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 A144678 Related to enumeration of quantum states (see reference for precise definition). 7
 1, 2, 3, 4, 7, 10, 13, 16, 22, 28, 34, 40, 50, 60, 70, 80, 95, 110, 125, 140, 161, 182, 203, 224, 252, 280, 308, 336, 372, 408, 444, 480, 525, 570, 615, 660, 715, 770, 825, 880, 946, 1012, 1078, 1144, 1222, 1300, 1378, 1456, 1547, 1638, 1729, 1820, 1925, 2030, 2135 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The Gi2 triangle sums of the triangle A159797 are linear sums of shifted versions of the sequence given above, i.e., Gi2(n) = a(n-1) + 2*a(n-2) + 2*a(n-3) + 3*a(n-4) + a(n-5). For the definitions of the Gi2 and other triangle sums see A180662. [Johannes W. Meijer, May 20 2011] Partial sums of 1,1,1,1, 3,3,3,3, 6,6,6,6,..., the quadruplicated A000217. - R. J. Mathar, Aug 25 2013 Number of partitions of n into two different parts of size 4 and two different parts of size 1. a(4) = 7: 4, 4', 1111, 1111', 111'1', 11'1'1', 1'1'1'1'. - Alois P. Heinz, Dec 22 2021 LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 Brian O'Sullivan and Thomas Busch, Spontaneous emission in ultra-cold spin-polarised anisotropic Fermi seas, arXiv 0810.0231v1 [quant-ph], 2008. [Eq 10b, lambda=4] Index entries for linear recurrences with constant coefficients, signature (2,-1,0,2,-4,2,0,-1,2,-1). FORMULA From Johannes W. Meijer, May 20 2011: (Start) a(n) = A190718(n-3) + A190718(n-2) + A190718(n-1) + A190718(n). a(n-3) + a(n-2) + a(n-1) + a(n) = A122046(n+3). G.f.: 1/((x-1)^4*(x^3+x^2+x+1)^2). (End) a(n) = A009531(n+5)/16 + (n+5)*(2*n^2+20*n+33+3*(-1)^n)/192 . - R. J. Mathar, Jun 20 2013 a(n) = Sum_{i=1..n+8} floor(i/4) * floor((n+8-i)/4). - Wesley Ivan Hurt, Jul 21 2014 From Alois P. Heinz, Dec 22 2021: (Start) G.f.: 1/((1-x)*(1-x^4))^2. a(n) = Sum_{j=0..floor(n/4)} (j+1)*(n-4*j+1). (End) MAPLE n:=80; lambda:=4; S10b:=[]; for ii from 0 to n do x:=floor(ii/lambda); snc:=1/6*(x+1)*(x+2)*(3*ii-2*x*lambda+3); S10b:=[op(S10b), snc]; od: S10b; A144678 := proc(n) option remember; local k; sum(A190718(n-k), k=0..3) end: A190718:= proc(n) binomial(floor(n/4)+3, 3) end: seq(A144678(n), n=0..54); # Johannes W. Meijer, May 20 2011 MATHEMATICA a[n_] = (r = Mod[n, 4]; (4+n-r)(8+n-r)(3+n+2r)/96); Table[a[n], {n, 0, 54}] (* Jean-François Alcover, Sep 02 2011 *) LinearRecurrence[{2, -1, 0, 2, -4, 2, 0, -1, 2, -1}, {1, 2, 3, 4, 7, 10, 13, 16, 22, 28}, 60] (* G. C. Greubel, Oct 18 2021 *) PROG (PARI) Vec(1/(x-1)^4/(x^3+x^2+x+1)^2+O(x^99)) \\ Charles R Greathouse IV, Jun 20 2013 (Magma) R:=PowerSeriesRing(Integers(), 60); Coefficients(R!( 1/((1-x)*(1-x^4))^2 )); // G. C. Greubel, Oct 18 2021 (Sage) def A144678_list(prec): P. = PowerSeriesRing(ZZ, prec) return P( 1/((1-x)*(1-x^4))^2 ).list() A144678_list(60) # G. C. Greubel, Oct 18 2021 CROSSREFS Cf. A000292, A006918, A144677, A144679, A190718. Sequence in context: A062042 A107817 A008811 * A309678 A279225 A073149 Adjacent sequences: A144675 A144676 A144677 * A144679 A144680 A144681 KEYWORD nonn,easy AUTHOR N. J. A. Sloane, Feb 06 2009 STATUS approved

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Last modified June 21 12:41 EDT 2024. Contains 373544 sequences. (Running on oeis4.)