|
|
A302537
|
|
a(n) = (n^2 + 13*n + 2)/2.
|
|
1
|
|
|
1, 8, 16, 25, 35, 46, 58, 71, 85, 100, 116, 133, 151, 170, 190, 211, 233, 256, 280, 305, 331, 358, 386, 415, 445, 476, 508, 541, 575, 610, 646, 683, 721, 760, 800, 841, 883, 926, 970, 1015, 1061, 1108, 1156, 1205, 1255, 1306, 1358, 1411, 1465, 1520, 1576
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
COMMENTS
|
Binomial transform of [1, 7, 1, 0, 0, 0, ...].
Numbers m > 0 such that 8*m + 161 is a square.
|
|
REFERENCES
|
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley, 1994.
|
|
LINKS
|
|
|
FORMULA
|
a(n) = binomial(n + 1, 2) + 6*n + 1 = binomial(n, 2) + 7*n + 1.
a(n) = a(n-1) + n + 6.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3, where a(0) = 1, a(1) = 8 and a(2) = 16.
a(n) = 2*a(n-1) - a(n-2) + 1.
G.f.: (5*x^2 - 5*x - 1)/(x - 1)^3.
E.g.f.: (1/2)*(x^2 + 14*x + 2)*exp(x).
Sum_{n>=0} 1/a(n) = 24097/45220 + 2*Pi*tan(sqrt(161)*Pi/2) / sqrt(161) = 1.4630922534498496... - Vaclav Kotesovec, Apr 11 2018
|
|
EXAMPLE
|
Illustration of initial terms (by the formula a(n) = A052905(n) + 3*n):
. o
. o o
. o o o o
. o o o o o o
. o o o o o o o o o
. o o o o o o o o o o o o
. o o o o o o o o o o . . . . . o
. o o o o o o o . . . . o o . . . . . o
. o o o o o . . . o o . . . . o o . . . . . o
. o o o . . o o . . . o o . . . . o o . . . . . o
. o o . o o . . o o . . . o o . . . . o o . . . . . o
. o o o . o o . . o o . . . o o . . . . o o . . . . . o
. o o o o o o o o o o o o o o o o o o o o o o o o o o o o
. o o o o o o o o o o o o o o o o o o o o o
. o o o o o o o o o o o o o o o o o o o o o
. o o o o o o o o o o o o o o o o o o o o o
----------------------------------------------------------------------
. 1 8 16 25 35 46 58
|
|
MAPLE
|
a := n -> (n^2 + 13*n + 2)/2;
seq(a(n), n = 0 .. 100);
|
|
MATHEMATICA
|
Table[(n^2 + 13 n + 2)/2, {n, 0, 100}]
CoefficientList[ Series[(5x^2 - 5x - 1)/(x - 1)^3, {x, 0, 50}], x] (* or *)
|
|
PROG
|
(Maxima) makelist((n^2 + 13*n + 2)/2, n, 0, 100);
(PARI) a(n) = (n^2 + 13*n + 2)/2; \\ Altug Alkan, Apr 12 2018
|
|
CROSSREFS
|
Sequences whose n-th terms are of the form binomial(n, 2) + n*k + 1:
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|