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 A143975 a(n) = floor(n*(n+3)/3). 3
 1, 3, 6, 9, 13, 18, 23, 29, 36, 43, 51, 60, 69, 79, 90, 101, 113, 126, 139, 153, 168, 183, 199, 216, 233, 251, 270, 289, 309, 330, 351, 373, 396, 419, 443, 468, 493, 519, 546, 573, 601, 630, 659, 689, 720, 751, 783, 816, 849, 883, 918, 953, 989, 1026, 1063, 1101 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Fourth diagonal of A143974, associated with counting unit squares in a lattice. LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..3000 Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1). FORMULA a(n) = floor(n*(n+3)/3). From R. J. Mathar, Oct 05 2009: (Start) a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5). G.f.: x*(-1 - x - x^2 + x^3)/( (1 + x + x^2) * (x-1)^3). (End) 9*a(n) = 3*n^2 + 9*n - 2 + A099837(n+3). - R. J. Mathar, Apr 26 2022 Sum_{n>=1} 1/a(n) = 4/3 + (tan((sqrt(13)+2)*Pi/6) - cot((sqrt(13)+1)*Pi/6)) * Pi/sqrt(13). - Amiram Eldar, Oct 01 2022 E.g.f.: (exp(x)*(3*x*(4 + x) - 2) + 2*exp(-x/2)*cos(sqrt(3)*x/2))/9. - Stefano Spezia, Oct 24 2022 EXAMPLE Main diagonal of A143974: (0,1,3,5,8,12,...) = A000212; 2nd diagonal: (0,2,4,6,10,14,18,...) = A128422; 3rd diagonal: (1,2,5,8,11,16,21,...) = A032765; 4th diagonal: (1,3,6,9,13,18,23,...) = A143975. MATHEMATICA a[n_] := Floor[n*(n+3)/3]; Array[a, 60] (* Amiram Eldar, Oct 01 2022 *) PROG (Magma) [Floor(n*(n+3)/3): n in [1..60]]; // Vincenzo Librandi, May 08 2011 CROSSREFS Cf. A099837, A143974. Sequence in context: A323622 A154287 A092847 * A005488 A048202 A014785 Adjacent sequences: A143972 A143973 A143974 * A143976 A143977 A143978 KEYWORD nonn,easy AUTHOR Clark Kimberling, Sep 06 2008 STATUS approved

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Last modified August 13 17:56 EDT 2024. Contains 375144 sequences. (Running on oeis4.)