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 A248851 a(n) = ( 2*n*(2*n^2 + 9*n + 14) + (-1)^n - 1 )/16. 4
 0, 3, 10, 22, 41, 68, 105, 153, 214, 289, 380, 488, 615, 762, 931, 1123, 1340, 1583, 1854, 2154, 2485, 2848, 3245, 3677, 4146, 4653, 5200, 5788, 6419, 7094, 7815, 8583, 9400, 10267, 11186, 12158, 13185, 14268, 15409, 16609, 17870, 19193, 20580, 22032 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Consider a grid of small triangles of side 1 forming a regular polygon with side n*(n+2); a(n) is the number of equilateral triangles of side length >= 1 in this figure which are oriented with the sides of the figure. This sequence gives the number of triangles of all sizes in a (n^2+2*n)-iamond with a 4*n-gon configuration. Equals (1/2)*Sum_{j=0..n-1} (n-j)*(n+1-j) + (-1 + (1/8)*Sum_{j=0..(2*n+1-(-1)^n)/4} (2*n+3+(-1)^n-4*j)*(2*n+3-(-1)^n-4*j)) numbers of triangles in a direction and in the opposite direction. LINKS Colin Barker, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1). FORMULA G.f.: x*(x^3-2*x^2+x+3) / ((x-1)^4*(x+1)). - Colin Barker, Mar 03 2015 a(n) = 3*a(n-1)-2*a(n-2)-2*a(n-3)+3*a(n-4)-a(n-5). - Colin Barker, Mar 03 2015 EXAMPLE From third comment: a(0)=0, a(1)=1+2, a(2)=4+6, a(3)=10+12, a(4)=20+21, a(5)=35+33. MATHEMATICA CoefficientList[Series[x (x^3 - 2 x^2 + x + 3) / ((x - 1)^4(x + 1)), {x, 0, 50}], x] (* Vincenzo Librandi, Mar 21 2015 *) LinearRecurrence[{3, -2, -2, 3, -1}, {0, 3, 10, 22, 41}, 50] (* Harvey P. Dale, Jan 17 2023 *) PROG (PARI) concat(0, Vec(x*(x^3-2*x^2+x+3)/((x-1)^4*(x+1)) + O(x^100))) \\ Colin Barker, Mar 03 2015 (Magma) [(4*n^3+18*n^2+28*n-(1-(-1)^n)) div 16: n in [0..50]]; // Vincenzo Librandi, Mar 21 2015 CROSSREFS Cf. A117143, A000292, A002623, A045947, A005563, A008586. Sequence in context: A122795 A140066 A006503 * A023554 A294414 A299336 Adjacent sequences: A248848 A248849 A248850 * A248852 A248853 A248854 KEYWORD nonn,easy AUTHOR Luce ETIENNE, Mar 03 2015 EXTENSIONS Typo in formula fixed by Vincenzo Librandi, Mar 21 2015 Name rewritten using the closed form by Bruno Berselli, Apr 19 2015 STATUS approved

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Last modified June 22 15:32 EDT 2024. Contains 373587 sequences. (Running on oeis4.)