A099392 a(n) = floor((n^2 - 2*n + 3)/2).

1, 1, 3, 5, 9, 13, 19, 25, 33, 41, 51, 61, 73, 85, 99, 113, 129, 145, 163, 181, 201, 221, 243, 265, 289, 313, 339, 365, 393, 421, 451, 481, 513, 545, 579, 613, 649, 685, 723, 761, 801, 841, 883, 925, 969, 1013, 1059, 1105, 1153, 1201, 1251, 1301, 1353, 1405

Offset: 1

Author

Ralf Stephan following a suggestion from Luke Pebody, Oct 20 2004

Keywords

• nonn
• easy

Links

• Guenther Schrack, Table of n, a(n) for n = 1..10010
• Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Crossrefs

Differs from A085913 at n = 61. Apart from leading term, identical to A080827.

Cf. A000217, A001844, A002522, A007494, A007590, A058331 (bisections).

From Guenther Schrack, Apr 17 2018: (Start)

First differences: A052928.

Partial sums: A212964(n) + n for n > 0.

Also A058331 and A001844 interleaved. (End)

Cf. A014601, A033683, A047838, A074148, A109613, A183575, A290743.

Programs

Mathematica

Array[Floor[(#^2 - 2 # + 3)/2] &, 54] (* or *)
Rest@ CoefficientList[Series[x (-1 + x - x^2 - x^3)/((1 + x) (x - 1)^3), {x, 0, 54}], x] (* Michael De Vlieger, Apr 21 2018 *)

PARI

a(n)=(n^2+3)\2-n \\ Charles R Greathouse IV, Aug 01 2013

Formula

a(n) = ceiling(n^2/2)-n+1. - Paul Barry, Jul 16 2006; index shifted by R. J. Mathar, Jul 29 2007

a(n) = ceiling(A002522(n-1)/2). - Branko Curgus, Sep 02 2007

From R. J. Mathar, Feb 20 2011: (Start)

G.f.: x *( -1+x-x^2-x^3 ) / ( (1+x)*(x-1)^3 ).

a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).

a(n+1) = (3 + 2*n^2 + (-1)^n)/4. (End)

a(n) = A007590(n-1) + 1 for n >= 2. - Richard R. Forberg, Aug 01 2013

a(n) = A000217(n) - A007494(n-1). - Bui Quang Tuan, Mar 27 2015

From Guenther Schrack, Apr 17 2018: (Start)

a(n) = (2*n^2 - 4*n + 5 -(-1)^n)/4.

a(n+2) = a(n) + 2*n for n > 0.

a(n) = 2*A033683(n-1) - 1 for n > 0.

a(n) = A047838(n-1) + 2 for n > 2.

a(n) = A074148(n-1) - n + 2 for n > 1.

a(n) = A183575(n-3) + 3 for n > 3.

a(n) = 2*A290743(n-1) - 3 for n > 0.

a(n) = 2*A290743(n-2) + A109613(n-5) for n > 4.

a(n) = A074148(n) - A014601(n-1) for n > 0. (End)

Sum_{n>=1} 1/a(n) = tanh(Pi/2)*Pi/2 + coth(Pi/sqrt(2))*Pi/(2*sqrt(2)) + 1/2. - Amiram Eldar, Sep 16 2022

E.g.f.: ((2 - x + x^2)*cosh(x) + (3 - x + x^2)*sinh(x) - 2)/2. - Stefano Spezia, Jan 28 2024