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%I #13 Jun 30 2023 15:07:32
%S 55,66,78,91,105,120,136,153,171,190,210,231,252,273,294,315,336,357,
%T 378,399,420,441,462,483,504,525,546,567,588,609,630,651,672,693,714,
%U 735,756,777,798,819,840,861,882,903,924,945,966,987,1008,1029,1050,1071,1092,1113,1134,1155
%N a(n) = sum of natural numbers m such that n - 10 <= m <= n + 10.
%C Generalization: If a(n,k) = sum of natural numbers m such that n - k <= m <= n + k (k >= 1) then a(n,k) = (k + n)*(k + n + 1)/2 = A000217(k+n) for 0 <= n <= k, a(n,k) = a(n-1,k) +2k + 1 = ((k + n - 1)*(k + n)/2) + 2k + 1 = A000217(k+n-1) +2k +1 for n >= k + 1 (see, e.g., A008486). a(n) = (10 + n)*(11 + n)/2 = A000217(10+n) for 0 <= n <= 10, a(n) = a(n-1) + 21 for n >= 11.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2, -1).
%F G.f.: (55 - 99*x + 45*x^2 - x^12)/(1 - x)^3. - _G. C. Greubel_, Jul 13 2016
%t CoefficientList[Series[(55 - 99*x + 45*x^2 - x^12)/(1 - x)^3, {x, 0, 50}] , x] (* _G. C. Greubel_, Jul 13 2016 *)
%o (PARI) a(n)=if(n>9,21*n,(n+10)*(n+11)/2) \\ _Charles R Greathouse IV_, Jul 13 2016
%K nonn,easy
%O 0,1
%A _Jaroslav Krizek_, Nov 18 2009
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