A143975 a(n) = floor(n*(n+3)/3).

1, 3, 6, 9, 13, 18, 23, 29, 36, 43, 51, 60, 69, 79, 90, 101, 113, 126, 139, 153, 168, 183, 199, 216, 233, 251, 270, 289, 309, 330, 351, 373, 396, 419, 443, 468, 493, 519, 546, 573, 601, 630, 659, 689, 720, 751, 783, 816, 849, 883, 918, 953, 989, 1026, 1063, 1101

Offset

1,2

Comments

Fourth diagonal of A143974, associated with counting unit squares in a lattice.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..3000

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

Formula

a(n) = floor(n*(n+3)/3).

From R. J. Mathar, Oct 05 2009: (Start)

a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5).

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

9*a(n) = 3*n^2 + 9*n - 2 + A099837(n+3). - R. J. Mathar, Apr 26 2022

Sum_{n>=1} 1/a(n) = 4/3 + (tan((sqrt(13)+2)*Pi/6) - cot((sqrt(13)+1)*Pi/6)) * Pi/sqrt(13). - Amiram Eldar, Oct 01 2022

E.g.f.: (exp(x)*(3*x*(4 + x) - 2) + 2*exp(-x/2)*cos(sqrt(3)*x/2))/9. - Stefano Spezia, Oct 24 2022

Example

Main diagonal of A143974: (0,1,3,5,8,12,...) = A000212;
2nd diagonal: (0,2,4,6,10,14,18,...) = A128422;
3rd diagonal: (1,2,5,8,11,16,21,...) = A032765;
4th diagonal: (1,3,6,9,13,18,23,...) = A143975.

Mathematica

a[n_] := Floor[n*(n+3)/3]; Array[a, 60] (* Amiram Eldar, Oct 01 2022 *)

Prog

(Magma) [Floor(n*(n+3)/3): n in [1..60]]; // Vincenzo Librandi, May 08 2011

Crossrefs

Cf. A099837, A143974.

Sequence in context: A323622 A154287 A092847 * A005488 A048202 A014785

Adjacent sequences: A143972 A143973 A143974 * A143976 A143977 A143978

Keyword

nonn,easy

Author

Clark Kimberling, Sep 06 2008

Status

approved