|
|
A278438
|
|
Numbers m such that T(m) + 2*T(m+1) is a square, where T = A000217.
|
|
2
|
|
|
7, 799, 78407, 7683199, 752875207, 73774087199, 7229107670407, 708378777612799, 69413891098384007, 6801852948864019999, 666512175097575576007, 65311391306613542428799, 6399849835873029582446407, 627119972524250285537319199, 61451357457540654953074835207
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
COMMENTS
|
It is well known that T(m) + k*T(m+1) is always a square for k=1. For k=3, the nonnegative values of m are the terms of A278310.
Square roots of T(m) + 2*T(m+1) are listed by A168520 (after 0).
Negative values of m for which T(m) + 2*T(m+1) is a square: -1, -2, -82, -7922, -776162, ...
|
|
LINKS
|
|
|
FORMULA
|
O.g.f.: x*(7 + 106*x - x^2)/((1 - x)*(1 - 98*x + x^2)).
E.g.f.: (exp((5-2*sqrt(6))^2*x) + exp((5+2*sqrt(6))^2*x) - 14*exp(x))/12 + 1.
a(n) = 99*a(n-1) - 99*a(n-2) + a(n-3) for n>3.
a(n) = 98*a(n-1) - a(n-2) + 112 for n>2.
a(n) = a(-n) = ((5 + 2*sqrt(6))^(2*n) + (5 - 2*sqrt(6))^(2*n))/12 - 7/6.
Lim_{n -> infinity} a(n)/a(n-1) = (5 + 2*sqrt(6))^2.
|
|
MAPLE
|
P:=proc(q) local n; for n from 1 to q do if type(sqrt((3*n^2+7*n+4)/2), integer) then print(n); fi; od; end: P(10^9); # Paolo P. Lava, Nov 25 2016
|
|
MATHEMATICA
|
Table[((5 + 2 Sqrt[6])^(2 n) + (5 - 2 Sqrt[6])^(2 n))/12 - 7/6, {n, 1, 20}]
RecurrenceTable[{a[1] == 7, a[2] == 799, a[n] == 98 a[n - 1] - a[n - 2] + 112}, a, {n, 1, 20}]
|
|
PROG
|
(Sage)
a, b = 7, 799
yield a
while True:
yield b
a, b = b, 98*b - a + 112
(Magma) Iv:=[7, 799]; [n le 2 select Iv[n] else 98*Self(n-1)-Self(n-2)+112: n in [1..20]];
(PARI) Vec(x*(7 + 106*x - x^2)/((1 - x)*(1 - 98*x + x^2)) + O(x^20)) \\ Colin Barker, Nov 27 2016
|
|
CROSSREFS
|
Cf. A278310: numbers m such that T(m) + 3*T(m+1) is a square.
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|