

A271937


a(n) = (7/4)*n^2 + (5/2)*n + (7 + (1)^n)/8.


1



1, 5, 13, 24, 39, 57, 79, 104, 133, 165, 201, 240, 283, 329, 379, 432, 489, 549, 613, 680, 751, 825, 903, 984, 1069, 1157, 1249, 1344, 1443, 1545, 1651, 1760, 1873, 1989, 2109, 2232, 2359, 2489, 2623, 2760, 2901, 3045, 3193, 3344, 3499, 3657, 3819, 3984, 4153
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OFFSET

0,2


COMMENTS

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 quasipolynomial (nP is the nfold dilation of P). See Wikipedia in Links section.
From Bob Selcoe, Sep 10 2016: (Start)
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.
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)


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Wikipedia, Examples of Ehrhart QuasiPolynomials.
Index entries for linear recurrences with constant coefficients, signature (2,0,2,1).


FORMULA

O.g.f.: (1 + 3*x + 3*x^2)/((1  x)^3*(1 + x)).
E.g.f.: (7 + 34*x + 14*x^2)*exp(x)/8 + exp(x)/8.
a(n) = 2*a(n1)  2*a(n3) + a(n4).
a(2*k) = k*(7*k + 5) + 1, a(2*k+1) = (k + 1)*(7*k + 5).
From Bob Selcoe, Sep 10 2016 (Start):
a(n) = (n+1)^2 + A006578(n).
a(n) = a(n1) + A047346(n+1).
a(n) = Sum_{j=0..n} floor((2n+3j+2)/2).
(End)


EXAMPLE

a(1) = 5; the 5 partitions are: {3,2}; {3,1,1}; {2,2,1}; {2,1,1,1}; {1,1,1,1,1}.
a(3) = 24: floor(8/2) + floor(11/2) + floor(14/2) + floor(17/2) = 4+5+7+8 = 24.


MATHEMATICA

Table[(7/4) n^2 + (5/2) n + (7 + (1)^n)/8, {n, 0, 50}]


PROG

(MAGMA) [(7/4)*n^2+(5/2)*n+(7+(1)^n)/8: n in [0..50]];
(PARI) Vec((1+3*x+3*x^2)/((1x)^3*(1+x)) + O(x^99)) \\ Altug Alkan, Sep 10 2016


CROSSREFS

First bisection (after 1) is A168235.
Second bisection is A135703 (without 0).
Cf. A006578, A047346.
Sequence in context: A119248 A114998 A140090 * A121511 A283750 A156679
Adjacent sequences: A271934 A271935 A271936 * A271938 A271939 A271940


KEYWORD

nonn,easy


AUTHOR

Vincenzo Librandi, Apr 20 2016


EXTENSIONS

Edited and extended by Bruno Berselli, Apr 20 2016


STATUS

approved



