A278438 Numbers m such that T(m) + 2*T(m+1) is a square, where T = A000217.

7, 799, 78407, 7683199, 752875207, 73774087199, 7229107670407, 708378777612799, 69413891098384007, 6801852948864019999, 666512175097575576007, 65311391306613542428799, 6399849835873029582446407, 627119972524250285537319199, 61451357457540654953074835207

Offset: 1

Author

Bruno Berselli, Nov 23 2016

Keywords

• nonn
• easy

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

• Colin Barker, Table of n, a(n) for n = 1..500
• Index entries for linear recurrences with constant coefficients, signature (99,-99,1).

Crossrefs

Subsequence of A056220.

Cf. A000217, A001078, A001079, A122652, A168520, A278620.

Cf. A278310: numbers m such that T(m) + 3*T(m+1) is a square.

Programs

Magma

Iv:=[7, 799]; [n le 2 select Iv[n] else 98*Self(n-1)-Self(n-2)+112: n in [1..20]];

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}]
LinearRecurrence[{99,-99,1},{7,799,78407},20] (* Harvey P. Dale, Oct 18 2024 *)

PARI

Vec(x*(7 + 106*x - x^2)/((1 - x)*(1 - 98*x + x^2)) + O(x^20)) \\ Colin Barker, Nov 27 2016

Sage

def A278438():
    a, b = 7, 799
    yield a
    while True:
        yield b
        a, b = b, 98*b - a + 112
a = A278438(); print([next(a) for  in range(15)]) # _Peter Luschny, Nov 24 2016

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.

a(n) = A001079(2*n)/6 - 7/6.

a(n) = 2*A001078(n)^2 - 1 = A122652(n)^2/2 - 1.

a(n) = -A278620(n+1) + 106*A278620(n) + 7*A278620(n-1).

Lim_{n -> infinity} a(n)/a(n-1) = (5 + 2*sqrt(6))^2.