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A128282 For definition see Comment lines . 1
1, 2, 2, 3, 4, 3, 5, 6, 6, 5, 7, 8, 9, 8, 7, 10, 11, 12, 12, 11, 10, 13, 14, 15, 16, 15, 14, 13, 17, 18, 19, 20, 20, 19, 18, 17, 21, 22, 23, 24, 25, 24, 23, 22, 21, 26, 27, 28, 29, 30, 30, 29, 28, 27, 26, 31, 32, 33, 34, 35, 36, 35, 34, 33, 32, 31, 37, 38, 39, 40, 41, 42, 42, 41 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Left half triangle is A000027 (natural numbers): 1 ; 2 ; 3, 4 ; 5, 6 ; 7, 8, 9 ; 10, 11, 12 ; 13, 14, 15, 16 ; 17, 18, 19, 20 ;...

LINKS

Table of n, a(n) for n=0..73.

FORMULA

T(n,k) = T(n,n-k).

T(2*n,n) = (n+1)^2 = A000290(n+1).

T(n,0) = T(n,n) = A033638(n+1).

From Yu-Sheng Chang, May 25 2020: (Start)

O.g.f.: F(z,v) = (z/((-z+1)^3*(z+1))-v^2*z/((-v*z+1)^3*(v*z+1)))/(1-v)+1/((-z+1)*(-v*z+1)*(-v*z^2+1)).

T(n,k) = [v^k] (1/8)*(1-v^(n+1))*(2*(n+1)^2-1-(-1)^n)/(1-v) + (v^(2+n)+(1/2*((sqrt(v)-1)^2*(-1)^n-(sqrt(v)+1)^2))*v^((1/2)*n+1/2)+1)/(1-v)^2.

T(n,k) = 1 + (1/4)*n*(n+1) + min(k, n-k) + (1/2)*ceiling((1/2)*n). (End)

EXAMPLE

Triangle begins:

   1;

   2,  2;

   3,  4,  3;

   5,  6,  6,  5;

   7,  8,  9,  8,  7;

  10, 11, 12, 12, 11, 10;

  13, 14, 15, 16, 15, 14, 13;

  17, 18, 19, 20, 20, 19, 18, 17;

  ...

MAPLE

A := proc(n, k) ## n >= 0 and k = 0 .. n

    1+(1/4)*n*(n+1)+min(k, n-k)+(1/2)*ceil((1/2)*n)

end proc: # Yu-Sheng Chang, May 25 2020

CROSSREFS

Cf. A000027, A000290, A033638 (1st column and right diagonal).

Sequence in context: A205002 A290735 A165634 * A146985 A132993 A106408

Adjacent sequences:  A128279 A128280 A128281 * A128283 A128284 A128285

KEYWORD

nonn,tabl

AUTHOR

Philippe Deléham, May 03 2007

STATUS

approved

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Last modified April 17 14:32 EDT 2021. Contains 343063 sequences. (Running on oeis4.)