OFFSET
0,2
COMMENTS
Previous name was: Modified triangular numbers.
Binomial transform is A084266.
Partial sums give A064843. - N. J. A. Sloane, Jul 20 2008
Starting with "1" = triangle A171608 * the odd integers, (1, 3, 5, ...). - Gary W. Adamson, Dec 12 2009
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).
FORMULA
E.g.f.: cosh(x) + exp(x)*(2x+x^2/2).
a(n) = (n^2+3*n+1)/2+(-1)^n/2.
G.f.: ( -1-2*x^2+x^3 ) / ( (1+x)*(x-1)^3 ). - R. J. Mathar, Nov 26 2012
From Wesley Ivan Hurt, Mar 21 2015: (Start)
a(n) = 2*a(n-1)-2*a(n-3)+a(n-4).
a(n) = Sum_{i=0..n+1} i-(-1)^i. (End)
a(2*n) = A000384(n+1); a(2*n-1) = A014105(n)-1; a(2*n-1) = A014107(n+1), for all integers n. - Hartmut F. W. Hoft, Feb 02 2022
MAPLE
MATHEMATICA
CoefficientList[Series[(-1 - 2 x^2 + x^3) / ((1 + x) (x - 1)^3), {x, 0, 60}], x] (* Vincenzo Librandi, Aug 15 2013 *)
PROG
(Magma) [(n^2+3*n+1)/2+(-1)^n/2: n in [0..60]]; // Vincenzo Librandi, Aug 15 2013
(PARI) vector(100, n, (n^2+n-1-(-1)^n)/2) \\ Derek Orr, Mar 22 2015
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, May 31 2003
EXTENSIONS
New name from Joerg Arndt, Aug 15 2013
STATUS
approved