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A089926
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a(n) = 12*a(n-1) + a(n-2), a(0)=1, a(1)=6.
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1
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1, 6, 73, 882, 10657, 128766, 1555849, 18798954, 227143297, 2744518518, 33161365513, 400680904674, 4841332221601, 58496667563886, 706801342988233, 8540112783422682, 103188154744060417, 1246797969712147686
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OFFSET
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0,2
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COMMENTS
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The family of recurrences a(n) = 2*k*a(n-1) + a(n-2), a(0)=1, a(1)=k has solution a(n) = ((k+sqrt(k^2+1))^n + (k-sqrt(k^2+1))^n)/2; a(n) = Sum_{j=0..floor(n/2)} C(n,2k)*(k^2+1)^jk^(n-2j); a(n) = T(n,ki)*(-i)^n; e.g.f. exp(kx)*cosh(sqrt(k^2+1)*x).
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LINKS
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FORMULA
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E.g.f.: exp(6x)*cosh(sqrt(37)x);
a(n) = ((6+sqrt(37))^n + (6-sqrt(37))^n)/2;
a(n) = Sum_{k=0..floor(n/2)} C(n, 2k)*37^k*6^(n-2k).
a(n) = T(n, 6i)*(-i)^n with T(n, x) Chebyshev's polynomials of the first kind (see A053120) and i^2 = -1.
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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