A160682 -id:A160682 - cp's OEIS frontend

A161588 The list of the k values in the common solutions to the 2 equations 11k+1=A^2, 15k+1=B^2.

0, 13, 2184, 364728, 60907405, 10171171920, 1698524803248, 283643470970509, 47366761127271768, 7909965464783414760, 1320916865857702993165, 220585206632771616443808, 36836408590807002243122784, 6151459649458136602985061133, 1027256925050918005696262086440

Offset: 1

Paul Weisenhorn, Jun 14 2009

The 2 equations are equivalent to the Pell equation x^2-165*y^2=1,

with x=(165*k+13)/2 and y=A*B/2, case C=11 in A160682.

Index entries for linear recurrences with constant coefficients, signature (168,-168,1).

Cf. A160682, A085260 (sequence of A), A126816 (sequence of B).

t:=0: for n from 0 to 1000000 do a:=sqrt(11*n+1): b:=sqrt(15*n+1):
if (trunc(a)=a) and (trunc(b)=b) then t:=t+1: print(t,n,a,b): end if: end do:

k(t+3)=168*(k(t+2)-k(t+1))+k(t).
k(t)=((13+w)*((167+13*w)/2)^(t-1)+(13-w)*((167-13*w)/2)^(t-1))/330 where w=sqrt(165).
k(t) = floor of ((13+w)*((167+13*w)/2)^(t-1))/330;
G.f.: -13*x^2/((x-1)*(x^2-167*x+1)).

Edited, extended by R. J. Mathar, Sep 02 2009

A269028 a(n) = 40*a(n - 1) - a(n - 2) for n>1, a(0) = 1, a(1) = 1.

Original entry on oeis.org

1, 1, 39, 1559, 62321, 2491281, 99588919, 3981065479, 159143030241, 6361740144161, 254310462736199, 10166056769303799, 406387960309415761, 16245352355607326641, 649407706263983649879, 25960062898203738668519, 1037753108221885563090881

Offset: 0

Ilya Gutkovskiy, Feb 18 2016

In general, the ordinary generating function for the recurrence relation b(n) = k*b(n - 1) - b(n - 2) with n>1 and b(0)=1, b(1)=1, is (1 - (k - 1)*x)/(1 - k*x +x^2). This recurrence gives the closed form b(n) = (2^( -n - 1)*((k - 2)*(k - sqrt(k^2 - 4))^n + sqrt(k^2 - 4)*(k - sqrt(k^2 - 4))^n - (k - 2)*(sqrt(k^2 - 4) + k)^n + sqrt(k^2 - 4)*(sqrt(k^2 - 4) + k)^n))/sqrt(k^2 - 4).

Index entries for linear recurrences with constant coefficients, signature (40,-1).

Cf. A001519, A001835, A001653, A049685, A070997, A070998, A072256, A078922, A160682, A007805, A075839, A157014, A159664, A159668, A157877, A238379, A097315.

[n le 2 select 1 else 40*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Feb 19 2016

Table[Cosh[n Log[20 + Sqrt[399]]] - Sqrt[19/21] Sinh[n Log[20 + Sqrt[399]]], {n, 0, 17}]
Table[(2^(-n - 2) (38 (40 - 2 Sqrt[399])^n + 2 Sqrt[399] (40 - 2 Sqrt[399])^n - 38 (40 + 2 Sqrt[399])^n + 2 Sqrt[399] (40 + 2 Sqrt[399])^n))/Sqrt[399], {n, 0, 17}]
LinearRecurrence[{40, -1}, {1, 1}, 17]

G.f.: (1 - 39*x)/(1 - 40*x + x^2).
a(n) = cosh(n*log(20 + sqrt(399))) - sqrt(19/21)*sinh(n*log(20 + sqrt(399))).
a(n) = (2^(-n - 2)*(38*(40 - 2*sqrt(399))^n + 2*sqrt(399)*(40 - 2*sqrt(399))^n - 38*(40 + 2*sqrt(399))^n + 2*sqrt(399)*(40 + 2*sqrt(399))^n))/sqrt(399).
Sum_{n>=0} 1/a(n) = 2.0262989201139499769986...

cp's OEIS Frontend

A161588 The list of the k values in the common solutions to the 2 equations 11k+1=A^2, 15k+1=B^2.

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A269028 a(n) = 40*a(n - 1) - a(n - 2) for n>1, a(0) = 1, a(1) = 1.

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cp's OEIS Frontend

A161588 The list of the k values in the common solutions to the 2 equations 11*k+1=A^2, 15*k+1=B^2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Formula

Extensions

A269028 a(n) = 40*a(n - 1) - a(n - 2) for n>1, a(0) = 1, a(1) = 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A161588 The list of the k values in the common solutions to the 2 equations 11k+1=A^2, 15k+1=B^2.