A008642 - OEIS

%I #116 Apr 07 2023 11:27:09

%S 1,1,2,2,4,4,6,6,9,9,12,12,16,16,20,20,25,25,30,30,36,36,42,42,49,49,

%T 56,56,64,64,72,72,81,81,90,90,100,100,110,110,121,121,132,132,144,

%U 144,156,156,169,169,182,182,196,196,210,210,225,225

%N Quarter-squares repeated.

%C The area of the largest rectangle whose perimeter is not greater than n. - _Dmitry Kamenetsky_, Aug 30 2006

%C Also number of partitions of n into parts 1, 2 or 4. - _Reinhard Zumkeller_, Aug 12 2011

%C Let us consider a rectangle composed of unit squares. Then count how many squares are necessary to surround this rectangle by a layer whose width is 1 unit. And repeat this surrounding ad libitum. This sequence, prepended by 4 zeros and with offset 0, gives the number of rectangles that need 2*n unit squares in one of their surrounding layers. - _Michel Marcus_, Sep 19 2015

%C a(n) is the number of nonnegative integer solutions (x,y,z) for n-2 <= 2*x + 3*y + 4*z <= n. For example, the two solutions for 1 <= 2*x + 3*y + 4*z <= 3 are (1,0,0) and (0,1,0). - _Ran Pan_, Oct 07 2015

%C Conjecture: Consider the number of compositions of n>=4*k+8 into odd parts, where the order of the parts 1,3,..,2k+1 does not count. Then, as k approaches infinity, a(n-4*k-8) is equal to the number of these restricted compositions minus A000009(n), the number of strict partitions of n. - _Gregory L. Simay_, Aug 12 2016

%C From _Gus Wiseman_, May 17 2019: (Start)

%C Also the number of length-3 integer partitions of n + 4 whose largest part is greater than the sum of the other two. These are unordered triples that cannot be the sides of a triangle. For example, the a(1) = 1 through a(10) = 9 partitions are (A = 10, B = 11, C = 12):

%C   (311)  (411)  (421)  (521)  (522)  (622)  (632)  (732)   (733)   (833)

%C                 (511)  (611)  (531)  (631)  (641)  (741)   (742)   (842)

%C                               (621)  (721)  (722)  (822)   (751)   (851)

%C                               (711)  (811)  (731)  (831)   (832)   (932)

%C                                             (821)  (921)   (841)   (941)

%C                                             (911)  (A11)   (922)   (A22)

%C                                                            (931)   (A31)

%C                                                            (A21)   (B21)

%C                                                            (B11)   (C11)

%C (End)

%C This sequence, prepended by four 0's and with offset 0, is the number of partitions of n into four parts whose smallest two parts are equal. - _Wesley Ivan Hurt_, Jan 05 2021

%C This sequence, prepended by four 0's and with offset 0, is the number of incongruent obtuse triangles formed from the vertices of a regular n-gon. - _Frank M Jackson_, Nov 27 2022

%D D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 105.

%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 112, D(n).

%H Vincenzo Librandi, <a href="/A008642/b008642.txt">Table of n, a(n) for n = 0..1000</a>

%H Sara C. Billey, Matjaž Konvalinka, and Joshua P. Swanson, <a href="https://arxiv.org/abs/1905.00975">Asymptotic normality of the major index on standard tableaux</a>, arXiv:1905.00975 [math.CO], 2019.

%H Ran Pan, <a href="http://www.math.ucsd.edu/~projectp/warmups/eU.html">Exercise U</a>, Project P.

%H Harold N. Ward, <a href="https://arxiv.org/abs/2201.00389">A Normal Graph Algebra</a>, arXiv:2201.00389 [math.CO], 2022.

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1,1,-1,-1,1).

%F G.f.: 1/((1-x)*(1-x^2)*(1-x^4)).

%F a(n) = (2*n^2 + 14*n + 21 + (2*n + 7)*(-1)^n)/32 + ((1 + (-1)^n)/2 - (1 - (-1)^n)*i/2)*i^n/8, with i = sqrt(-1).

%F a(n) = floor(((n+1)*((-1)^n+n+6)+9)/16). - _Tani Akinari_, Jun 16 2013

%F a(n) = Sum_{i=1..floor((n+6)/2)} floor((n+6-2*i-(n mod 2))/4). - _Wesley Ivan Hurt_, Mar 31 2014

%F a(0)=1, a(1)=1, a(2)=2, a(3)=2, a(4)=4, a(5)=4, a(6)=6; for n>6, a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4) - a(n-5) - a(n-6) + a(n-7). - _Harvey P. Dale_, Jun 03 2015

%F a(n) = floor(floor(n/2+2)^2/4) = floor(floor(n/2+2)^2/2)/2. - _Bruno Berselli_, Mar 03 2016

%F E.g.f.: ((14 + 7*x + x^2)*cosh(x) + 2*(cos(x) + sin(x)) + (7 + 9*x + x^2)*sinh(x))/16. - _Stefano Spezia_, Mar 05 2023

%F a(n) = floor((n + 4)/4)*floor((n + 6)/4). - _Ridouane Oudra_, Apr 01 2023

%p seq((7/8+(-1)^k/8 + k + k^2/4)$2, k=0..100); # _Robert Israel_, Oct 08 2015

%t CoefficientList[Series[1/((1-x)(1-x^2)(1-x^4)), {x, 0, 70}], x] (* _Vincenzo Librandi_, Apr 02 2014 *)

%t LinearRecurrence[{1,1,-1,1,-1,-1,1},{1,1,2,2,4,4,6}, 70] (* _Harvey P. Dale_, Jun 03 2015 *)

%t Table[Floor[((n + 1) ((-1)^n + n + 6) + 9)/16], {n, 0, 70}] (* _Michael De Vlieger_, Aug 14 2016 *)

%o (PARI) Vec(1/((1-x)*(1-x^2)*(1-x^4)) + O(x^70)) \\ _Michel Marcus_, Mar 31 2014

%o (PARI) vector(70, n, n--; floor(((n+1)*((-1)^n+n+6)+9)/16)) \\ _Altug Alkan_, Oct 08 2015

%o (Magma) [Floor(((n+1)*((-1)^n+n+6)+9)/16): n in [0..70]]; // _Vincenzo Librandi_, Apr 02 2014

%o (Sage) [floor(floor(n/2+2)^2/2)/2 for n in (0..70)] # _Bruno Berselli_, Mar 03 2016

%Y Cf. A002620.

%Y Cf. A001399, A005044 (triangles without self-intersections), A069905, A124278, A266223, A325686, A325689, A325690, A325691, A325695.

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_