A141046 a(n) = 4*n^4.

0, 4, 64, 324, 1024, 2500, 5184, 9604, 16384, 26244, 40000, 58564, 82944, 114244, 153664, 202500, 262144, 334084, 419904, 521284, 640000, 777924, 937024, 1119364, 1327104, 1562500, 1827904, 2125764, 2458624, 2829124, 3240000, 3694084, 4194304, 4743684, 5345344

Offset: 0

Author

Fredrik Johansson, Jul 31 2008

Keywords

• easy
• nonn

Comments

Nonnegative integers a(n) such that (-a(n))^(1/4) is a Gaussian integer, since (n + n*i)^4 = -4*n^4

For n > 1, a(n) + k^4 is not prime for any k. - Derek Orr, May 31 2014

Suppose the vertices of a triangle are (T(n), T(n+j)), (T(n+2*j), T(n+3*j)) and (T(n+4*j), T(n+5*j)) where T(n) is the n-th triangular number. Then the area of this triangle will be a(j). - Charlie Marion, Mar 06 2021

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A000290, A000583, A001105, A005843, A008586.

Programs

Haskell

a141046 = (* 4) . (^ 4)  -- Reinhard Zumkeller, Jan 25 2012

Mathematica

Table[4 n^4, {n, 0, 20}]

PARI

a(n)=4*n^4 \\ Charles R Greathouse IV, Jan 26 2012

Formula

a(n) = 4*n^4.

a(n) = A008586(A000583(n)) = A000290(A005843(A000290(n))). - Reinhard Zumkeller, Jan 25 2012

G.f.: 4*x*(1 + x)*(1 + 10*x + x^2)/(1 - x)^5. - Chai Wah Wu, Jun 22 2016

From G. C. Greubel, Jun 22 2016: (Start)

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).

E.g.f.: 4*x*(1 + 7*x + 6*x^2 + x^3)*exp(x). (End)

a(n) = A001105(n)^2. - Bruce J. Nicholson, Apr 03 2017

From Amiram Eldar, Jan 29 2021: (Start)

Sum_{n>=1} 1/a(n) = Pi^4/360.

Sum_{n>=1} (-1)^(n+1)/a(n) = 7*Pi^4/2880.

Product_{n>=1} (1 + 1/a(n)) = 2*cosh(Pi/2)^2/Pi^2.

Product_{n>=1} (1 - 1/a(n)) = 2*sin(Pi/sqrt(2))*sinh(Pi/sqrt(2))/Pi^2. (End)