A041027 -id:A041027 - cp's OEIS frontend

A077420 Bisection of Chebyshev sequence T(n,3) (odd part) with Diophantine property.

1, 33, 1121, 38081, 1293633, 43945441, 1492851361, 50713000833, 1722749176961, 58522759015841, 1988051057361633, 67535213191279681, 2294209197446147521, 77935577499977736033, 2647515425801796877601

Offset: 0

Wolfdieter Lang, Nov 29 2002

(3*a(n))^2 - 2*(2*b(n))^2 = 1 with companion sequence b(n)= A046176(n+1), n>=0 (special solutions of Pell equation).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Z. Cerin and G. M. Gianella, On sums of squares of Pell-Lucas Numbers, INTEGERS 6 (2006) #A15
Tanya Khovanova, Recursive Sequences
S. Vidhyalakshmi, V. Krithika, and K. Agalya, On The Negative Pell Equation y^2 = 72x^2 - 8, International Journal of Emerging Technologies in Engineering Research (IJETER) Volume 4, Issue 2, February (2016).
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (34,-1).

Cf. A056771 (even part).
Row 34 of array A094954.
Row 3 of array A188646.
Cf. similar sequences listed in A238379.
Similar sequences of the type cosh((2*n+1)*arccosh(k))/k are listed in A302329. This is the case k=3.

I:=[1,33]; [n le 2 select I[n] else 34*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 22 2011

LinearRecurrence[{34,-1},{1,33},20] (* Vincenzo Librandi, Nov 22 2011 *)
a[c_, n_] := Module[{},
   p := Length[ContinuedFraction[ Sqrt[ c]][[2]]];
   d := Denominator[Convergents[Sqrt[c], n p]];
   t := Table[d[[1 + i]], {i, 0, Length[d] - 1, p}];
   Return[t];
] (* Complement of A041027 *)
a[18, 20] (* Gerry Martens, Jun 07 2015 *)

makelist(expand(((1+sqrt(2))^(4*n+2)+(1-sqrt(2))^(4*n+2))/6),n,0,14);  /* _Bruno Berselli, Nov 22 2011 */

Vec((1-x)/(1-34*x+x^2)+O(x^99)) \\ Charles R Greathouse IV, Nov 22 2011

a(n) = 34*a(n-1) - a(n-2), a(-1)=1, a(0)=1.
a(n) = T(2*n+1, 3)/3 = S(n, 34) - S(n-1, 34), with S(n, x) := U(n, x/2), resp. T(n, x), Chebyshev's polynomials of the second, resp. first, kind. See A049310 and A053120. S(-1, x)=0, S(n, 34)= A029547(n), T(n, 3)=A001541(n).
G.f.: (1-x)/(1-34*x+x^2).
a(n) = sqrt(8*A046176(n+1)^2 + 1)/3.
a(n) = (k^n)+(k^(-n))-a(n-1) = A003499(2*n)-a(n-1), where k = (sqrt(2)+1)^4 = 17+12*sqrt(2) and a(0)=1. - Charles L. Hohn, Apr 05 2011
a(n) = a(-n-1) = A029547(n)-A029547(n-1) = ((1+sqrt(2))^(4n+2)+(1-sqrt(2))^(4n+2))/6. - Bruno Berselli, Nov 22 2011

A088317 a(n) = 8*a(n-1) + a(n-2), starting with a(0) = 1 and a(1) = 4.

Original entry on oeis.org

1, 4, 33, 268, 2177, 17684, 143649, 1166876, 9478657, 76996132, 625447713, 5080577836, 41270070401, 335241141044, 2723199198753, 22120834731068, 179689877047297, 1459639851109444, 11856808685922849, 96314109338492236, 782369683393860737, 6355271576489378132, 51624542295308885793

Offset: 0

Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Nov 06 2003

G. C. Greubel, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (8,1).

Cf. A002018, A002190, A013192, A028576, A041024, A041027.

Cf. A053120, A058153, A058155, A072754, A075132.

[n le 2 select 4^(n-1) else 8*Self(n-1) +Self(n-2): n in [1..31]]; // G. C. Greubel, Dec 13 2022

LinearRecurrence[{8,1},{1,4},30] (* or *) With[{c=Sqrt[17]},Simplify/@ Table[1/2 (c-4)((c+4)^n-(4-c)^n (33+8c)),{n,30}]] (* Harvey P. Dale, May 07 2012 *)

a[0]:1$ a[1]:4$ a[n]:=8*a[n-1]+a[n-2]$ A088317(n):=a[n]$
makelist(A088317(n),n,0,20); /* Martin Ettl, Nov 12 2012 */

