A108766 - OEIS

%I #26 Jun 02 2026 16:24:45

%S -1,1,7,19,39,69,111,167,239,329,439,571,727,909,1119,1359,1631,1937,

%T 2279,2659,3079,3541,4047,4599,5199,5849,6551,7307,8119,8989,9919,

%U 10911,11967,13089,14279,15539,16871,18277,19759,21319,22959,24681,26487,28379,30359,32429,34591,36847,39199,41649

%N a(n) = A007290(n+2) - 1 = 2*C(n+2,3) - 1.

%C Note (in reference to FAMP program code): 1kbasejrokseq = A005286 (Number of permutations of [n+3] with three inversions), 1ibasekrokseq = A004006 = C(n,1) + C(n,2) + C(n,3) (from second term).

%C Floretion Algebra Multiplication Program, FAMP Code: a(n) = -1tesrok[(- 'j + 'k - 'ii' - 'ij' - 'ik')(- 'i + 'j - 'kk' - 'ki' - 'kj'), Roktype: Y[sqa.Findk()] = Y[sqa.Findk()] + p (internal program code).

%H Harvey P. Dale, <a href="/A108766/b108766.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F a(n) = n*(n+2)*(n+1)/3 - 1 = 2*A000292(n) - 1.

%F G.f.: (-1 + 5*x - 3*x^2 + x^3)/(x-1)^4.

%F From _Elmo R. Oliveira_, Jun 02 2026: (Start)

%F E.g.f.: exp(x)*(-3 + 6*x + 6*x^2 + x^3)/3.

%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)

%p a[0]:=-1:a[1]:=1:for n from 2 to 50 do a[n]:=a[n-1]+n^2+n od: seq(a[n], n=0..49);

%t s1=s2=s3=s4=0;lst={};Do[a=n+(n+2);s1+=a;s2+=s1;s3+=s2;s4+=s3;AppendTo[lst,s2-1],{n,0,5!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Apr 04 2009 *)

%t (* Alternative: *)

%t 2 Binomial[Range[2,60],3]-1 (* _Harvey P. Dale_, Aug 19 2021 *)

%Y Cf. A000292, A004006, A005286, A007290.

%K easy,sign,changed

%O 0,3

%A _Creighton Dement_, Jun 24 2005