OFFSET
1,3
COMMENTS
Also, integers m such that 21*(3*m-1)^2 - 48 is a square. - Max Alekseyev, May 23 2022
a(n) is prime exactly for n = 3, 4, 5, 8, 16, 20, 22, 23, 58, 302, 386, 449, 479, 880 up to 1000. - Tomohiro Yamada, Dec 23 2018
Similarly, positive integers m,k with m|(1+k+^2) and k|(1-m+m^2) are consecutive terms of A061646, where m has an even index. - Max Alekseyev, May 23 2022
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..1471
Esther Banaian and Archan Sen, A Generalization of Markov Numbers, arXiv:2210.07366 [math.CO], 2022. See Table 1 p. 12.
T. Cai, Z. Shen and L. Jia, A congruence involving harmonic sums modulo p^alpha q^beta, arXiv preprint arXiv:1503.02798 [math.NT], 2015.
W. W. Chao, Problem 2981, Crux Mathematicorum, 30 (2004), p. 430.
Yasuaki Gyoda, Positive integer solutions to (x+y)^2+(y+z)^2+(z+x)^2=12xyz, arXiv:2109.09639 [math.NT], 2021. See Remark 3.3 p. 6.
W. H. Mills, A system of quadratic Diophantine equations. Pacific J. Math. 3:1 (1953), 209-220.
Index entries for linear recurrences with constant coefficients, signature (6,-6,1).
FORMULA
Recurrence: a(1)=a(2)=1 and a(n+1)=(1+a(n)+a(n)^2)/a(n-1) for n>2.
G.f.: x(1 - 5x + 3x^2) / [(1-x)(1 - 5x + x^2)]; a(n) = 2 * A089817(n-3) + 1, n>2. - Conjectured by Ralf Stephan, Jan 14 2005, proved by Max Alekseyev, Aug 03 2006
a(n) = 6a(n-1)-6a(n-2)+a(n-3), a(n) = 5a(n-1)-a(n-2)-1. - Floor van Lamoen, Aug 01 2006
a(n) = (4/3 - (2/7)*sqrt(21))*((5 + sqrt(21))/2)^n + (4/3 + (2/7)*sqrt(21))*((5 - sqrt(21))/2)^n + 1/3. - Floor van Lamoen, Aug 04 2006
For n>1, a(n) = (2 * A004253(n-1) + 1) / 3. - Max Alekseyev, May 23 2022
EXAMPLE
a(5) = 61 because (1 + a(4) + a(4)^2)/a(3) = (1 + 13 + 169)/3 = 61.
MAPLE
seq(coeff(series(x*(1-5*x+3*x^2)/((1-x)*(1-5*x+x^2)), x, n+1), x, n), n = 1 .. 30); # Muniru A Asiru, Dec 28 2018
MATHEMATICA
Rest@ CoefficientList[Series[x (1 - 5 x + 3 x^2)/((1 - x) (1 - 5 x + x^2)), {x, 0, 28}], x] (* or *)
RecurrenceTable[{a[n] == (1 + a[n - 1] + a[n - 1]^2)/a[n - 2], a[1] == a[2] == 1}, a, {n, 1, 28}] (* or *)
RecurrenceTable[{a[n] == 5 a[n - 1] - a[n - 2] - 1, a[1] == a[2] == 1}, a, {n, 1, 28}] (* or *)
LinearRecurrence[{6, -6, 1}, {1, 1, 3}, 28] (* Michael De Vlieger, Aug 28 2016 *)
PROG
(PARI) Vec(x*(1-5*x+3*x^2)/((1-x)*(1-5*x+x^2)) + O(x^30)) \\ Michel Marcus, Aug 03 2016
(PARI) a(n)=([0, 1, 0; 0, 0, 1; 1, -6, 6]^n*[3; 1; 1])[1, 1] \\ Charles R Greathouse IV, Aug 28 2016
(Magma) [n le 2 select 1 else 5*Self(n-1)-Self(n-2)-1: n in [1..30]]; // Vincenzo Librandi, Dec 25 2018
(GAP) a:=[1, 1];; for n in [3..30] do a[n]:=5*a[n-1]-a[n-2]-1; od; Print(a); # Muniru A Asiru, Dec 28 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
M. Benito, O. Ciaurri and E. Fernandez (oscar.ciaurri(AT)dmc.unirioja.es), Jan 13 2005
STATUS
approved