A049685 -id:A049685 - cp's OEIS frontend

A173731 a(n) = a(n-1) * (11a(n-1) - a(n-2)) / (a(n-1) + 4a(n-2)), a(0) = a(1) = 0, a(2) = 1.

0, 0, 1, 11, 88, 638, 4466, 30856, 212135, 1455685, 9981840, 68428140, 469043796, 3214953456, 22035826813, 151036348463, 1035219958696, 7095506886986, 48633337477670, 333337879614520, 2284731883069955, 15659785467455305

Offset: 0

Michael Somos, Feb 23 2010

nonn

			x^2 + 11*x^3 + 88*x^4 + 638*x^5 + 4466*x^6 + 30856*x^7 + 212135*x^8 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (11,-33,33,-11,1).

Cf. A172511

[(4+Fibonacci(4*n+1)/3+Fibonacci(4*n + 3)/3-5* Fibonacci(2*n+1)) / 20: n in [0..25]]; // Vincenzo Librandi, Nov 30 2016

Table[(4 + Fibonacci[4*n + 1]/3 + Fibonacci[4*n + 3]/3 - 5*Fibonacci[2*n + 1])/20, {n, 0, 25}] (* or *) LinearRecurrence[{11, -33, 33, -11, 1}, {0, 0, 1, 11, 88}, 25] (* G. C. Greubel, Nov 29 2016 *)

{a(n) = (4 + fibonacci(4*n + 1)/3 + fibonacci(4*n + 3)/3 - 5 * fibonacci(2*n + 1)) / 20}

a(n) = (4 + A049685(n) - 5 * A122367(n)) / 20 = a(1 - n).
G.f.: x^2 / ((1 - x) * (1 - 3*x + x^2) * (1 - 7*x + x^2)) = ( 4 / (1 - x) - 5 * (1 - x) / (1 - 3*x + x^2) + (1 - x) / (1 - 7*x + x^2) ) / 20.
From G. C. Greubel, Nov 29 2016: (Start)
a(n) = 11*a(n-1) - 33*a(n-2) + 33*a(n-3) - 11*a(n-4) + a(n-5).
a(n) = (12 + Fibonacci(4*n + 1) + Fibonacci(4*n + 3) - 15*Fibonacci[2*n + 1] ) / 60. (End)

A333718 a(n) = L(8*n+4)/7, where L=A000032 (the Lucas sequence).

Original entry on oeis.org

1, 46, 2161, 101521, 4769326, 224056801, 10525900321, 494493258286, 23230657239121, 1091346396980401, 51270050000839726, 2408601003642486721, 113152977121196036161, 5315781323692571212846, 249728569236429650967601, 11731926972788501024264401, 551150839151823118489459246

Offset: 0

Greg Dresden and Tracy Z. Wu, Sep 03 2020

a(n) is the denominator of the continued fraction [3*sqrt(5), 3*sqrt(5),..., 3*sqrt(5)] with 2n+1 terms.

a(n) = (2/7)*T(2*n+1, 7/2), where T(n,x) denotes the n-th Chebyshev polynomial of the first kind. - Peter Bala, Jul 08 2022

			The continued fraction [3*sqrt(5), 3*sqrt(5), 3*sqrt(5)] with 2*1 + 1 terms equals 141*sqrt(5)/46, and 46 is our a(1) term.

Index entries for linear recurrences with constant coefficients, signature (47,-1).

Cf. A000032, A049685, first differences of A049668.

Mathematica
```
Table[LucasL[8 n + 4]/7, {n, 0, 20}]
```

a(n) = 47*a(n-1) - a(n-2) for n>2.

G.f.: (1-x)/(1-47*x+x^2). - R. J. Mathar, Sep 03 2020

A143790 Integer Quotients of Lucas Numbers; a rectangular array by downward antidiagonals.

