|
|
A082044
|
|
Main diagonal of A082043: a(n) = n^4 + 2*n^2 + 1.
|
|
15
|
|
|
1, 4, 25, 100, 289, 676, 1369, 2500, 4225, 6724, 10201, 14884, 21025, 28900, 38809, 51076, 66049, 84100, 105625, 131044, 160801, 195364, 235225, 280900, 332929, 391876, 458329, 532900, 616225, 708964, 811801, 925444, 1050625, 1188100
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
COMMENTS
|
a(n) = longest side b of all integer-sided triangles with sides a <= b <= c and inradius n >= 1. Triangle has sides (n^2+2, n^4+2*n^2+1, n^4+3*n^2+1).
|
|
LINKS
|
|
|
FORMULA
|
a(n) = n^4 + 2*n^2 + 1.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Harvey P. Dale, Feb 27 2015
G.f.: (1 - x + 15*x^2 + 5*x^3 + 4*x^4 ) / (1 - x)^5. - Michael Somos, Apr 17 2017
Sum_{n>=0} 1/a(n) = Pi^2*csch(Pi)^2/4 + Pi*coth(Pi)/4 + 1/2.
Sum_{n>=0} (-1)^n/a(n) = Pi^2*csch(Pi)*coth(Pi)/4 + Pi*csch(Pi)/4 + 1/2. (End)
E.g.f.: (1 + 3*x + 9*x^2 + 6*x^3 + x^4)*exp(x). - G. C. Greubel, Dec 24 2022
|
|
EXAMPLE
|
G.f. = 1 + 4*x + 25*x^2 + 100*x^3 + 289*x^4 + 676*x^5 + 1369*x^6 + ...
|
|
MAPLE
|
|
|
MATHEMATICA
|
Fibonacci[3, Range[0, 40]]^2 (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {1, 4, 25, 100, 289}, 40] (* Harvey P. Dale, Feb 27 2015 *)
|
|
PROG
|
(SageMath) [(n^2+1)^2 for n in range(41)] # G. C. Greubel, Dec 24 2022
|
|
CROSSREFS
|
See A120062 for sequences related to integer-sided triangles with integer inradius n.
|
|
KEYWORD
|
easy,nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|