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A109900 The (n,r)-th term of the following triangle is T(n)-T(r) for r = 0 to n. The n-th row contains n+1 terms. T(n) = the n-th triangular number = n(n+1)/2. Sequence contains the sum of terms at a 45-degree angle. 1
0, 1, 3, 8, 15, 27, 42, 64, 90, 125, 165, 216, 273, 343, 420, 512, 612, 729, 855, 1000, 1155, 1331, 1518, 1728, 1950, 2197, 2457, 2744, 3045, 3375, 3720, 4096, 4488, 4913, 5355, 5832, 6327, 6859, 7410, 8000, 8610, 9261, 9933, 10648, 11385, 12167, 12972 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Initial terms match those of A047866 with a difference of +1 or -1 in some cases. A047866: 0, 1, 3, 8, 15, 27, 42, 63, 90, 124, 165, 215, ...

LINKS

Table of n, a(n) for n=0..46.

FORMULA

a(2n+1) = (n+1)^3; a(2n) = (2n+1)*T(n) = (2n+1)*(n+1)*n/2, where T=A000217. - R. J. Mathar, Feb 11 2008

a(n) = A034828(n+1). - R. J. Mathar, Aug 18 2008

G.f.: x*(1+x+x^2)/(1-2*x-x^2+4*x^3-x^4-2*x^5+x^6). - Colin Barker, Jan 04 2012

a(n) = (2*n^3+6*n^2+5*n+1-(n+1)*(-1)^n)/16. - Luce ETIENNE, May 12 2015

EXAMPLE

The (n,r)-th term of the following triangle is T(n)-T(r) for r = 0 to n. The n-th row contains n+1 terms.

   0

   1  0

   3  2  0

   6  5  3  0

  10  9  7  4  0

  15 14 12  9  5  0

  21 20 18 15 11  6  0

  28 27 ...

  36 ...

Sequence contains the sum of terms at a 45-degree angle.

a(5) = 15 + 9 + 3 = 27.

MAPLE

A109900 := proc(n) if n mod 2 = 1 then ( (n+1)/2)^3 ; else (n+1)*(n/2+1)*(n/2)/2 ; fi ; end: seq(A109900(n), n=0..80) ; # R. J. Mathar, Feb 11 2008

CROSSREFS

Cf. A047866.

Sequence in context: A241565 A047866 A080183 * A034828 A081276 A210979

Adjacent sequences:  A109897 A109898 A109899 * A109901 A109902 A109903

KEYWORD

nonn,easy

AUTHOR

Amarnath Murthy, Jul 13 2005

EXTENSIONS

Corrected and extended by R. J. Mathar, Feb 11 2008

STATUS

approved

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Last modified August 27 07:38 EDT 2016. Contains 275904 sequences.