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%I #48 Sep 08 2022 08:46:16
%S 1,5,13,24,39,57,79,104,133,165,201,240,283,329,379,432,489,549,613,
%T 680,751,825,903,984,1069,1157,1249,1344,1443,1545,1651,1760,1873,
%U 1989,2109,2232,2359,2489,2623,2760,2901,3045,3193,3344,3499,3657,3819,3984,4153
%N a(n) = (7/4)*n^2 + (5/2)*n + (7 + (-1)^n)/8.
%C Let P be a polygon with vertices (0,0), (0,2), (1,1) and (0,3/2). The number of integer points in nP is counted by this quasi-polynomial (nP is the n-fold dilation of P). See Wikipedia in Links section.
%C From _Bob Selcoe_, Sep 10 2016: (Start)
%C a(n) = the number of partitions in reverse lexicographic order starting with n 3's followed by n 2's; i.e., the number of partitions summing to 5n such that no part > 3 and the number of 3's digits <= the number of 2's digits.
%C First differences are A047346(n+1); second differences are 4 when n is even and 3 when n is odd (i.e., A010702(n+1); third differences are 1 when n is even and -1 when n is odd. (End)
%H Vincenzo Librandi, <a href="/A271937/b271937.txt">Table of n, a(n) for n = 0..1000</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Ehrhart_polynomial#Examples_of_Ehrhart_Quasi-Polynomials">Examples of Ehrhart Quasi-Polynomials</a>.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).
%F O.g.f.: (1 + 3*x + 3*x^2)/((1 - x)^3*(1 + x)).
%F E.g.f.: (7 + 34*x + 14*x^2)*exp(x)/8 + exp(-x)/8.
%F a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
%F a(2*k) = k*(7*k + 5) + 1, a(2*k+1) = (k + 1)*(7*k + 5).
%F From _Bob Selcoe_, Sep 10 2016 (Start):
%F a(n) = (n+1)^2 + A006578(n).
%F a(n) = a(n-1) + A047346(n+1).
%F a(n) = Sum_{j=0..n} floor((2n+3j+2)/2).
%F (End)
%e a(1) = 5; the 5 partitions are: {3,2}; {3,1,1}; {2,2,1}; {2,1,1,1}; {1,1,1,1,1}.
%e a(3) = 24: floor(8/2) + floor(11/2) + floor(14/2) + floor(17/2) = 4+5+7+8 = 24.
%t Table[(7/4) n^2 + (5/2) n + (7 + (-1)^n)/8, {n, 0, 50}]
%o (Magma) [(7/4)*n^2+(5/2)*n+(7+(-1)^n)/8: n in [0..50]];
%o (PARI) Vec((1+3*x+3*x^2)/((1-x)^3*(1+x)) + O(x^99)) \\ _Altug Alkan_, Sep 10 2016
%Y First bisection (after 1) is A168235.
%Y Second bisection is A135703 (without 0).
%Y Cf. A006578, A047346.
%K nonn,easy
%O 0,2
%A _Vincenzo Librandi_, Apr 20 2016
%E Edited and extended by _Bruno Berselli_, Apr 20 2016
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