A084156 Binomial transform of sinh(x)*cosh(sqrt(3)*x).
0, 1, 2, 13, 44, 181, 662, 2521, 9368, 35113, 130922, 489061, 1824836, 6811741, 25420670, 94875313, 354076208, 1321442641, 4931681234, 18405321661, 68689566044, 256353060613, 956722558310, 3570537526921, 13325427195080, 49731172316281, 185599261007162
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
- Index entries for linear recurrences with constant coefficients, signature (4,2,-12,3).
Programs
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Magma
I:=[0,1,2,13]; [n le 4 select I[n] else 4*Self(n-1)+2*Self(n-2)-12*Self(n-3)+3*Self(n-4): n in [1..30]]; // Vincenzo Librandi, Feb 13 2014
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Maple
seq((2*simplify(ChebyshevT(n,2)) - (1+(-1)^n)*3^(n/2))/4, n = 0..30); # G. C. Greubel, Oct 10 2022
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Mathematica
LinearRecurrence[{4,2,-12,3},{0,1,2,13},30] (* Harvey P. Dale, Feb 01 2014 *) -
SageMath
def A084156(n): return (chebyshev_T(n, 2) - ((n+1)%2)*3^(n/2))/2 [A084156(n) for n in range(31)] # G. C. Greubel, Oct 10 2022
Formula
a(n) = 4*a(n-1) + 2*a(n-2) - 12*a(n-3) + 3*a(n-4).
a(n) = ((2+sqrt(3))^n + (2-sqrt(3))^n - (sqrt(3))^n - (-sqrt(3))^n)/4.
G.f.: x*(1-2*x+3*x^2)/((1-3*x^2)(1-4*x+x^2)).
E.g.f. : exp(x)*sinh(x)*cosh(sqrt(3)*x).
a(n) = (2*ChebyshevT(n, 2) - (1+(-1)^n)*3^(n/2))/4 = (A001075(n) - A254006(n))/2. - G. C. Greubel, Oct 10 2022