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%I #8 Jul 31 2015 18:00:23
%S -1,-2,-3,-2,3,14,33,62,103,158,229,318,427,558,713,894,1103,1342,
%T 1613,1918,2259,2638,3057,3518,4023,4574,5173,5822,6523,7278,8089,
%U 8958,9887,10878,11933,13054,14243,15502,16833,18238,19719,21278,22917,24638,26443,28334,30313,32382,34543,36798
%N a(n) = (1/3)*n^3 - n^2 - (1/3)*n - 1.
%C It is interesting that this sequence is generated using the same rules as those given for A108618 (version "jes"; the initial seed is the floretion given in program code, below). In reference to those rules, we have: **Loop 0** + .5'i + .5i' + .5'ik' + .5'ji' + e **Loop 1** + 1.5'i - .5'j + 1.5i' - .5k' + .5'ii' + 1.5'ik' + 1.5'ji' + .5'kj' + 3e **Loop 2** + 3.5'i - 2'j + 3.5i' - 2k' + 2'ii' + 3.5'ik' + 3.5'ji' + 2'kj' + 5e **Loop 3** + 6.5'i - 5.5'j + 6.5i' - 5.5k' + 5.5'ii' + 6.5'ik' + 6.5'ji' + 5.5'kj' + 7e **Loop 4** + 10.5'i - 12'j + 10.5i' - 12k' + 12'ii' + 10.5'ik' + 10.5'ji' + 12'kj' + 9e **Loop 5** + 15.5'i - 22.5'j + 15.5i' - 22.5k' + 22.5'ii' + 15.5'ik' + 15.5'ji' + 22.5'kj' + 11e **Loop 6** + 21.5'i - 38'j + 21.5i' - 38k' + 38'ii' + 21.5'ik' + 21.5'ji' + 38'kj' + 13e **Loop 7** + 28.5'i - 59.5'j + 28.5i' - 59.5k' + 59.5'ii' + 28.5'ik' + 28.5'ji' + 59.5'kj' + 15e **Loop 8** + 36.5'i - 88'j + 36.5i' - 88k' + 88'ii' + 36.5'ik' + 36.5'ji' + 88'kj' + 17e **Loop 9** + 45.5'i - 124.5'j + 45.5i' - 124.5k' + 124.5'ii' + 45.5'ik' + 45.5'ji' + 124.5'kj' + 19e. a(n) is calculated by adding the real number coefficients of 'i, 'j and 'k (which is always 0 here) from the n-th loop and multiplying the result by -2.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4, -6, 4, -1).
%F a(n) = A006527(n) - A002061(n+1), g.f. (2*x-1)*(x^2+1)/(x-1)^4
%F a(0)=-1, a(1)=-2, a(2)=-3, a(3)=-2, a(n)=4*a(n-1)-6*a(n-2)+4*a(n-3)- a(n-4). - _Harvey P. Dale_, Jul 21 2013
%p seriestolist(series((2*x-1)*(x^2+1)/(x-1)^4, x=0,50)); -or- Floretion Algebra Multiplication Program, FAMP Code: -2jessumseq[ + .5'i + .5i' + .5'ik' + .5'ji' + e], Sumtype: sum[Y[15]] = sum[ * ]. Note: 2ibasesumseq = A002061, apart from initial term, -2jbasesumseq = A006527.
%t Table[n^3/3-n^2-n/3-1,{n,0,50}] (* or *) LinearRecurrence[{4,-6,4,-1},{-1,-2,-3,-2},60] (* _Harvey P. Dale_, Jul 21 2013 *)
%Y Cf. A006527, A002061.
%K easy,sign
%O 0,2
%A _Creighton Dement_, Aug 01 2005
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