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 A008642 Quarter-squares repeated. 14
 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, 56, 56, 64, 64, 72, 72, 81, 81, 90, 90, 100, 100, 110, 110, 121, 121, 132, 132, 144, 144, 156, 156, 169, 169, 182, 182, 196, 196, 210, 210, 225, 225 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS The area of the largest rectangle whose perimeter is not greater than n. - Dmitry Kamenetsky, Aug 30 2006 Also number of partitions of n into parts 1, 2 or 4. - Reinhard Zumkeller, Aug 12 2011 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 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 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 From Gus Wiseman, May 17 2019: (Start) 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):   (311)  (411)  (421)  (521)  (522)  (622)  (632)  (732)   (733)   (833)                 (511)  (611)  (531)  (631)  (641)  (741)   (742)   (842)                               (621)  (721)  (722)  (822)   (751)   (851)                               (711)  (811)  (731)  (831)   (832)   (932)                                             (821)  (921)   (841)   (941)                                             (911)  (A11)   (922)   (A22)                                                            (931)   (A31)                                                            (A21)   (B21)                                                            (B11)   (C11) (End) 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 REFERENCES D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 105. L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 112, D(n). LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 Sara C. Billey, Matjaž Konvalinka, and Joshua P. Swanson, Asymptotic normality of the major index on standard tableaux, arXiv:1905.00975 [math.CO], 2019. Ran Pan, Exercise U, Project P. Index entries for linear recurrences with constant coefficients, signature (1,1,-1,1,-1,-1,1). FORMULA G.f.: 1/((1-x)*(1-x^2)*(1-x^4)). 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). a(n) = floor(((n+1)*((-1)^n+n+6)+9)/16). - Tani Akinari, Jun 16 2013 a(n) = Sum_{i=1..floor((n+6)/2)} floor((n+6-2*i-(n mod 2))/4). - Wesley Ivan Hurt, Mar 31 2014 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 a(n) = floor(floor(n/2+2)^2/4) = floor(floor(n/2+2)^2/2)/2. - Bruno Berselli, Mar 03 2016 MAPLE seq((7/8+(-1)^k/8 + k + k^2/4)\$2, k=0..100); # Robert Israel, Oct 08 2015 MATHEMATICA CoefficientList[Series[1/((1-x)(1-x^2)(1-x^4)), {x, 0, 70}], x] (* Vincenzo Librandi, Apr 02 2014 *) LinearRecurrence[{1, 1, -1, 1, -1, -1, 1}, {1, 1, 2, 2, 4, 4, 6}, 70] (* Harvey P. Dale, Jun 03 2015 *) Table[Floor[((n + 1) ((-1)^n + n + 6) + 9)/16], {n, 0, 70}] (* Michael De Vlieger, Aug 14 2016 *) PROG (PARI) Vec(1/((1-x)*(1-x^2)*(1-x^4)) + O(x^70)) \\ Michel Marcus, Mar 31 2014 (PARI) vector(70, n, n--; floor(((n+1)*((-1)^n+n+6)+9)/16)) \\ Altug Alkan, Oct 08 2015 (MAGMA) [Floor(((n+1)*((-1)^n+n+6)+9)/16): n in [0..70]]; // Vincenzo Librandi, Apr 02 2014 (Sage) [floor(floor(n/2+2)^2/2)/2 for n in (0..70)] # Bruno Berselli, Mar 03 2016 CROSSREFS Cf. A002620. Cf. A001399, A005044 (triangles without self-intersections), A069905, A124278, A266223, A325686, A325689, A325690, A325691, A325695. Sequence in context: A278296 A332305 A340282 * A001364 A029010 A060027 Adjacent sequences:  A008639 A008640 A008641 * A008643 A008644 A008645 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified April 16 18:53 EDT 2021. Contains 343050 sequences. (Running on oeis4.)