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A145069 a(n) = n*(n^2 + 3*n + 5)/3. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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

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).

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

EXAMPLE

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.

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 *)

PROG

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

(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

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.

Sequence in context: A172112 A227347 A068043 * A293350 A256525 A192973

Adjacent sequences:  A145066 A145067 A145068 * A145070 A145071 A145072

KEYWORD

nonn,easy

AUTHOR

Vladimir Joseph Stephan Orlovsky, Sep 30 2008

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

STATUS

approved

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Last modified July 9 01:30 EDT 2020. Contains 335537 sequences. (Running on oeis4.)