Original entry on oeis.org

1, 3, 1, 4, 6, 1, 7, 41, 19, 1, 11, 281, 341, 46, 1, 18, 1926, 6119, 2161, 124, 1, 29, 13201, 109801, 101521, 15251, 321, 1, 47, 90481, 1970299, 4769326, 1875749, 103361, 844, 1, 76, 620166, 35355581, 224056801, 230701876, 33281921, 711491, 2206, 1, 123

Offset: 1

Clark Kimberling, Sep 01 2008

Every integer-valued quotient of Lucas numbers (excluding L(0)=2) is in this array.
(Row 1) = A000032 except for a(0)
(Row 2) = A049685
(Row 3) = A049629
(Column 2) = A110391 except for initial terms

			Q(3,2)=L(9)/L(3)=76/4=19.

Cf. A000032.

Row 1: L(k)/L(1), where L(k)=A000032(k) = k-th Lucas number, for k>=1;

Row n: L(2nk-n)/L(n) for n>=2, k>=1.

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...

A278475 a(n) = floor(phi^7*a(n-1)) for n>0, a(0) = 1, where phi is the golden ratio (A001622).

Original entry on oeis.org

1, 29, 841, 24417, 708933, 20583473, 597629649, 17351843293, 503801085145, 14627583312497, 424703717147557, 12331035380591649, 358024729754305377, 10395048198255447581, 301814422479162285225, 8763013300093961719105, 254429200125204052139269, 7387209816931011473757905

Offset: 0

Ilya Gutkovskiy, Nov 23 2016

In general, the ordinary generating function for the recurrence relation b(n) = floor(phi^k*b(n - 1)) with n>0 and b(0) = 1, is (1 - x)/(1 - (phi^k + (-phi)^(-k))*x + x^2) if k is even, and (1 - x - x^2)/((1 - x)*(1 - (phi^k + (-phi)^(-k))*x - x^2)) if k is odd.

Index entries for linear recurrences with constant coefficients, signature (30,-28,-1).

Cf. A001622.

Cf. similar sequences with recurrence relation b(n) = floor(phi^k*b(n-1)) for n>0, b(0) = 1: A000012 (k = 1), A001519 (k = 2), A024551 (k = 3), A049685 (k = 4), A214993 (k = 5), A007805 (k = 6), this sequence (k = 7).

RecurrenceTable[{a[0] == 1, a[n] == Floor[GoldenRatio^7 a[n - 1]]}, a, {n, 17}]
LinearRecurrence[{30, -28, -1}, {1, 29, 841}, 18]

Vec( (1 - x - x^2)/((1 - x)*(1 - 29*x - x^2)) + O(x^50) ) \\ G. C. Greubel, Nov 24 2016

G.f.: (1 - x - x^2)/((1 - x)*(1 - 29*x - x^2)).
a(n) = 30*a(n-1) - 28*a(n-2) - a(n-3).
a(n) = ((-29 - 13*sqrt(5))^(-n)*(-7*(407 + 182*sqrt(5))*2^(n+3) + 13*(1885 + 843*sqrt(5))*(-29 - 13*sqrt(5))^n + 28*(25319 + 11323*sqrt(5))*(-843 - 377*sqrt(5))^n))/(377*(1885 + 843*sqrt(5))).

cp's OEIS Frontend

A173731 a(n) = a(n-1) * (11*a(n-1) - a(n-2)) / (a(n-1) + 4*a(n-2)), a(0) = a(1) = 0, a(2) = 1.

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A333718 a(n) = L(8*n+4)/7, where L=A000032 (the Lucas sequence).

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A143790 Integer Quotients of Lucas Numbers; a rectangular array by downward antidiagonals.

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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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Magma

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A278475 a(n) = floor(phi^7*a(n-1)) for n>0, a(0) = 1, where phi is the golden ratio (A001622).

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Mathematica

PARI

Formula

A173731 a(n) = a(n-1) * (11a(n-1) - a(n-2)) / (a(n-1) + 4a(n-2)), a(0) = a(1) = 0, a(2) = 1.