A088317=BinaryRecurrenceSequence(8,1,1,4)
[A088317(n) for n in range(31)] # G. C. Greubel, Dec 13 2022

a(n) = ( (4+sqrt(17))^n + (4-sqrt(17))^n )/2.
a(n) = A086594(n)/2.
Lim_{n -> oo} a(n+1)/a(n) = 4 + sqrt(17).
From Paul Barry, Nov 15 2003: (Start)
E.g.f.: exp(4*x)*cosh(sqrt(17)*x).
a(n) = Sum_{k=0..floor(n/2)} C(n, 2*k)*17^k*4^(n-2*k).
a(n) = (-i)^n * T(n, 4*i) with T(n, x) Chebyshev's polynomials of the first kind (see A053120) and i^2=-1. (End)
a(n) = A041024(n-1), n>0. - R. J. Mathar, Sep 11 2008
G.f.: (1-4*x)/(1-8*x-x^2). - Philippe Deléham, Nov 16 2008 and Nov 20 2008
a(n) = (1/2)*((33+8*sqrt(17))*(4-sqrt(17))^(n+2) + (33-8*sqrt(17))*(4+sqrt(17))^(n+2)). - Harvey P. Dale, May 07 2012

A041026 Numerators of continued fraction convergents to sqrt(18).

Original entry on oeis.org

4, 17, 140, 577, 4756, 19601, 161564, 665857, 5488420, 22619537, 186444716, 768398401, 6333631924, 26102926097, 215157040700, 886731088897, 7309005751876, 30122754096401, 248291038523084

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Index entries for linear recurrences with constant coefficients, signature (0,34,0,-1).

Cf. A010474, A041027.

Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[18],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 17 2011 *)
Numerator[Convergents[Sqrt[18],20]] (* or *) LinearRecurrence[{0,34,0,-1},{4,17,140,577},20] (* Harvey P. Dale, Jun 12 2014 *)
a0[n_] := ((-4-3*Sqrt[2])/(17+12*Sqrt[2])^n+(-4+3*Sqrt[2])*(17+12*Sqrt[2])^n)/2 // Simplify
a1[n_] := (1/(17+12*Sqrt[2])^n+(17+12*Sqrt[2])^n)/2 // Simplify
Flatten[MapIndexed[{a0[#], a1[#]} &,Range[20]]] (* Gerry Martens, Jul 11 2015 *)

G.f.: (4+17*x+4*x^2-x^3)/(1-34*x^2+x^4). - Colin Barker, Jan 02 2012
From Gerry Martens, Jul 11 2015: (Start)
Interspersion of 2 sequences [a0(n),a1(n)] for n>0:
a0(n) = ((-4-3*sqrt(2))/(17+12*sqrt(2))^n+(-4+3*sqrt(2))*(17+12*sqrt(2))^n)/2.
a1(n) = (1/(17+12*sqrt(2))^n+(17+12*sqrt(2))^n)/2. (End)

A241021 Smallest prime numbers p of length n having a decimal expansion x(1)x(2)... x(n) such that there exists an index j where x(j) = 1 and x(i) = 9 for i<>j, or 0 if no such prime exists.

Original entry on oeis.org

19, 199, 1999, 99991, 199999, 9999991, 19999999, 0, 9199999999, 99999199999, 991999999999, 9919999999999, 99999999991999, 919999999999999, 9999999999999199, 99919999999999999, 0, 9991999999999999999, 99999199999999999999, 0, 9991999999999999999999

Offset: 2

Michel Lagneau, Apr 15 2014

The corresponding indices of the decimal digit 1 are 1, 1, 1, 5, 1, 7, 1, 0, 2, 6, 3, 3, 11, 2, 14, 4, 0, 4, 6, 0, 4, ... (A241018).

Michel Lagneau, Table of n, a(n) for n = 2..150

Cf. A241018, A241019, A241020, A241022, A041027.

with(numtheory):nn:=80:T:=array(1..nn):
   for n from 2 to nn do:
     for i from 1 to n do:
     T[i]:=9:
     od:
       ii:=0:
       for j from 1 to n while(ii=0)do:
       T[j]:=1:s:=sum('T[i]*10^(n-i)', 'i'=1..n):
         if type(s,prime)=true
         then
         ii:=1: printf(`%d, `,s):
         else
         T[j]:=9:
         fi:
         od:
          if ii=0
           then
           printf(`%d, `,0):
           else
          fi:
     od:

Table[SelectFirst[FromDigits/@Table[Insert[PadRight[{},k,9],1,n],{n,k+1}],PrimeQ],{k,30}]/.Missing["NotFound"]->0 (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 10 2017 *)

cp's OEIS Frontend

A077420 Bisection of Chebyshev sequence T(n,3) (odd part) with Diophantine property.

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A088317 a(n) = 8*a(n-1) + a(n-2), starting with a(0) = 1 and a(1) = 4.

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A041026 Numerators of continued fraction convergents to sqrt(18).

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A241021 Smallest prime numbers p of length n having a decimal expansion x(1)x(2)... x(n) such that there exists an index j where x(j) = 1 and x(i) = 9 for i<>j, or 0 if no such prime exists.

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