This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A008642 #116 Apr 07 2023 11:27:09
%S A008642 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 A008642 56,56,64,64,72,72,81,81,90,90,100,100,110,110,121,121,132,132,144,
%U A008642 144,156,156,169,169,182,182,196,196,210,210,225,225
%N A008642 Quarter-squares repeated.
%C A008642 The area of the largest rectangle whose perimeter is not greater than n. - _Dmitry Kamenetsky_, Aug 30 2006
%C A008642 Also number of partitions of n into parts 1, 2 or 4. - _Reinhard Zumkeller_, Aug 12 2011
%C A008642 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 A008642 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 A008642 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 A008642 From _Gus Wiseman_, May 17 2019: (Start)
%C A008642 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 A008642 (311) (411) (421) (521) (522) (622) (632) (732) (733) (833)
%C A008642 (511) (611) (531) (631) (641) (741) (742) (842)
%C A008642 (621) (721) (722) (822) (751) (851)
%C A008642 (711) (811) (731) (831) (832) (932)
%C A008642 (821) (921) (841) (941)
%C A008642 (911) (A11) (922) (A22)
%C A008642 (931) (A31)
%C A008642 (A21) (B21)
%C A008642 (B11) (C11)
%C A008642 (End)
%C A008642 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 A008642 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 A008642 D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 105.
%D A008642 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 112, D(n).
%H A008642 Vincenzo Librandi, <a href="/A008642/b008642.txt">Table of n, a(n) for n = 0..1000</a>
%H A008642 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 A008642 Ran Pan, <a href="http://www.math.ucsd.edu/~projectp/warmups/eU.html">Exercise U</a>, Project P.
%H A008642 Harold N. Ward, <a href="https://arxiv.org/abs/2201.00389">A Normal Graph Algebra</a>, arXiv:2201.00389 [math.CO], 2022.
%H A008642 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1,1,-1,-1,1).
%F A008642 G.f.: 1/((1-x)*(1-x^2)*(1-x^4)).
%F A008642 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 A008642 a(n) = floor(((n+1)*((-1)^n+n+6)+9)/16). - _Tani Akinari_, Jun 16 2013
%F A008642 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 A008642 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 A008642 a(n) = floor(floor(n/2+2)^2/4) = floor(floor(n/2+2)^2/2)/2. - _Bruno Berselli_, Mar 03 2016
%F A008642 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 A008642 a(n) = floor((n + 4)/4)*floor((n + 6)/4). - _Ridouane Oudra_, Apr 01 2023
%p A008642 seq((7/8+(-1)^k/8 + k + k^2/4)$2, k=0..100); # _Robert Israel_, Oct 08 2015
%t A008642 CoefficientList[Series[1/((1-x)(1-x^2)(1-x^4)), {x, 0, 70}], x] (* _Vincenzo Librandi_, Apr 02 2014 *)
%t A008642 LinearRecurrence[{1,1,-1,1,-1,-1,1},{1,1,2,2,4,4,6}, 70] (* _Harvey P. Dale_, Jun 03 2015 *)
%t A008642 Table[Floor[((n + 1) ((-1)^n + n + 6) + 9)/16], {n, 0, 70}] (* _Michael De Vlieger_, Aug 14 2016 *)
%o A008642 (PARI) Vec(1/((1-x)*(1-x^2)*(1-x^4)) + O(x^70)) \\ _Michel Marcus_, Mar 31 2014
%o A008642 (PARI) vector(70, n, n--; floor(((n+1)*((-1)^n+n+6)+9)/16)) \\ _Altug Alkan_, Oct 08 2015
%o A008642 (Magma) [Floor(((n+1)*((-1)^n+n+6)+9)/16): n in [0..70]]; // _Vincenzo Librandi_, Apr 02 2014
%o A008642 (Sage) [floor(floor(n/2+2)^2/2)/2 for n in (0..70)] # _Bruno Berselli_, Mar 03 2016
%Y A008642 Cf. A002620.
%Y A008642 Cf. A001399, A005044 (triangles without self-intersections), A069905, A124278, A266223, A325686, A325689, A325690, A325691, A325695.
%K A008642 nonn,easy
%O A008642 0,3
%A A008642 _N. J. A. Sloane_