Manuel López Holgueras - cp's OEIS frontend

User: Manuel López Holgueras

Manuel López Holgueras has authored 2 sequences.

A275155 a(1) = 18; a(n) = 3a(n - 1) + 2sqrt(2a(n - 1)(a(n - 1) - 14)) - 14 for n > 1.

18, 64, 338, 1936, 11250, 65536, 381938, 2226064, 12974418, 75620416, 440748050, 2568867856, 14972459058, 87265886464, 508622859698, 2964471271696, 17278204770450, 100704757350976, 586950339335378, 3420997278661264, 19939033332632178, 116213202717131776

Offset: 1

Manuel López Holgueras, Jul 18 2016

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (7,-7,1).

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(2*x*(9-31*x+8*x^2)/((1-x)*(1-6*x+x^2)))); // G. C. Greubel, Sep 30 2018

NestList[3 # + 2 Sqrt[2 # (# - 14)] - 14 &, 18, 18] (* Michael De Vlieger, Jul 19 2016 *)
CoefficientList[Series[2*x*(9-31*x+8*x^2)/((1-x)*(1-6*x+x^2)), {x, 0, 50}], x] (* G. C. Greubel, Sep 30 2018 *)

a(n)=([0,1,0; 0,0,1; 1,-7,7]^(n-1)*[18;64;338])[1,1] \\ Charles R Greathouse IV, Jul 20 2016

Vec(2*x*(9-31*x+8*x^2)/((1-x)*(1-6*x+x^2)) + O(x^30)) \\ Colin Barker, Jul 21 2016

a(n+1) = 3*a(n) + 2*sqrt(2*a(n)*(a(n) - 14)) - 14.
From Colin Barker, Jul 21 2016: (Start)
a(n) = (14+(9-4*sqrt(2))*(3+2*sqrt(2))^n + (3-2*sqrt(2))^n*(9+4*sqrt(2)))/2.
a(n) = 7*a(n-1) -7*a(n-2) +a(n-3) for n>3.
G.f.: 2*x*(9-31*x+8*x^2) / ((1-x)*(1-6*x+x^2)). (End)

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

Original entry on oeis.org

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

Manuel López Holgueras, Jul 17 2016

nonn

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

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).

Cf. A055997.

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

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);

NestList[3 # + 2 Sqrt[2 # (# - 7)] - 7 &, 8, 23] (* Michael De Vlieger, Jul 18 2016 *)

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

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)

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User: Manuel López Holgueras

A275155 a(1) = 18; a(n) = 3a(n - 1) + 2sqrt(2a(n - 1)(a(n - 1) - 14)) - 14 for n > 1.

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

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cp's OEIS Frontend

User: Manuel López Holgueras

A275155 a(1) = 18; a(n) = 3*a(n - 1) + 2*sqrt(2*a(n - 1)*(a(n - 1) - 14)) - 14 for n > 1.

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Magma

Mathematica

PARI

PARI

Formula

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

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Magma

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Formula

A275155 a(1) = 18; a(n) = 3a(n - 1) + 2sqrt(2a(n - 1)(a(n - 1) - 14)) - 14 for n > 1.

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