A084263 a(n) = (-1)^n/2+(n^2+n+1)/2.

1, 1, 4, 6, 11, 15, 22, 28, 37, 45, 56, 66, 79, 91, 106, 120, 137, 153, 172, 190, 211, 231, 254, 276, 301, 325, 352, 378, 407, 435, 466, 496, 529, 561, 596, 630, 667, 703, 742, 780, 821, 861, 904, 946, 991, 1035, 1082, 1128, 1177, 1225, 1276, 1326, 1379, 1431

Offset: 0

Author

Paul Barry, May 31 2003

Keywords

• easy
• nonn

Comments

Old name was "Modified triangular numbers".

Starting with offset 1 = row sums of an infinite lower triangular matrix with alternate columns of (1, 3, 5, 7, ...) and (1, 0, 0, 0, ...) (see example). - Gary W. Adamson, May 14 2010

Examples

			From _Gary W. Adamson_, May 14 2010: (Start)
First few rows of the triangle with row sums = A084263 =
1;
3, 1;
5, 0, 1;
7, 0, 3, 1;
9, 0, 5, 0, 1;
11, 0, 7, 0, 3, 1;
...
Example: a(4) = 11 = (7 + 0 + 3 + 1). (End)
		

Links

• Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Crossrefs

Partial sums of A004442.

Cf. A000217, A002522, A059841, A084264, A084570.

Programs

Magma

[(-1)^n/2+(n^2+n+1)/2 : n in [0..80]]; // Wesley Ivan Hurt, Dec 23 2021

Mathematica

Table[(-1)^n/2 + (n^2 + n + 1)/2, {n, 0, 100}] (* Wesley Ivan Hurt, Dec 23 2021 *)
LinearRecurrence[{2,0,-2,1},{1,1,4,6},60] (* Harvey P. Dale, Jun 10 2023 *)

PARI

a(n)=n*(n+1)/2+(1-n)%2 \\ Charles R Greathouse IV, Jun 04 2013

Formula

E.g.f.: cosh(x)+exp(x)*(x+x^2/2).

a(n) = Sum_{k=0..n} k+(-1)^k.

a(n) = A000217(n)+A059841(n). Partial sums are A084570. Binomial transform is A084264.

G.f.: (1-x+2*x^2)/((1-x)^3*(1+x)). - R. J. Mathar, Apr 02 2008

a(0) = 1, a(n) = n^2 - a(n-1) + 1 for n >= 1. - Richard R. Forberg, Jun 05 2013

a(n) = 1 + floor(n/2) + floor(n^2/2). - Wesley Ivan Hurt, Jun 15 2013

a(n) + a(n+1) = A002522(n+1). - R. J. Mathar, May 21 2018

a(n) = 2*a(n-1)-2*a(n-3)+a(n-4). - Wesley Ivan Hurt, Dec 23 2021

Extensions

Name changed by Wesley Ivan Hurt, Dec 23 2021