A168105 - OEIS

%I #13 Nov 16 2016 12:56:36

%S 21,28,36,45,55,66,78,91,104,117,130,143,156,169,182,195,208,221,234,

%T 247,260,273,286,299,312,325,338,351,364,377,390,403,416,429,442,455,

%U 468,481,494,507,520,533,546,559,572,585,598,611,624,637,650,663,676,689,702,715,728,741

%N a(n) = sum of natural numbers m such that n - 6 <= m <= n + 6.

%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="/A168105/b168105.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) = (6 + n)*(7 + n)/2 = A000217(6+n) for 0 <= n <= 6, a(n) = a(n-1) + 13 for n >= 7.

%F G.f.: (21  - 35*x + 15*x^2 - x^8)/(1 - x)^3. - _G. C. Greubel_, Jul 13 2016

%t CoefficientList[Series[(21  - 35*x + 15*x^2 - x^8)/(1 - x)^3, {x,0,50}], x] (* _G. C. Greubel_, Jul 13 2016 *)

%o (PARI) a(n)=if(n>5,13*n,n*(n+13)/2+21) \\ _Charles R Greathouse IV_, Jul 13 2016

%K nonn,easy

%O 0,1

%A _Jaroslav Krizek_, Nov 18 2009