|
|
A142463
|
|
a(n) = 2*n^2 + 2*n - 1.
|
|
31
|
|
|
-1, 3, 11, 23, 39, 59, 83, 111, 143, 179, 219, 263, 311, 363, 419, 479, 543, 611, 683, 759, 839, 923, 1011, 1103, 1199, 1299, 1403, 1511, 1623, 1739, 1859, 1983, 2111, 2243, 2379, 2519, 2663, 2811, 2963, 3119, 3279, 3443, 3611, 3783, 3959, 4139, 4323, 4511, 4703, 4899, 5099
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
COMMENTS
|
Numbers k such that 2*k + 3 is a square.
The terms a(n) give the values for c of indefinite binary quadratic forms [a, b, c] = [2, 4n+2, a(n)] of discriminant D = 12, where a and c can be switched. The positive numbers represented by these forms are given in A084917. - Klaus Purath, Aug 31 2023
|
|
LINKS
|
|
|
FORMULA
|
a(n) = a(n-1) + 4*n.
G.f.: (1 - 6*x + x^2)/(1-x)^3.
a(n) = 4*C(n+1,2) - 1. (End)
a(n) = 3*( Sum_{k=1..n} k^5 )/( Sum_{k=1..n} k^3 ), n > 0. - Gary Detlefs, Oct 18 2011
a(n) = 2*A005563(n) - A005408(n). See Hexagonic Diamonds illustration.
a(n) = A016945(n-1) + A001105(n-1). See Hexagonic Rectangles illustration.
a(n) = A004767(n-1) + A046092(n-1). See Hexagonic Crosses illustration.
a(n) = A002378(n) + A028387(n-1). See Hexagonic Columns illustration. (End)
Sum_{n>=0} 1/a(n) = tan(sqrt(3)*Pi/2)*Pi/(2*sqrt(3)). - Amiram Eldar, Sep 16 2022
|
|
MAPLE
|
|
|
MATHEMATICA
|
|
|
PROG
|
(Magma) [2*n^2+2*n-1: n in [0..100]]
(Sage) [2*n^2 +2*n -1 for n in (0..50)] # G. C. Greubel, Mar 01 2021
|
|
CROSSREFS
|
|
|
KEYWORD
|
sign,easy
|
|
AUTHOR
|
|
|
EXTENSIONS
|
Edited by the Associate Editors of the OEIS, Sep 02 2009
|
|
STATUS
|
approved
|
|
|
|