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A053186
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Square excess of n: difference between n and largest square <= n.
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28
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0, 0, 1, 2, 0, 1, 2, 3, 4, 0, 1, 2, 3, 4, 5, 6, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13
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OFFSET
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0,4
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COMMENTS
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Comment from David W. Wilson, Jan 05 2009: (Start)
More generally we may consider sequences defined by:
a(n) = n^j - (largest k^th power <= n^j),
a(n) = n^j - (largest k^th power < n^j),
a(n) = (largest k^th power >= n^j) - n^j,
a(n) = (largest k^th power > n^j) - n^j,
for small values of j and k.
The present entry is the first of these with j = 1 and n = 2.
It might be interesting to add further examples to the OEIS. (End)
a(A000290(n)) = 0; a(A005563(n)) = 2*n. [Reinhard Zumkeller, May 20 2009]
0 ^ a(n) = A010052(n). [Reinhard Zumkeller, Feb 12 2012]
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REFERENCES
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S. H. Weintraub, An interesting recursion, Amer. Math. Monthly, 111 (No. 6, 2004), 528-530.
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LINKS
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_Reinhard Zumkeller_, Table of n, a(n) for n = 0..10000
H. Bottomley, Illustration of A000196, A048760, A053186
M. Somos, Sequences used for indexing triangular or square arrays
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FORMULA
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a(n) = n-A048760(n) =n-floor(sqrt(n))^2
a(n)=f(n,1) with f(n,m) = if n<m then n else f(n-m,m+2). [From Reinhard Zumkeller, May 20 2009]
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MAPLE
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S:=proc(n) if issqr(n) then RETURN(0); fi; n-(floor(sqrt(n)))^2; end;
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MATHEMATICA
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f[n_] := n - (Floor@ Sqrt@ n)^2; Table[f@ n, {n, 0, 94}] [From Robert G. Wilson v, Jan 23 2009]
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PROG
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(PARI) a(n)=if(n<0, 0, n-sqrtint(n)^2)
(Haskell)
a053186 n = n - a048760 n
a053186_list = f 0 0 (map fst $ iterate (\(y, z) -> (y+z, z+2)) (0, 1))
where f e x ys'@(y:ys) | x < y = e : f (e + 1) (x + 1) ys'
| x == y = 0 : f 1 (x + 1) ys
-- Reinhard Zumkeller, Apr 27 2012
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CROSSREFS
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Cf. A002262, A048760. A071797(n)=1+a(n-1).
A002262. [From Reinhard Zumkeller, May 20 2009]
Cf. A048760, A000196.
Sequence in context: A201076 A201079 A049260 * A066628 A218601 A135317
Adjacent sequences: A053183 A053184 A053185 * A053187 A053188 A053189
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KEYWORD
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easy,nonn
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AUTHOR
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Henry Bottomley, Mar 01 2000
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STATUS
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approved
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