# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a282513 Showing 1-1 of 1 %I A282513 #77 Aug 10 2024 19:25:49 %S A282513 0,1,3,5,8,12,17,22,28,35,43,51,60,70,81,92,104,117,131,145,160,176, %T A282513 193,210,228,247,267,287,308,330,353,376,400,425,451,477,504,532,561, %U A282513 590,620,651,683,715,748,782,817,852,888,925,963 %N A282513 a(n) = floor((3*n + 2)^2/24 + 1/3). %C A282513 List of quadruples: 2*n*(3*n+1), (2*n+1)*(3*n+1), 6*n^2+8*n+3, (n+1)*(6*n+5). These terms belong to the sequences A033580, A033570, A126587 and A049452, respectively. See links for all the permutations. %C A282513 After 0, subsequence of A025767. %C A282513 It seems that a(n) is the smallest number of cells that need to be painted in a (n+1) X (n+1) grid, such that it has no unpainted hexominoes (see link to Kamenetsky and Pratt). - _Rob Pratt_, _Dmitry Kamenetsky_, Aug 30 2020 %H A282513 Harvey P. Dale, Table of n, a(n) for n = 0..1000 %H A282513 Luce ETIENNE, Permutations %H A282513 Dmitry Kamenetsky and Rob Pratt, 10x10 grid with no unpainted hexominoes, Puzzling StackExchange, October 2019. %H A282513 Rob Pratt, Optimal solution for n=11. %H A282513 Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,1). %F A282513 G.f.: x*(1 + x + x^3)/((1 + x)*(1 + x^2)*(1 - x)^3). %F A282513 a(n) = 2*a(n-1) - a(n-2) + a(n-4) - 2*a(n-5) + a(n-6) for n>5. %F A282513 a(n) = floor((3*n + 2)^2/24 + 2/3). %F A282513 a(n) = (6*n^2 + 8*n + 3 + (-1)^n - 2*((-1)^((2*n - 1 + (-1)^n)/4) + (-1)^((2*n + 1 - (-1)^n)/4)))/16. Therefore: %F A282513 a(2*k) = (6*k^2 + 4*k + 1 - (-1)^k)/4, %F A282513 a(2*k+1) = (k + 1)*(3*k + 2)/2. %F A282513 a(n) = (6*n^2 + 8*n + 3 + cos(n*Pi) - 4*cos(n*Pi/2))/16. %F A282513 a(n) = (3*n + 2)^2/24 + 1/3 + (-6 + (1 + (-1)^n)*(1 + 2*i^((n+1)*(n+2))))/16, where i=sqrt(-1). %F A282513 a(n) = A130519(n+3)+A130519(n+2)+A130519(n). - _R. J. Mathar_, Jun 23 2021 %e A282513 Rectangular array with four columns: %e A282513 . 0, 1, 3, 5; %e A282513 . 8, 12, 17, 22; %e A282513 . 28, 35, 43, 51; %e A282513 . 60, 70, 81, 92; %e A282513 . 104, 117, 131, 145, etc. %e A282513 From _Rob Pratt_, Aug 30 2020: (Start) %e A282513 For n = 3, painting only 2 cells would leave an unpainted hexomino, but painting the following 3 cells avoids all unpainted hexominoes: %e A282513 . . . %e A282513 . . X %e A282513 X X . %e A282513 (End) %t A282513 Table[Floor[(3 n + 2)^2/24 + 1/3], {n, 0, 50}] (* or *) CoefficientList[Series[x (1 + x + x^3)/((1 + x) (1 + x^2) (1 - x)^3), {x, 0, 50}], x] (* or *) Table[(6 n^2 + 8 n + 3 + Cos[n Pi] - 4 Cos[n Pi/2])/16, {n, 0, 50}] (* or *) Table[(3 n + 2)^2/24 + 1/3 + (-6 + (1 + (-1)^n) (1 + 2 I^((n + 1) (n + 2))))/16, {n, 0, 50}] (* _Michael De Vlieger_, Feb 17 2017 *) %t A282513 LinearRecurrence[{2,-1,0,1,-2,1},{0,1,3,5,8,12},60] (* _Harvey P. Dale_, Aug 10 2024 *) %o A282513 (PARI) a(n)=(3*n^2 + 4*n + 4)\8 \\ _Charles R Greathouse IV_, Feb 17 2017 %o A282513 (Magma) [(3*n^2+4*n+4) div 8: n in [0..50]]; // _Bruno Berselli_, Feb 17 2017 %Y A282513 Cf. A033436: floor((3*n)^2/24 + 1/3). %Y A282513 Cf. A000326, A000567, A025767, A033570, A033580, A049452, A064412, A126587, A222017, A269064, A274221. %Y A282513 Cf. A130519. %Y A282513 Minimum number of painted cells in other n-ominoes: A337501, A337502, A337503. %K A282513 nonn,easy %O A282513 0,3 %A A282513 _Luce ETIENNE_, Feb 17 2017 %E A282513 Corrected and extended by _Bruno Berselli_, Feb 17 2017 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE