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%I #55 Dec 04 2023 01:38:37
%S 1,2,4,5,9,9,15,16,21,23,33,29,41,45,51,52,68,65,83,81,91,99,121,109,
%T 128,138,152,152,180,168,198,199,217,231,253,234,270,286,308,298,338,
%U 326,368,372,384,404,450,422,463,470,500,506,558,546,584,576,610,636
%N Antidiagonal sums of the numerical array defined by M(n,k) = 1 + (k-1) mod n.
%C Previous name was "Sum of the element of the antidiagonals of the numerical array M(m,n) defined as follows. First row (M11, M12, ..., M1n): 1, 1, 1, 1, 1, 1, ... (all 1's). Second row (M21, M22, ..., M2n): 1, 2, 1, 2, 1, 2, ... (sequence 1, 2 repeated). Third row (M31, M32, ..., M3n): 1, 2, 3, 1, 2, 3, 1, 2, 3, ... (sequence 1, 2, 3 repeated). Fourth row (M41, M42, ..., M4n): 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, ... (sequence 1, 2, 3, 4 repeated). And so on."
%C Then the sequence is M(1,1), M(1,2) + M(2,1), M(1,3) + M(2,2) + M(3,1), etc., a(n) = Sum_{i=1..n} M(i, n-i+1).
%C The successive determinants of the arrays are the factorial numbers (A000142). - _Robert G. Wilson v_
%H Robert Israel, <a href="/A111490/b111490.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = n + A004125(n). - _Juri-Stepan Gerasimov_, Aug 30 2009
%F a(n) = Sum_{i=1..n+1} (n mod i). - _Wesley Ivan Hurt_, Dec 05 2014
%F G.f.: 2*x/(1-x)^3 - (1-x)^(-1)*Sum_{k>=1} k*x^k/(1-x^k). - _Robert Israel_, Oct 11 2015
%F a(n) = (1 - Pi^2/12) * n^2 + O(n*log(n)). - _Amiram Eldar_, Dec 04 2023
%e Considering the 6 X 6 array:
%e 1, 1, 1, 1, 1, 1
%e 1, 2, 1, 2, 1, 2
%e 1, 2, 3, 1, 2, 3
%e 1, 2, 3, 4, 1, 2
%e 1, 2, 3, 4, 5, 1
%e 1, 2, 3, 4, 5, 6
%e The third element of the sequence is 1+2+1=4.
%e The fifth element of the sequence is 1+2+3+2+1=9.
%p A111490:=n->add(n mod i, i=1..n+1): seq(A111490(n), n=1..100); # _Wesley Ivan Hurt_, Dec 05 2014
%t t = Table[Flatten@Table[Range@n, {m, Ceiling[99/n]}], {n, 99}]; f[n_] := Sum[ t[[i, n - i + 1]], {i, n}]; Array[f, 58] (* _Robert G. Wilson v_, Nov 22 2005 *)
%t (* to view table *) Table[Flatten@Table[Range@n, {m, Ceiling[40/n]}], {n, 10}] // TableForm
%o (PARI) vector(100, n, n + sum(k=2, n, n % k)) \\ _Altug Alkan_, Oct 12 2015
%o (PARI) a(n) = sum(k=1, n, 2*k-sigma(k)); \\ _Michel Marcus_, Oct 11 2015
%o (Python)
%o from math import isqrt
%o def A111490(n): return n*(n+1)+((s:=isqrt(n))**2*(s+1)-sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1))>>1) # _Chai Wah Wu_, Nov 01 2023
%Y Cf. A000142, A004125.
%Y Partial sums of A033879. - _Gionata Neri_, Sep 10 2015
%K nonn,easy
%O 1,2
%A _Paolo P. Lava_ and _Giorgio Balzarotti_, Nov 21 2005
%E Edited and extended by _Robert G. Wilson v_, Nov 22 2005
%E Name changed by _Michel Marcus_, Sep 23 2013
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