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A003983 Array read by antidiagonals with T(n,k) = min(n,k). 34
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; text; internal format)
OFFSET

1,5

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. - Artur Jasinski, Mar 07 2010

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)

LINKS

Reinhard Zumkeller, Rows n=1..100 of triangle, flattened

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). - Leonid Bedratyuk, Dec 13 2009

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

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) # Leonid Bedratyuk, Dec 13 2009

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, Mar 07 2010 *)

Flatten[Table[Min[n-k+1, k], {n, 1, 14}, {k, 1, n}]] (* Jean-François Alcover, Feb 23 2012 *)

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

(PARI) T(n, k) = min(n, k) \\ Charles R Greathouse IV, Feb 06 2017

CROSSREFS

Cf. A002620, A001399, A087062, A115236, A115237, A124258, A006752, A120268, A173945, A173947, A173948, A173949, A173953, A173954, A173955, A173973, A173982-A173986, A004197.

Sequence in context: A113453 A245851 A230596 * A087062 A204026 A110537

Adjacent sequences:  A003980 A003981 A003982 * A003984 A003985 A003986

KEYWORD

tabl,nonn,easy,nice

AUTHOR

Marc LeBrun

EXTENSIONS

More terms from Larry Reeves (larryr(AT)acm.org), Nov 08 2000

Entry revised by N. J. A. Sloane, Dec 05 2006

STATUS

approved

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Last modified May 25 16:02 EDT 2017. Contains 287039 sequences.