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A276026 a(n) = Sum_{k=0..7} (n + k)^2. 0

%I

%S 140,204,284,380,492,620,764,924,1100,1292,1500,1724,1964,2220,2492,

%T 2780,3084,3404,3740,4092,4460,4844,5244,5660,6092,6540,7004,7484,

%U 7980,8492,9020,9564,10124,10700,11292,11900,12524,13164,13820,14492,15180,15884,16604,17340,18092,18860,19644,20444,21260,22092,22940

%N a(n) = Sum_{k=0..7} (n + k)^2.

%C Sums of eight consecutive squares.

%C More generally, the ordinary generating function for the sums of m consecutive squares of nonnegative integers is m*(1 - 2*x + 13*x^2 + 2*m^2 (1 - 2*x + x^2) - 3*m*(1 - 4*x + 3*x^2))/(6*(1 - x)^3).

%H <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1)

%F O.g.f.: 4*(35 - 54*x + 23*x^2)/(1 - x)^3.

%F E.g.f.: 4*(35 + 16*x + 2*x^2)*exp(x).

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

%F a(-n) = a(n-7).

%F a(n) = n^2 + (n+1)^2 + (n+2)^2 + (n+3)^2 + (n+4)^2 + (n+5)^2 + (n+6)^2 + (n+7)^2.

%F a(n) = 8*n^2 + 56*n + 140.

%e a(0) = 0^2 + 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 = 140;

%e a(1) = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 8^2 = 204;

%e a(2) = 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 8^2 + 9^2 = 284, etc.

%t Table[8 n^2 + 56 n + 140, {n, 0, 50}]

%t LinearRecurrence[{3, -3, 1}, {140, 204, 284}, 51]

%o (PARI) a(n)=8*n^2+56*n+140 \\ _Charles R Greathouse IV_, Jun 17 2017

%Y Cf. A001844, A005918, A027575, A027578, A027865, A260637.

%K nonn,easy

%O 0,1

%A _Ilya Gutkovskiy_, Aug 16 2016

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Last modified October 21 23:02 EDT 2018. Contains 316431 sequences. (Running on oeis4.)