A084264 -id:A084264 - cp's OEIS frontend

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

Paul Barry, May 31 2003

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

			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)

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

Partial sums of A004442.

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

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

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 *)

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

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

Name changed by Wesley Ivan Hurt, Dec 23 2021

A104270 a(n) = 2^(n - 2)*(binomial(n,2) + 2).

Original entry on oeis.org

1, 3, 10, 32, 96, 272, 736, 1920, 4864, 12032, 29184, 69632, 163840, 380928, 876544, 1998848, 4521984, 10158080, 22675456, 50331648, 111149056, 244318208, 534773760, 1166016512, 2533359616, 5486149632, 11844714496, 25501368320, 54760833024, 117306294272, 250718715904

Offset: 1

Ralf Stephan, Apr 17 2005

Cardinality of set of crossing-similarity classes.
Total number of hexagonal systems with n hexagons that cannot be placed in a cage of size n-1. - Parthasarathy Nambi, Sep 06 2006
a(n+1) is equal to n! times the determinant of the n X n matrix whose (i,j)-entry is KroneckerDelta[i,j](((i+2)/(i)) - 1) + 1. - John M. Campbell, May 20 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..237
M. Klazar, On identities concerning the numbers of crossings and nestings of two edges in matchings
Tosic R., Masulovic D., Stojmenovic I., Brunvoll J., Cyvin B. N. and Cyvin S. J., Enumeration of polyhex hydrocarbons to h = 17, J. Chem. Inf. Comput. Sci., 1995, 35, 181-187, Table 1 (with an error at h=16).
Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

Equals (1/2) A053730. Partial sums of A084264.

[2^(n-2)*(Binomial(n,2)+2): n in [1..30]]; // Vincenzo Librandi, May 24 2011

Table[n!*Det[Array[KroneckerDelta[#1,#2](((#1+2)/(#1))-1)+1 &, {n,n}]], {n, 1, 10}] (* John M. Campbell, May 20 2011 *)
LinearRecurrence[{6,-12,8},{1,3,10},30] (* Harvey P. Dale, Jul 03 2017 *)

a(n)=(binomial(n,2)+2)<<(n-2) \\ Charles R Greathouse IV, May 24 2011

G.f.: x*(1 - 3*x + 4*x^2)/(1-2*x)^3. - Colin Barker, Apr 01 2012

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions