

A109900


The (n,r)th term of the following triangle is T(n)T(r) for r = 0 to n. The nth row contains n+1 terms. T(n) = the nth triangular number = n(n+1)/2. Sequence contains the sum of terms at a 45degree 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
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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)/(12*xx^2+4*x^3x^42*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 nth 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 45degree 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,changed


AUTHOR

Amarnath Murthy, Jul 13 2005


EXTENSIONS

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


STATUS

approved



