A165568 a(n) = -1 - 2*n + n^2 + 2*n^3 + n^4.

-1, 1, 31, 137, 391, 889, 1751, 3121, 5167, 8081, 12079, 17401, 24311, 33097, 44071, 57569, 73951, 93601, 116927, 144361, 176359, 213401, 255991, 304657, 359951, 422449, 492751, 571481, 659287, 756841, 864839, 984001, 1115071, 1258817, 1416031, 1587529, 1774151

Offset: 0

Author

Paul Curtz, Sep 22 2009

Keywords

• sign
• easy

Comments

Consider the Lyman spectrum of Hydrogen A005563(n)/A000290(n+1) = n*(n+2)/(n+1)^2 = 0/1, 3/4, 8/9, 15/16, ...

The first differences of these fractions are 3/4, 5/36, 7/144, 9/400, 11/900, 13/1764, 15/3136, ... = (2n+1)/(n*(n+1))^2.

Adding numerator and denominator of these first differences yields 1 + 2n + n^2 + 2n^3 + n^4 = A165563(n) = 3+4, 5+36, 7+144, ... = 1 + 2n + n^2*(n+1)^2 = A144396(n) + A035287(n+1) = A005408(n) + A035287(n+1).

Subtracting numerator from denominator, on the other hand, yields this sequence here: a(n) = A035287(n+1) - A005408(n).

Links

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

Programs

Magma

[-1 -2*n +n^2 +2*n^3 +n^4: n in [0..40]]; // Vincenzo Librandi, May 21 2011

Formula

a(-1-n) = A165563(n). A165563(-1-n) = a(n).

a(n) = A165563(n) - 2 - 4*n = A165563(n) - A016825(n).

a(n) + A165563 + a(n) = 2*n^2*(1+n)^2 = 2*A035287(n+1).

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

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + 24.

G.f.: (-1 + 6*x + 16*x^2 + 2*x^3 + x^4)/(1-x)^5.

Extensions

Edited and extended by R. J. Mathar, Feb 02 2010