cp's OEIS Frontend

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

%I A145069 #45 Sep 08 2022 08:45:38
%S A145069 0,3,10,23,44,75,118,175,248,339,450,583,740,923,1134,1375,1648,1955,
%T A145069 2298,2679,3100,3563,4070,4623,5224,5875,6578,7335,8148,9019,9950,
%U A145069 10943,12000,13123,14314,15575,16908,18315,19798,21359,23000,24723,26530
%N A145069 a(n) = n*(n^2 + 3*n + 5)/3.
%C A145069 Old name was: Partial sums of A002061, starting at n=2.
%C A145069 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
%H A145069 Vincenzo Librandi, <a href="/A145069/b145069.txt">Table of n, a(n) for n = 0..1000</a> (corrected by Ray Chandler, Jan 19 2019)
%H A145069 Cleve Moler, <a href="http://www.netlib.org/linpack/sgefa.f">LINPACK subroutine sgefa.f</a>, University of New Mexico, Argonne National Lab, 1978.
%H A145069 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A145069 G.f.: x*(3-2*x+x^2)/(1-x)^4.
%F A145069 a(n) = Sum_{j=2..n+1} A002061(j).
%F A145069 a(n) = a(n-1) + n^2 + n + 1 for n > 0, with a(0) = 0.
%F A145069 a(n) = n*(n^2+3*n+5)/3. - _Bruno Berselli_, Apr 01 2011
%F A145069 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - _Vincenzo Librandi_, Jun 30 2012
%F A145069 a(n) = Sum_{i=1..n} 3i+(n-i)^2. - _Wesley Ivan Hurt_, Aug 21 2014
%F A145069 a(n) = A007290(n+2) + n. - _Hugo Pfoertner_, Mar 28 2018
%e A145069 a(2) = a(1) + 2^2 + 2 + 1 = 3 + 4 + 2 + 1 = 10.
%e A145069 a(3) = a(2) + 3^2 + 3 + 1 = 10 + 9 + 3 + 1 = 23.
%p A145069 A145069:=n->n*(n^2+3*n+5)/3: seq(A145069(n), n=0..100); # _Wesley Ivan Hurt_, Aug 21 2014
%t A145069 lst={};s=0;Do[s+=n^2+n+1;AppendTo[lst,s-1],{n,0,5!}];lst
%t A145069 CoefficientList[Series[x(3-2*x+x^2)/(1-x)^4,{x,0,40}],x] (* _Vincenzo Librandi_, Jun 30 2012 *)
%t A145069 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 *)
%o A145069 (PARI) {a=0; for(n=1, 42, print1(a, ", "); a=a+n^2+n+1)} \\ adapted by _Michel Marcus_, Aug 23 2014
%o A145069 (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
%Y A145069 Cf. A002061 (n^2 - n + 1).
%Y A145069 Cf. A028387 (n + (n+1)^2).
%Y A145069 Cf. A077415 (zero followed by partial sums of A028387, starting at n=1).
%Y A145069 Cf. A007290.
%K A145069 nonn,easy
%O A145069 0,2
%A A145069 _Vladimir Joseph Stephan Orlovsky_, Sep 30 2008
%E A145069 Edited by _Klaus Brockhaus_, Oct 21 2008
%E A145069 G.f. adapted to the offset by _Bruno Berselli_, Apr 01 2011
%E A145069 Name, offset, and formulas changed by _Wesley Ivan Hurt_, Aug 21 2014