A275151 a(1) = 8; a(n) = 3*a(n-1) + 2*sqrt(2*a(n-1)*(a(n-1)-7)) - 7 for n > 1.

8, 25, 128, 729, 4232, 24649, 143648, 837225, 4879688, 28440889, 165765632, 966152889, 5631151688, 32820757225, 191293391648, 1114939592649, 6498344164232, 37875125392729, 220752408192128, 1286639323760025, 7499083534368008, 43707861882448009, 254748087760320032, 1484780664679472169

Offset: 1

Author

Manuel López Holgueras, Jul 17 2016

Keywords

• nonn

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).

Crossrefs

Cf. A055997.

Programs

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

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 *)

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

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)