|
|
A158447
|
|
a(n) = 10*n^2 - 1.
|
|
3
|
|
|
9, 39, 89, 159, 249, 359, 489, 639, 809, 999, 1209, 1439, 1689, 1959, 2249, 2559, 2889, 3239, 3609, 3999, 4409, 4839, 5289, 5759, 6249, 6759, 7289, 7839, 8409, 8999, 9609, 10239, 10889, 11559, 12249, 12959, 13689, 14439, 15209, 15999, 16809, 17639
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
COMMENTS
|
Sequence found by reading the line from 9, in the direction 9, 39, ..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. - Omar E. Pol, Jul 18 2012
|
|
LINKS
|
|
|
FORMULA
|
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f: x*(9 + 12*x - x^2)/(1 - x)^3.
Sum_{n>=1} 1/a(n) = (1 - (Pi/sqrt(10))*cot(Pi/sqrt(10)))/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = ((Pi/sqrt(10))*csc(Pi/sqrt(10)) - 1)/2.
Product_{n>=1} (1 + 1/a(n)) = (Pi/sqrt(10))*csc(Pi/sqrt(10)).
Product_{n>=1} (1 - 1/a(n)) = csc(Pi/sqrt(10))*sin(Pi/sqrt(5))/sqrt(2). (End)
|
|
MAPLE
|
|
|
MATHEMATICA
|
Table[10n^2-1, {n, 50}]
LinearRecurrence[{3, -3, 1}, {9, 39, 89}, 50] (* Harvey P. Dale, Dec 08 2017 *)
|
|
PROG
|
(Magma) I:=[9, 39, 89]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]];
(PARI) a(n) = 10*n^2 - 1.
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|