A145069 a(n) = n*(n^2 + 3*n + 5)/3.

0, 3, 10, 23, 44, 75, 118, 175, 248, 339, 450, 583, 740, 923, 1134, 1375, 1648, 1955, 2298, 2679, 3100, 3563, 4070, 4623, 5224, 5875, 6578, 7335, 8148, 9019, 9950, 10943, 12000, 13123, 14314, 15575, 16908, 18315, 19798, 21359, 23000, 24723, 26530

Offset: 0

Author

Vladimir Joseph Stephan Orlovsky, Sep 30 2008

Keywords

• nonn
• easy

Comments

Old name was: Partial sums of A002061, starting at n=2.

Number of floating point dot operations (multiplications and divisions) in the factorization of an (n+1) X (n+1) real matrix by Gaussian elimination as, e.g., implemented in LINPACK subroutines sgefa.f or dgefa.f. The number of multiplications alone is given by A007290. The number of additions is given by A000330. - Hugo Pfoertner, Mar 28 2018

Examples

			a(2) = a(1) + 2^2 + 2 + 1 = 3 + 4 + 2 + 1 = 10.
a(3) = a(2) + 3^2 + 3 + 1 = 10 + 9 + 3 + 1 = 23.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000 (corrected by Ray Chandler, Jan 19 2019)
• Cleve Moler, LINPACK subroutine sgefa.f, University of New Mexico, Argonne National Lab, 1978.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A002061 (n^2 - n + 1).

Cf. A028387 (n + (n+1)^2).

Cf. A077415 (zero followed by partial sums of A028387, starting at n=1).

Cf. A007290.

Programs

Magma

I:=[0, 3, 10, 23]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Jun 30 2012

Maple

A145069:=n->n*(n^2+3*n+5)/3: seq(A145069(n), n=0..100); # Wesley Ivan Hurt, Aug 21 2014

Mathematica

lst={};s=0;Do[s+=n^2+n+1;AppendTo[lst,s-1],{n,0,5!}];lst
CoefficientList[Series[x(3-2*x+x^2)/(1-x)^4,{x,0,40}],x] (* Vincenzo Librandi, Jun 30 2012 *)
Table[n (n^2+3n+5)/3,{n,0,50}] (* or *) LinearRecurrence[{4,-6,4,-1},{0,3,10,23},50] (* Harvey P. Dale, Sep 10 2016 *)

PARI

{a=0; for(n=1, 42, print1(a, ", "); a=a+n^2+n+1)} \\ adapted by Michel Marcus, Aug 23 2014

Formula

G.f.: x*(3-2*x+x^2)/(1-x)^4.

a(n) = Sum_{j=2..n+1} A002061(j).

a(n) = a(n-1) + n^2 + n + 1 for n > 0, with a(0) = 0.

a(n) = n*(n^2+3*n+5)/3. - Bruno Berselli, Apr 01 2011

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jun 30 2012

a(n) = Sum_{i=1..n} 3i+(n-i)^2. - Wesley Ivan Hurt, Aug 21 2014

a(n) = A007290(n+2) + n. - Hugo Pfoertner, Mar 28 2018

Extensions

Edited by Klaus Brockhaus, Oct 21 2008

G.f. adapted to the offset by Bruno Berselli, Apr 01 2011

Name, offset, and formulas changed by Wesley Ivan Hurt, Aug 21 2014