A153177 -id:A153177 - cp's OEIS frontend

A110391 a(n) = L(3*n)/L(n), where L(n) = Lucas number.

1, 4, 6, 19, 46, 124, 321, 844, 2206, 5779, 15126, 39604, 103681, 271444, 710646, 1860499, 4870846, 12752044, 33385281, 87403804, 228826126, 599074579, 1568397606, 4106118244, 10749957121, 28143753124, 73681302246, 192900153619, 505019158606, 1322157322204

Offset: 0

Amarnath Murthy, Jul 27 2005

Subsidiary sequences: a(n) = L((2k+1)*n)/L(n) for k = 2,3, etc. This is the sequence for k = 1.

			a(1) = L(3)/L(1) = 4/1 = 4.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,2,-1).

Cf. A000204, A000032, A002878, A114525, A153173, A153175, A153177, A153179, A153180.

[Lucas(3*n)/Lucas(n): n in [0..30]]; // G. C. Greubel, Dec 17 2017

with(combinat): L:=n->fibonacci(n+2)-fibonacci(n-2): seq(L(3*n)/L(n),n=0..30); # Emeric Deutsch, Jul 31 2005

Table[LucasL[3 n]/LucasL[n], {n, 0, 27}] (* Michael De Vlieger, Mar 18 2015 *)
LinearRecurrence[{2,2,-1},{1,4,6},40] (* Harvey P. Dale, Aug 20 2020 *)

Vec((1+2*x-4*x^2)/((1+x)*(x^2-3*x+1)) + O(x^30)) \\ Colin Barker, Jun 03 2016

for(n=0,30, print1((fibonacci(3*n+1) + fibonacci(3*n-1))/( fibonacci(n+1) + fibonacci(n-1)), ", ")) \\ G. C. Greubel, Dec 17 2017

From R. J. Mathar, Oct 18 2010: (Start)
a(n) = A005248(n) - (-1)^n.
a(n) = +2*a(n-1) +2*a(n-2) -a(n-3).
G.f.: ( 1+2*x-4*x^2 ) / ( (1+x)*(x^2-3*x+1) ). (End)
Exp( Sum_{n >= 1} a(n)*t^n/n ) = 1 + 4*t + 11*t^2 + 29*t^3 + ... is the o.g.f. for A002878. This is the case x = 1 of the general result exp( Sum_{n >= 1} L(3*n,x)/L(n,x)*t^n/n ) = Sum_{n >= 0} L(2*n + 1,x)*t^n, where L(n,x) is the n-th Lucas polynomial of A114525. - Peter Bala, Mar 18 2015
a(n) = 2^(-n)*(-(-2)^n+(3-sqrt(5))^n+(3+sqrt(5))^n). - Colin Barker, Jun 03 2016

Corrected and extended by Emeric Deutsch and Erich Friedman, Jul 31 2005

A153175 a(n) = L(7*n)/L(n) where L(n) = Lucas number A000204(n).

Original entry on oeis.org

29, 281, 6119, 101521, 1875749, 33281921, 599786069, 10745088481, 192933544679, 3461223997001, 62114818827629, 1114566304366081, 20000347407134669, 358889844987430121, 6440029487834912999, 115561554399692896321

Offset: 1

Artur Jasinski, Dec 20 2008

nonn

All numbers in this sequence are:
congruent to 9 mod 10 (iff n is odd),
congruent to 1 mod 10 (iff n is even).

G. C. Greubel, Table of n, a(n) for n = 1..795
Index entries for linear recurrences with constant coefficients, signature (13,104,-260,-260,104,13,-1).

Cf. A000032, A000204, A110391, A153173.

Cf. A153177, A153179, A153180. [From R. J. Mathar, Oct 22 2010]

[Lucas(7*n)/Lucas(n): n in [0..30]]; // G. C. Greubel, Dec 21 2017

Table[LucasL[7*n]/LucasL[n], {n, 1, 50}]

{lucas(n) = fibonacci(n+1) + fibonacci(n-1)};
for(n=0,30, print1( lucas(7*n)/lucas(n), ", ")) \\ G. C. Greubel, Dec 21 2017

From R. J. Mathar, Oct 22 2010: (Start)
a(n) = +13*a(n-1) +104*a(n-2) -260*a(n-3) -260*a(n-4) +104*a(n-5) +13*a(n-6) -a(n-7).
G.f.: -x*(-29+96*x+550*x^2-290*x^3-200*x^4+16*x^5+x^6) / ( (1+x)*(x^2-3*x+1)*(x^2-18*x+1)*(x^2+7*x+1) ).
a(n) = A005248(n) +A087215(n) -(-1)^n*A056854(n) - (-1)^n. (End)

A153179 a(n) = L(11*n)/L(n) where L(n) = A000204(n).

