A084265 a(n) = (n^2 + 3*n + 1 + (-1)^n) / 2.

1, 2, 6, 9, 15, 20, 28, 35, 45, 54, 66, 77, 91, 104, 120, 135, 153, 170, 190, 209, 231, 252, 276, 299, 325, 350, 378, 405, 435, 464, 496, 527, 561, 594, 630, 665, 703, 740, 780, 819, 861, 902, 946, 989, 1035, 1080, 1128, 1175, 1225, 1274, 1326, 1377, 1431, 1484

Offset: 0

Author

Paul Barry, May 31 2003

Keywords

• easy
• nonn

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).

Crossrefs

Cf. A084263.

Cf. A171608.

Cf. A000384, A014105, A014107.

Programs

Magma

[(n^2+3*n+1)/2+(-1)^n/2: n in [0..60]]; // Vincenzo Librandi, Aug 15 2013

Maple

A084265:=n->(n^2+3*n+1)/2+(-1)^n/2: seq(A084265(n),n=0..100); # Wesley Ivan Hurt, Mar 21 2015

Mathematica

CoefficientList[Series[(-1 - 2 x^2 + x^3) / ((1 + x) (x - 1)^3), {x, 0, 60}], x] (* Vincenzo Librandi, Aug 15 2013 *)

PARI

vector(100,n,(n^2+n-1-(-1)^n)/2) \\ Derek Orr, Mar 22 2015

Formula

a(n) = A000217(n)+A059841(n)+n.

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

Extensions

New name from Joerg Arndt, Aug 15 2013