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A111490
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Antidiagonal sums of the numerical array defined by M(n,k) = 1 + (k-1) mod n.
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22
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1, 2, 4, 5, 9, 9, 15, 16, 21, 23, 33, 29, 41, 45, 51, 52, 68, 65, 83, 81, 91, 99, 121, 109, 128, 138, 152, 152, 180, 168, 198, 199, 217, 231, 253, 234, 270, 286, 308, 298, 338, 326, 368, 372, 384, 404, 450, 422, 463, 470, 500, 506, 558, 546, 584, 576, 610, 636
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OFFSET
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1,2
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COMMENTS
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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."
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).
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LINKS
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FORMULA
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G.f.: 2*x/(1-x)^3 - (1-x)^(-1)*Sum_{k>=1} k*x^k/(1-x^k). - Robert Israel, Oct 11 2015
a(n) = (1 - Pi^2/12) * n^2 + O(n*log(n)). - Amiram Eldar, Dec 04 2023
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EXAMPLE
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Considering the 6 X 6 array:
1, 1, 1, 1, 1, 1
1, 2, 1, 2, 1, 2
1, 2, 3, 1, 2, 3
1, 2, 3, 4, 1, 2
1, 2, 3, 4, 5, 1
1, 2, 3, 4, 5, 6
The third element of the sequence is 1+2+1=4.
The fifth element of the sequence is 1+2+3+2+1=9.
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MAPLE
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MATHEMATICA
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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 *)
(* to view table *) Table[Flatten@Table[Range@n, {m, Ceiling[40/n]}], {n, 10}] // TableForm
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PROG
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(PARI) vector(100, n, n + sum(k=2, n, n % k)) \\ Altug Alkan, Oct 12 2015
(PARI) a(n) = sum(k=1, n, 2*k-sigma(k)); \\ Michel Marcus, Oct 11 2015
(Python)
from math import isqrt
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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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