OFFSET
1,2
COMMENTS
Essentially the same as A051047.
It appears that a(n) = A046175(n)-A046174(n), that is, the triangular index of the n-th pentagonal triangular number minus its pentagonal index. - Jonathan Vos Post, Feb 28 2011
Sequence lists the nonnegative x solutions when (x + 1)*(3*x + 1) is a square. Positive x solutions when (x - 1)*(3*x - 1) is a square are in A011922. - Bruno Berselli, Feb 20 2018
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..870
A. Baker and H. Davenport, The Equations 3x^2-2=y^2 and 8x^2-7=z^2, Quart. J. Math., Oxford, Vol. 20, No. 1 (1969), 29-137.
Andrej Dujella and Attila Pethő, A generalization of a theorem of Baker and Davenport, Quart. J. Math., Oxford, Ser. (2), Vol. 49, No. 3 (1998), 291-306; alternative link.
Andrej Dujella, The Problem of Diophantus and Davenport, References.
Andrej Dujella, Publications of Andrej Dujella.
Zrinka Franušić, On the Extension of the Diophantine Pair {1,3} in Z[surd d], J. Int. Seq., Vol. 13 (2010), Article 10.9.6.
Philip Gibbs, 1,3,8,120 ... A Diophantine Problem. [Wayback Machine link]
Philip Gibbs, Diophantine quadruples and Cayley's hyperdeterminant, arXiv:math/0107203 [math.NT], 2001.
Index entries for linear recurrences with constant coefficients, signature (15,-15,1).
FORMULA
a(n) = A046184(n+1) - 1.
a(n) = 14*a(n-1) - a(n-2) + 8.
a(n) = ((2 + sqrt(3))*(7 + 4*sqrt(3))^n + (2 - sqrt(3))*(7 - 4*sqrt(3))^n - 4)/6. - Joseph Biberstine (jrbibers(AT)indiana.edu), Apr 23 2006
a(n) = 8*A076139(n-1) = 4*A217855(n-1) = 2*A123480(n-1) = 8/3*A076140(n-1). - Peter Bala, Dec 31 2012
From Colin Barker, Jul 30 2013: (Start)
G.f.: -8*x^2 / ((x - 1)*(x^2 - 14*x + 1)).
a(n) = 15*a(n-1) - 15*a(n-2) + a(n-3). (End)
E.g.f.: (-4*exp(x) + (2 + sqrt(3))*exp((7-4*sqrt(3))*x) + (2 - sqrt(3))*exp((7+4*sqrt(3))*x))/6. - Ilya Gutkovskiy, Apr 28 2016
Sum_{n>=2} 1/a(n) = 1 - sqrt(3)/2 (A375069). - Amiram Eldar, Jan 26 2026
MATHEMATICA
f[n_] := FullSimplify[((Sqrt[3] + 2)*(7 + 4*Sqrt[3])^n - (Sqrt[3] - 2) (7 - 4 Sqrt[3])^n - 4)/6]; Array[f, 18, 0] (* Joseph Biberstine (jrbibers(AT)indiana.edu), Apr 23 2006 *)
Rest[CoefficientList[Series[-8*x^2/((x - 1)*(x^2 - 14*x + 1)), {x, 0, 50}], x]] (* G. C. Greubel, Jun 07 2017 *)
LinearRecurrence[{15, -15, 1}, {0, 8, 120}, 20] (* Harvey P. Dale, Jul 14 2024 *)
PROG
(PARI) my(x='x+O('x^50)); concat([0], Vec(-8*x^2/((x - 1)*(x^2 - 14*x + 1)))) \\ G. C. Greubel, Jun 07 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Andrej Dujella (duje(AT)math.hr)
STATUS
approved
