A052465 a(n) is the smallest positive integral solution k to 24*k == 1 (mod 11^n).
6, 116, 721, 14031, 87236, 1697746, 10555551, 205427261, 1277221666, 24856698576, 154543821581, 3007660527691, 18699802411296, 363926923850606, 2262676091766811, 44035157785923321, 273783807103784126, 5328254092096721836, 33127840659557879241, 644718745143703342151, 4008468719806503388156
Offset: 1
Examples
From _Petros Hadjicostas_, Jul 29 2020: (Start) A000041(a(1)) = A000041(6) = 11 == 0 (mod 11^1). A000041(a(2)) = A000041(116) = 1188908248 == 0 (mod 11^2). A000041(a(3)) = A000041(721) = 161061755750279477635534762 == 0 (mod 11^3). (End)
Links
- Vincenzo Librandi, Table of n, a(n) for n = 1..900
- A. O. L. Atkin, Proof of a Conjecture of Ramanujan, Glasgow Math. J. 8 (1967), 14-32.
- G. K. Patil, Ramanujan's Life And His Contributions In The Field Of Mathematics, International Journal of Scientific Research and Engineering Studies (IJSRES), 1(6) (2014), ISSN: 2349-8862.
- Eric Weisstein's World of Mathematics, Partition Function P Congruences.
- Index entries for linear recurrences with constant coefficients, signature (1,121,-121).
Programs
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Magma
I:=[6, 116, 721]; [n le 3 select I[n] else Self(n-1)+121*Self(n-2)-121*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Jul 01 2012
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Mathematica
Table[PowerMod[24, -1, 11^c], {c, 20}] CoefficientList[Series[(-121x^2+110x+6)/((1-x)(1-121*x^2)),{x,0,30}],x] (* Vincenzo Librandi, Jul 01 2012 *) LinearRecurrence[{1,121,-121},{6,116,721},20] (* Harvey P. Dale, Apr 27 2014 *) -
PARI
a(n) = lift(Mod(24, 11^n)^-1) \\ David A. Corneth, Jul 29 2020
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SageMath
def a(n): return 24.inverse_mod(11^n) print([a(n) for n in range(1, 22)]) # Peter Luschny, Jul 29 2020
Formula
G.f.: x*(-121*x^2 + 110*x + 6)/((1 - x)*(1 - 121*x^2)). - Vincenzo Librandi, Jul 01 2012
a(n) = a(n-1) + 121*a(n-2) - 121*a(n-3). - Vincenzo Librandi, Jul 01 2012
A000041(a(n)) == 0 (mod 11^n). - Petros Hadjicostas, Jul 29 2020
From Petros Hadjicostas, Aug 02 2020: (Start)
a(n) = (1 + 23*11^n)/24, if n is even, and a(n) = (1 + 13*11^n)/24, if n is odd.
a(n) - a(n-1) = 10*11^(n-1), if n is even >= 2, and 5*11^(n-1), if n is odd >= 3. (End)
Extensions
More terms from David A. Corneth, Jul 29 2020
Comments