqrt(2)-1)^n+(-sqrt(2+sqrt(2))+1)^n;
seq(simplify(A215503(i)),i=0..28);
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LinearRecurrence[{1,9,-5,-17,1,5,-1}, {7,1,19,13,111,121,763}, 50] (* G. C. Greubel, Apr 23 2018 *)
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x='x+O('x^30); Vec((7-6*x-45*x^2+20*x^3+51*x^4 -2*x^5-5*x^6)/( (1+x)*(1+2*x-x^2)*(1 -4*x +2*x^2 +4*x^3-x^4))) \\ G. C. Greubel, Apr 23 2018
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polsym(polrecip(((1+x)*(1+2*x-x^2)*(1-4*x+2*x^2+4*x^3-x^4))),33) \\ Joerg Arndt, Apr 29 2018
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def A215503(n) :
return (sqrt(2+sqrt(2))+1)^n+(-sqrt(2)-1)^n+(sqrt(2-sqrt(2))+1)^n+(-1)^n+(-sqrt(2-sqrt(2))+1)^n+(sqrt(2)-1)^n+(-sqrt(2+sqrt(2))+1)^n
[A215503(i).round() for i in (0..28)]
Showing 1-2 of 2 results.