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A189835 Number of representations of n as a*b + b*c + c*d + d*e where a, b, d, e>0, c>=0 are integers.

%I A189835 #26 Jul 16 2025 21:59:28
%S A189835 0,1,4,9,16,26,36,53,64,90,100,138,144,194,200,261,256,347,324,426,
%T A189835 416,522,484,658,576,746,712,882,784,1060,900,1173,1088,1314,1160,
%U A189835 1587,1296,1658,1544,1890,1600,2164,1764,2298,2096,2466,2116,2930,2304,2955,2696
%N A189835 Number of representations of n as a*b + b*c + c*d + d*e where a, b, d, e>0, c>=0 are integers.
%C A189835 Related to "Liouville's Last Theorem".
%C A189835 Inverse Möbius transform of A344485(n). - _Wesley Ivan Hurt_, Jul 16 2025
%H A189835 Reinhard Zumkeller, <a href="/A189835/b189835.txt">Table of n, a(n) for n = 1..10000</a>
%H A189835 R. P. Agarwal, <a href="https://www.ias.ac.in/article/fulltext/pmsc/103/03/0269-0293">Lambert series and Ramanujan</a>, Prod. Indian Acad. Sci. (Math. Sci.), v. 103, n. 3, 1993, pp. 269-293. see p. 273.
%H A189835 George E. Andrews, <a href="http://dx.doi.org/10.1007/BF01608779">Stacked lattice boxes</a>, Ann. Comb. 3 (1999), 115-130. See L_5(n).
%H A189835 E. T. Bell, <a href="https://doi.org/10.1090/S0002-9904-1936-06305-6">The form wx+xy+yz+zu</a>, Bull. Amer. Math. Soc., 42 (1936), 377-380.
%F A189835 G.f.: Sum_{k>0} (x^k + x^(2*k)) / (1 - x^k)^3 - k * x^k / (1 - x^k)^2.
%F A189835 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.
%F A189835 a(n) = Sum_{d|n} A344485(d). - _Wesley Ivan Hurt_, Jul 16 2025
%e A189835 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 + ...
%e A189835 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.
%p A189835 with(numtheory); f:=n->sigma[2](n)-n*sigma[0](n);
%t A189835 a[n_] := DivisorSigma[2, n] - n*DivisorSigma[0, n]; Table[a[n], {n, 51}] (* _Jean-François Alcover_, Aug 31 2011 *)
%o A189835 (PARI) {a(n) = if( n<1, 0, sigma( n, 2) - n * sigma( n, 0))}
%o A189835 (Haskell)
%o A189835 a189835 n = a001157 n - a038040 n  -- _Reinhard Zumkeller_, Jan 21 2014
%Y A189835 Cf. A001157, A038040, A191822, A344485.
%K A189835 nonn
%O A189835 1,3
%A A189835 _Michael Somos_, Apr 28 2011
%E A189835 Added references, comment, Maple program, cross-reference to A191822. - _N. J. A. Sloane_, Jun 17 2011