Original entry on oeis.org

199, 13201, 1970299, 224056801, 28374454999, 3450736132801, 426236170575799, 52337681992411201, 6441140796368008699, 792018481913198430001, 97420733208491869044199, 11981539981561372141075201

Offset: 1

Artur Jasinski, Dec 20 2008

nonn

All numbers in this sequence are:
congruent to 99 mod 100 (iff n is odd),
congruent to 1 mod 100 (iff n is even).

G. C. Greubel, Table of n, a(n) for n = 1..475
Index entries for linear recurrences with constant coefficients, signature (89,4895,-83215,-582505,1514513,1514513,-582505,-83215,4895,89,-1).

Cf. A000032, A000204, A110391, A153173, A153175, A153177.

[Lucas(11*n)/Lucas(n): n in [0..30]]; // G. C. Greubel, Dec 21 2017

Table[LucasL[11*n]/LucasL[n], {n, 1, 50}]

{lucas(n) = fibonacci(n+1) + fibonacci(n-1)};
for(n=0,30, print1( lucas(11*n)/lucas(n), ", ")) \\ G. C. Greubel, Dec 21 2017

From R. J. Mathar, Oct 22 2010: (Start)
a(n) = +89*a(n-1) +4895*a(n-2) -83215*a(n-3) -582505*a(n-4) +1514513*a(n-5) +1514513*a(n-6) -582505*a(n-7) -83215*a(n-8) +4895*a(n-9) +89*a(n-10) -a(n-11).
G.f.: -1 -1/(1+x) +(-2-47*x)/(x^2+47*x+1) +(2-3*x)/(x^2-3*x+1) +(-2-7*x)/(x^2+7*x+1) +(2-123*x)/(x^2-123*x+1) +(2-18*x)/(x^2-18*x+1).
a(n) = -(-1)^n -(-1)^n*A087265(n) +A005248(n) -(-1)^n*A056854(n) +A065705(n) +A087215(n). (End)

A153180 a(n) = L(13n)/L(n) where L(n) = Lucas number A000204(n).

Original entry on oeis.org

521, 90481, 35355581, 10525900321, 3489827263001, 1111126318086721, 359316586176453881, 115509240442846111681, 37216910406644366498621, 11980863523543017476802001, 3858153294795970321295258921

Offset: 1

Artur Jasinski, Dec 20 2008

nonn

All numbers in this sequence are:
congruent to 1 mod 10
congruent to 1 mod 100 (iff n is congruent to 0 mod 5),Q congruent to 21 mod 100 (iff n is congruent to 1 or 4 mod 5),
congruent to 81 mod 100 (iff n is congruent to 2 or 3 mod 5).Q

Index entries for linear recurrences with constant coefficients, signature (233, 33552, -1493064, -27372840, 186135312, 488605194, -488605194, -186135312, 27372840, 1493064, -33552, -233, 1).

A000032, A000204, A047221, A110391, A153173, A153175, A153177, A153179.

Table[LucasL[13 n]/LucasL[n], {n, 1, 150}]

a(n)= +233*a(n-1) +33552*a(n-2) -1493064*a(n-3) -27372840*a(n-4) +186135312*a(n-5) +488605194*a(n-6) -488605194*a(n-7) -186135312*a(n-8) +27372840*a(n-9) +1493064*a(n-10) -33552*a(n-11) -233*a(n-12) +a(n-13). G.f.: -1+ (-2-123*x)/(x^2+123*x+1) +(2-322*x)/(x^2-322*x+1) +(-2-3*x)/(x^2+3*x+1) +(2-7*x)/(x^2-7*x+1) +(2-47*x)/(x^2-47*x+1) -1/(x-1)+ (-2-18*x)/(x^2+18*x+1). [From R. J. Mathar, Oct 22 2010]

cp's OEIS Frontend

A110391 a(n) = L(3*n)/L(n), where L(n) = Lucas number.

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A153175 a(n) = L(7*n)/L(n) where L(n) = Lucas number A000204(n).

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A153179 a(n) = L(11*n)/L(n) where L(n) = A000204(n).

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Magma

Mathematica

PARI

Formula

A153180 a(n) = L(13n)/L(n) where L(n) = Lucas number A000204(n).

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