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A189835
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Number of representations of n as a*b + b*c + c*d + d*e where a, b, d, e>0, c>=0 are integers.
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3
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0, 1, 4, 9, 16, 26, 36, 53, 64, 90, 100, 138, 144, 194, 200, 261, 256, 347, 324, 426, 416, 522, 484, 658, 576, 746, 712, 882, 784, 1060, 900, 1173, 1088, 1314, 1160, 1587, 1296, 1658, 1544, 1890, 1600, 2164, 1764, 2298, 2096, 2466, 2116, 2930, 2304, 2955, 2696
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OFFSET
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1,3
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COMMENTS
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Related to "Liouville's Last Theorem".
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LINKS
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FORMULA
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G.f.: Sum_{k>0} (x^k + x^(2*k)) / (1 - x^k)^3 - k * x^k / (1 - x^k)^2.
a(n) = A001157(n) - A038040(n) = sigma( n, 2) - n * sigma( n, 0) where sigma( n, k) is the sum of the k-th powers of the divisors of n.
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EXAMPLE
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G.f. = x^2 + 4*x^3 + 9*x^4 + 16*x^5 + 26*x^6 + 36*x^7 + 53*x^8 + 64*x^9 + 90*x^10 + ...
a(3) = 4 since 3 = 1*1 + 1*0 + 0*1 + 1*2 = 1*1 + 1*0 + 0*2 + 2*1 = 1*2 + 2*0 + 0*1 + 1*1 = 2*1 + 1*0 + 0*1 + 1*1 are all 4 representations of 3.
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MAPLE
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with(numtheory); f:=n->sigma[2](n)-n*sigma[0](n);
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MATHEMATICA
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a[n_] := DivisorSigma[2, n] - n*DivisorSigma[0, n]; Table[a[n], {n, 51}] (* Jean-François Alcover, Aug 31 2011 *)
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PROG
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(PARI) {a(n) = if( n<1, 0, sigma( n, 2) - n * sigma( n, 0))}
(Haskell)
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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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