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A101368 The sequence solves the following problem: find all the pairs (i,j) such that i divides 1+j+j^2 and j divides 1+i+i^2. In fact, the pairs (a(n),a(n+1)), n>0, are all the solutions. 7
1, 1, 3, 13, 61, 291, 1393, 6673, 31971, 153181, 733933, 3516483, 16848481, 80725921, 386781123, 1853179693, 8879117341, 42542407011, 203832917713, 976622181553, 4679277990051, 22419767768701, 107419560853453, 514678036498563, 2465970621639361, 11815175071698241, 56609904736851843, 271234348612560973 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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

LINKS

Seiichi Manyama, Table of n, a(n) for n = 1..1471

T. Cai, Z. Shen, 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.

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

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

Cf. A001519, A276160.

Sequence in context: A239995 A319924 A108143 * A026704 A046748 A256333

Adjacent sequences:  A101365 A101366 A101367 * A101369 A101370 A101371

KEYWORD

nonn,easy

AUTHOR

M. Benito, O. Ciaurri and E. Fernandez (oscar.ciaurri(AT)dmc.unirioja.es), Jan 13 2005

STATUS

approved

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Last modified December 16 03:14 EST 2019. Contains 330013 sequences. (Running on oeis4.)