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A135836
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Column three of the triangular matrix listed by rows in A135835.
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2
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3, 22, 82, 254, 677, 1692, 3972, 9052, 19975, 43394, 92534, 195546, 408489, 848584, 1749544, 3594104, 7345547, 14976366, 30424986, 61706038, 124829101, 252226676, 508704716, 1025115156, 2062984719, 4149086938, 8336437438, 16742227730, 33599246513, 67406551968
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OFFSET
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1,1
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COMMENTS
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Column two of the associated matrix is A005803.
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LINKS
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FORMULA
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a(n) = (1/4)*(110 + 26*n + 2^(n+8) - (1-(-1)^n)*106*3^((n+1)/2) - (1+(-1)^n)*61*3^(1+n/2)).
a(2*n) = (1/2)*(55 + 26*n + 2^(2*n+7) - 61*3^(n+1)).
a(2*n+1) = (1/2)*(68 + 26*n + 4^(n+4) - 106*3^(n+1)).
G.f.: x*(3 + 10*x)/((1-x)^2*(1 - 2*x - 3*x^2 + 6*x^3)).
E.g.f.: (1/2)*( (55 + 13*x)*exp(x) + 128*exp(2*x) - 183*cosh(sqrt(3)*x) - 106*sqrt(3)*sinh(sqrt(3)*x) ). (End)
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MATHEMATICA
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LinearRecurrence[{4, -2, -10, 15, -6}, {3, 22, 82, 254, 677}, 40] (* G. C. Greubel, Feb 07 2022 *)
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PROG
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(Magma) [(1/12)*(330 +78*n +3*2^(n+8) -(1-(-1)^n)*106*3^((n+3)/2) -(1+(-1)^n)*61*3^(2 +n/2)): n in [1..40]]; // G. C. Greubel, Feb 07 2022
(Sage)
def a(n):
if (n%2==0): return (1/2)*(55 + 13*n + 2^(n+7) -61*3^(n/2+1))
else: return (1/2)*(55 + 13*n + 2^(n+7) - 106*3^((n+1)/2))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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