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A072031 Row sums of A072030.

%I A072031 #26 Jul 28 2024 00:15:45
%S A072031 1,4,7,14,15,28,29,40,43,64,53,86,79,90,99,134,109,160,133,160,171,
%T A072031 216,167,232,223,244,233,310,233,344,297,328,339,366,315,450,403,422,
%U A072031 387,522,401,560,473,490,539,636,485,648,569,634,609,758,603,726,655
%N A072031 Row sums of A072030.
%C A072031 From _Peter Bala_, Dec 09 2014: (Start)
%C A072031 Let A(x) = sum {n >= 1} x^n/(1 - x^n) be the Lambert series generating function of the divisor function tau(n) - see A000005. Then it appears that A(x)^2 is equal to the Lambert series sum {n >= 2} a(n-1)*x^n/(1 - x^n) (checked up to x^1000 using Luschny's formula for a(n)).
%C A072031 This conjecture is equivalent to the following identity for the divisor function: for n >= 2 there holds sum {k = 1..n} tau(k)*tau(n-k) = sum {d divides n} ( sum {k = 1..d} tau(k*(d - k)) ), where we take tau(0) = 0. (End)
%H A072031 James C. Alexander, <a href="http://web.archive.org/web/20070418131521/http://www.case.edu/artsci/math/alexander/pdf/fib.pdf">The Entropy of the Fibonacci Code</a> (1989).
%p A072031 A072031 := n -> add(numtheory[tau](j*(n+1-j)), j=1..n);
%p A072031 seq(A072031(i), i=1..55); # _Peter Luschny_, Sep 13 2012
%t A072031 T[n_, k_] := T[n, k] = Which[n < 1 || k < 1, 0, n == k, 1, n < k, T[k, n], True, 1 + T[k, n - k]];
%t A072031 a[n_] := Sum[T[n - k + 1, k],  {k, 1, n}];
%t A072031 Array[a, 55] (* _Jean-François Alcover_, Jun 03 2019 *)
%Y A072031 Row sums of A072030. Cf. A000005.
%K A072031 nonn,easy
%O A072031 1,2
%A A072031 _Michael Somos_, Jun 07 2002