%I #13 Jan 24 2021 14:40:02
%S 45,55,66,78,91,105,120,136,153,171,190,209,228,247,266,285,304,323,
%T 342,361,380,399,418,437,456,475,494,513,532,551,570,589,608,627,646,
%U 665,684,703,722,741,760,779,798,817,836,855,874,893,912,931,950,969,988,1007,1026,1045,1064
%N a(n) = sum of natural numbers m such that n - 9 <= m <= n + 9.
%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).
%H G. C. Greubel, <a href="/A168108/b168108.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F a(n) = (9 + n)*(10 + n)/2 = A000217(9+n) for 0 <= n <= 9, a(n) = a(n-1) + 19 for n >= 10.
%F G.f.: (45 - 80*x + 36*x^2 - x^11)/(1 - x)^3. - _G. C. Greubel_, Jul 13 2016
%t CoefficientList[Series[(45 - 80*x + 36*x^2 - x^11)/(1 - x)^3, {x, 0, 50}]
%t , x] (* _G. C. Greubel_, Jul 13 2016 *)
%t LinearRecurrence[{2,-1},{45,55,66,78,91,105,120,136,153,171,190},60] (* _Harvey P. Dale_, Jan 24 2021 *)
%o (PARI) a(n)=if(n>9,19*n,(n+9)*(n+10)/2) \\ _Charles R Greathouse IV_, Jul 13 2016
%K nonn,easy
%O 0,1
%A _Jaroslav Krizek_, Nov 18 2009