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

140, 204, 284, 380, 492, 620, 764, 924, 1100, 1292, 1500, 1724, 1964, 2220, 2492, 2780, 3084, 3404, 3740, 4092, 4460, 4844, 5244, 5660, 6092, 6540, 7004, 7484, 7980, 8492, 9020, 9564, 10124, 10700, 11292, 11900, 12524, 13164, 13820, 14492, 15180, 15884, 16604, 17340, 18092, 18860, 19644, 20444, 21260, 22092, 22940

Offset: 0

Author

Ilya Gutkovskiy, Aug 16 2016

Keywords

• nonn
• easy

Comments

Sums of eight consecutive squares.

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).

Examples

			a(0) = 0^2 + 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 = 140;
a(1) = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 8^2 = 204;
a(2) = 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 8^2 + 9^2 = 284, etc.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
• Index entries for two-way infinite sequences.

Crossrefs

Cf. A000290, A001844, A027575, A027578, A027865, A120328, A260637.

Cf. A001844, A005918.

Programs

Magma

[4*(2*n^2 +14*n +35): n in [0..60]]; // G. C. Greubel, Aug 24 2022

Mathematica

Table[8 n^2 + 56 n + 140, {n, 0, 50}]
LinearRecurrence[{3, -3, 1}, {140, 204, 284}, 51]

PARI

a(n)=8*n^2+56*n+140 \\ Charles R Greathouse IV, Jun 17 2017

SageMath

[4*(2*n^2 +14*n +35) for n in (0..60)] # G. C. Greubel, Aug 24 2022

Formula

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

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

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

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

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.

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

Sum_{n>=0} 1/a(n) = tanh(sqrt(21)*Pi/2)*Pi/(8*sqrt(21)) - 763/15180. - Amiram Eldar, Sep 15 2022