The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A275151 a(1) = 8; a(n) = 3*a(n-1) + 2*sqrt(2*a(n-1)*(a(n-1)-7)) - 7 for n > 1. 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 (list; graph; refs; listen; history; text; internal format)
 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 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 Cf. A055997. Sequence in context: A068315 A069586 A253237 * A302617 A042189 A305680 Adjacent sequences:  A275148 A275149 A275150 * A275152 A275153 A275154 KEYWORD nonn AUTHOR Manuel López Holgueras, Jul 17 2016 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified July 30 17:21 EDT 2021. Contains 346359 sequences. (Running on oeis4.)