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.

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

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.

Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

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

a(n) = Sum_{k=0..n} A001318(k). - Jacob Szlachetka, Dec 20 2021

Sum_{n>=1} 1/a(n) = 6 - 8*log(2) + zeta(3). - Amiram Eldar, Sep 17 2022

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

Mathematica

LinearRecurrence[{2, 1, -4, 1, 2, -1}, {0, 1, 3, 8, 15, 27}, 50] (* Amiram Eldar, Sep 17 2022 *)

Crossrefs

Cf. A000217, A001318, A034828, A047866.

Sequence in context: A359210 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