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 A118060 a(n) = 1681*n^2 - 984*n - 696. 3
 1, 4060, 11481, 22264, 36409, 53916, 74785, 99016, 126609, 157564, 191881, 229560, 270601, 315004, 362769, 413896, 468385, 526236, 587449, 652024, 719961, 791260, 865921, 943944, 1025329, 1110076, 1198185, 1289656, 1384489, 1482684, 1584241 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS In general, all sequences of equations which contain every positive integer in order exactly once (a pairwise equal summed, ordered partition of the positive integers) may be defined as follows: For all k, let x(k)=A001652(k) and z(k)=A001653(k). Then if we define a(n) to be (x(k)+z(k))n^2-(z(k)-1)n-x(k), the following equation is true: a(n)+(a(n)+1)+...+(a(n)+(x(k)+z(k))n+(2x(k)+z(k)-1)/2)=(a(n)+ (x(k)+z(k))n+(2x(k)+z(k)+1)/2)+...+(a(n)+2(x(k)+z(k))n+x(k)); a(n)+2(x(k)+z(k))n+x(k))=a(n+1)-1; e.g., in this sequence, x(4)=A001652(4)=696 and z(4)=A001653(4)=985; cf. A000290, A118057-A118059, A118061. LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..1000 Index entries for linear recurrences with constant coefficients, signature (3,-3,1). FORMULA a(n) = 3*a(n-1)-3*a(n-2)+a(n-3). G.f.: x*(1+4057*x-696*x^2)/(1-x)^3. - Colin Barker, Jul 01 2012 a(n)+(a(n)+1)+...+(a(n)+1681n+1188) = (a(n)+1681n+1189)+ ... +a(n+1)-1; a(n+1)-1 = a(n)+3362n+696. a(n)+(a(n)+1)+...+(a(n)+1681n+1188)=41(41n-12)(41n+29)(82n+17)/2; e.g., 11481+11482+...+17712=90965388=41*111*152*263/2. EXAMPLE a(3)=1681*3^2-984*3-696=11481, a(4)=1681*4^2-984*4-696=22264 and 11481+11482+...+17712=17713+...+22263 MATHEMATICA CoefficientList[Series[(1+4057*x-696*x^2)/(1-x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Jul 09 2012 *) LinearRecurrence[{3, -3, 1}, {1, 4060, 11481}, 40] (* Harvey P. Dale, Oct 28 2016 *) PROG (MAGMA) [1681*n^2 - 984*n - 696: n in [1..40]]; // Vincenzo Librandi, Jul 09 2012 (PARI) a(n)=1681*n^2-984*n-696 \\ Charles R Greathouse IV, Jun 17 2017 CROSSREFS Sequence in context: A020430 A163010 A069332 * A146894 A013688 A221452 Adjacent sequences:  A118057 A118058 A118059 * A118061 A118062 A118063 KEYWORD nonn,easy,less AUTHOR Charlie Marion, Apr 26 2006 EXTENSIONS Corrected by T. D. Noe, Nov 13 2006 STATUS approved

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Last modified July 30 22:16 EDT 2021. Contains 346365 sequences. (Running on oeis4.)