OFFSET
1,1
COMMENTS
Related to A055997.
If we solve X^2 + (X+7)^2 = (X+N)^2 over the positive integers we find that the solutions belong to three sequences:
1) The first is a(1) = 7; a(n) = 3*a(n-1) + 2*sqrt(2*a(n-1)*(a(n-1)-7)) - 7 for n > 1: 7, 14, 63, 350, 2023, 11774, 68607, 399854, 2330503, 13583150, 79168383, 461427134, ... We observe that a(n) = 7*A055997(n).
2) The second is this sequence.
3) The third is a(1) = 9; a(n) = 3*a(n-1) + 2*sqrt(2*a(n-1)*(a(n-1)-7))-7 for n > 1: 9, 32, 169, 968, 5625, 32768, 190969, 1113032, 6487209, 37810208, 220374025, 1284433928, 7486229529, 43632943232, 254311429849, 1482235635848, ...
There is a property of the formula:
If y = 3*x + 2*sqrt(2*x*(x-q)) - q then x = 3*y - 2*sqrt(2*y*(y-q)) - q.
Let F(X) = 3*x - 2*sqrt(2*x*(x-7)) - 7.
Let us use this function:
With the 1st sequence: With the 2nd: With the 3rd:
F(2023)=350 F(729)=128 F(968)=169
F(350)=63 F(128)=25 F(169)=32
F(63)=14 F(25)=8 F(32)=9
F(14)=7 F(8)=9 F(9)=8
F(7)=14 F(9)=8 F(8)=9
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
Yurii S. Bystryk, Vitalii L. Denysenko, and Volodymyr I. Ostryk, Lune and Lens Sequences, ResearchGate preprint, 2024. See pp. 54, 56.
Index entries for linear recurrences with constant coefficients, signature (7,-7,1).
FORMULA
a(n) = 3*a(n-1) + 2*sqrt(2*a(n-1)*(a(n-1)-7)) - 7, for n > 1, with a(1)=8.
Conjectures from Colin Barker, Jul 19 2016: (Start)
a(n) = (14 + (11-6*sqrt(2))*(3+2*sqrt(2))^n + (3-2*sqrt(2))^n*(11+6*sqrt(2)))/4.
a(n) = 7*a(n-1) - 7*a(n-2) + a(n-3) for n > 3.
G.f.: x*(8 - 31*x + 9*x^2) / ((1-x)*(1 - 6*x + x^2)). (End)
MAPLE
a:= proc(n) option remember; `if`(n=1, 8,
3*a(n-1)+2*isqrt(2*a(n-1)*(a(n-1)-7))-7)
end:
seq(a(n), n=1..25);
MATHEMATICA
NestList[3 # + 2 Sqrt[2 # (# - 7)] - 7 &, 8, 23] (* Michael De Vlieger, Jul 18 2016 *)
PROG
(PARI) m=30; v=concat([8], vector(m-1)); for(n=2, m, v[n] = floor(3*v[n-1] +2*sqrt(2*v[n-1]*(v[n-1]-7))-7)); v \\ G. C. Greubel, Oct 07 2018
(Magma) I:=[8]; [n le 1 select I[n] else Floor(3*Self(n-1) +2*Sqrt(2*Self(n-1)*(Self(n-1) - 7)) -7): n in [1..30]]; // G. C. Greubel, Oct 07 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Manuel López Holgueras, Jul 17 2016
STATUS
approved