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A003983
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Array read by antidiagonals with T(n,k) = min(n,k).
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29
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1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 3, 2, 1, 1, 2, 3, 4, 3, 2, 1, 1, 2, 3, 4, 4, 3, 2, 1, 1, 2, 3, 4, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 6, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 7, 7, 6, 5, 4, 3, 2, 1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,5
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COMMENTS
| Also, "correlation triangle" for the constant sequence 1. - Paul Barry, Jan 16 2006
Antidiagonal sums are in A002620.
As a triangle, row sums are A002620. T(2n,n)=n+1. Diagonal sums are A001399. Construction: Take antidiagonal triangle of MM^T where M is the sequence array for the constant sequence 1 (lower triangular matrix with all 1's). - Paul Barry, Jan 16 2006
Zeta[2,k/p]+Zeta[2,(p-k)/p]=(Pi/Sin[(Pi*a(n))/p])^2, where p=2,3,4, k=1..p-1 [From Artur Jasinski, Mar 07 2010]
Contribution from Franklin T. Adams-Watters, Sep 25 2011 (start) As a triangle, count up to ceiling(n/2) and back down again (repeating the central term when n is even).
When the first two instances of each number are removed from the sequence, the original sequence is recovered.
(end)
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LINKS
| Reinhard Zumkeller, Rows n=1..100 of triangle, flattened
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FORMULA
| Number triangle T(n, k)=sum{j=0..n, [j<=k][j<=n-k]}. - Paul Barry, Jan 16 2006
G.f.: 1/((1-x)*(1-x*y)*(1-x^2*y)) (Christian G. Bower, Jan 17 2006)
a(n)=min(floor( 1/2 + sqrt(2*n))-(2*n + round(sqrt(2*n)) - round(sqrt(2*n))^2)/2+1,(2*n + round(sqrt(2*n)) - round(sqrt(2*n))^2)/2) [From Leonid Bedratyuk (leonid.uk(AT)gmail.com), Dec 13 2009]
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EXAMPLE
| Triangle version begins
1,
1, 1,
1, 2, 1,
1, 2, 2, 1,
1, 2, 3, 2, 1,
1, 2, 3, 3, 2, 1,
1, 2, 3, 4, 3, 2, 1,
1, 2, 3, 4, 4, 3, 2, 1,
1, 2, 3, 4, 5, 4, 3, 2, 1
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MAPLE
| a(n)=min(floor( 1/2 + sqrt(2*n))-(2*n + round(sqrt(2*n)) - round(sqrt(2*n))^2)/2+1, (2*n + round(sqrt(2*n)) - round(sqrt(2*n))^2)/2) [From Leonid Bedratyuk (leonid.uk(AT)gmail.com), Dec 13 2009]
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MATHEMATICA
| aa = {}; Do[Do[AppendTo[aa, (p/Pi) ArcSin[Sqrt[1/((1/Pi^2) (Zeta[2, k/p] + Zeta[2, (p - k)/p]))]]], {k, 1, p - 1}], {p, 2, 50}]; N[aa, 50] (*Artur Jasinski*) [From Artur Jasinski, Mar 07 2010]
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PROG
| (Haskell)
a003983 n k = a003983_tabl !! (n-1) !! (k-1)
a003983_tabl = map a003983_row [1..]
a003983_row n = hs ++ drop m (reverse hs)
where hs = [1..n' + m]
(n', m) = divMod n 2
-- Reinhard Zumkeller, Aug 14 2011
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CROSSREFS
| Cf. A002620, A001399, A087062, A115236, A115237, A124258.
A006752, A120268, A173945, A173947, A173948, A173949, A173953, A173954, A173955, A173973, A173982-A173986. [From Artur Jasinski, Mar 07 2010]
Cf. A004197.
Sequence in context: A184720 A054526 A113453 * A087062 A204026 A110537
Adjacent sequences: A003980 A003981 A003982 * A003984 A003985 A003986
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KEYWORD
| tabl,nonn,easy,nice
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AUTHOR
| Marc LeBrun (mlb(AT)well.com)
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EXTENSIONS
| More terms from Larry Reeves (larryr(AT)acm.org), Nov 08 2000
Entry revised by N. J. A. Sloane (njas(AT)research.att.com), Dec 05 2006
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