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A144678
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Related to enumeration of quantum states (see reference for precise definition).
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7
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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
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OFFSET
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0,2
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COMMENTS
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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
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (2,-1,0,2,-4,2,0,-1,2,-1).
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FORMULA
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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) = Sum_{i=1..n+8} floor(i/4) * floor((n+8-i)/4). - Wesley Ivan Hurt, Jul 21 2014
G.f.: 1/((1-x)*(1-x^4))^2.
a(n) = Sum_{j=0..floor(n/4)} (j+1)*(n-4*j+1). (End)
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MAPLE
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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;
end:
binomial(floor(n/4)+3, 3)
end:
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MATHEMATICA
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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 *)
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PROG
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(Magma) R<x>:=PowerSeriesRing(Integers(), 60); Coefficients(R!( 1/((1-x)*(1-x^4))^2 )); // G. C. Greubel, Oct 18 2021
(Sage)
P.<x> = PowerSeriesRing(ZZ, prec)
return P( 1/((1-x)*(1-x^4))^2 ).list()
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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