A110391 -id:A110391 - cp's OEIS frontend

A157696 Define k(n) to be the sequence of integers such that k(n)F(n)=F(2n)(Fibonacci sequence) (A000204); in turn define g(n) to be the sequence of integers such that g(n)k(n)=k(3n)(A110391); finally a(n) is the sequence of integers such that a(n)g(n)=g(5n).

31, 2521, 97921, 4974481, 226965751, 10783342081, 504420084871, 23735900452321, 1114384154071681, 52364857850613001, 2459808940392975631, 115562692701892638721, 5428914300540041959471, 255044709450472227347881

Offset: 1

Carmine Suriano, Mar 04 2009

nonn

Indices 2 of F(i), 3 of k(i) and 5 of g(i) are the minimum integers that provide sequences of integers.

A000204, A110391, A000045

Contribution from R. J. Mathar, Oct 18 2010: (Start)
A005248 := proc(n) combinat[fibonacci](2*n-1)+combinat[fibonacci](2*n+1) ; end proc:
A110391 := proc(n) A005248(n)-(-1)^n ; end proc:
A157696 := proc(n) A110391(5*n)/A110391(n) ; end proc: seq(A157696(n),n=1..25) ; (End)

a(n) = A110391(5*n)/A110391(n) = 27*a(n-1) +904*a(n-2) +1660*a(n-3) -1660*a(n-4) -904*a(n-5) -27*a(n-6) +a(n-7). [From R. J. Mathar, Oct 18 2010]

a(1) replaced by 31 - R. J. Mathar, Oct 18 2010

A159583 Values of A110391(5n)/A110391(n).

Original entry on oeis.org

1, 31, 2521, 97921, 4974481, 226965751, 10783342081, 504420084871, 23735900452321, 1114384154071681, 52364857850613001, 2459808940392975631, 115562692701892638721, 5428914300540041959471, 255044709450472227347881

Offset: 0

John W. Layman, Apr 16 2009

nonn

It appears that from a(2)=2521 onward, this sequence is the same as A157696.

A110391, A157696

A153173 a(n) = L(5*n)/L(n) where L(n) = Lucas number A000204(n).

Original entry on oeis.org

11, 41, 341, 2161, 15251, 103361, 711491, 4868641, 33391061, 228811001, 1568437211, 10749853441, 73681573691, 505018447961, 3461454668501, 23725145626561, 162614613425891, 1114577020834241, 7639424866266611

Offset: 1

Artur Jasinski, Dec 20 2008

All numbers in this sequence are congruent to 1 mod 10.

G. C. Greubel, Table of n, a(n) for n = 1..1000
L. Carlitz, Problem B-185, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 8, No. 3 (1970), p. 325; Lucas Ratio I, Solution to Problem B-185 by C. B. A. Peck, ibid., Vol. 9, No. 1 (1971), p. 109.
L. Carlitz, Problem B-186, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 8, No. 3 (1970), p. 326; Lucas Ratio II, Solution to Problem B-186 by John Wessner, ibid., Vol. 9, No. 1 (1971), pp. 109-110.
Index entries for linear recurrences with constant coefficients, signature (5,15,-15,-5,1).

Cf. A000032, A000045, A000204, A005248, A056854, A085695, A110391.

I:=[11, 41, 341, 2161, 15251]; [n le 5 select I[n] else 5*Self(n-1)+15*Self(n-2)-15*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..30]]; // G. C. Greubel, Dec 21 2017

Table[LucasL[5*n]/LucasL[n], {n, 1, 50}]
CoefficientList[Series[x*(11-14*x-29*x^2+6*x^3+x^4)/((1-x)*(x^2-7*x+1)*(x^2+3*x+1)), {x,0,50}], x] (* G. C. Greubel, Dec 21 2017 *)
a[ n_] := 1 + 5*Fibonacci[n]*Fibonacci[3*n]; (* Michael Somos, Apr 23 2022 *)

{L(n)=fibonacci(n-1)+fibonacci(n+1)}; a(n) = L(5*n)/L(n) \\ Charles R Greathouse IV, Jun 11 2015

my(x='x+O('x^30)); Vec(x*(11-14*x-29*x^2+6*x^3+x^4 )/((1-x)*(x^2-7*x +1)*(x^2+3*x+1))) \\ G. C. Greubel, Dec 21 2017

{a(n) = 1 + 5*fibonacci(n)*fibonacci(3*n)}; /* Michael Somos, Apr 23 2022 */

From R. J. Mathar, Oct 22 2010: (Start)
a(n) = 5*a(n-1) + 15*a(n-2) - 15*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: -x*(11-14*x-29*x^2+6*x^3+x^4)/( (x-1)*(x^2-7*x+1)*(x^2+3*x+1) ).
a(n) = 1 + A056854(n) - (-1)^n*A005248(n). (End)
From Amiram Eldar, Feb 02 2022: (Start)
a(n) = Lucas(2*n)^2 - (-1)^n*Lucas(2*n) - 1 (Carlitz, Problem B-185).
a(n) = (Lucas(2*n) - 3*(-1)^n)^2 + (-1)^n*(5*Fibonacci(n))^2 (Carlitz, Problem B-186). (End)
a(n) = a(-n) = 1 + 10*A085695(n) = 5 + L(n-1)*L(n)^2*L(n+1) for all n in Z. - Michael Somos, Apr 23 2022

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)

A153177 a(n) = L(9*n)/L(n) where L(n) = Lucas number A000204(n).

Original entry on oeis.org

76, 1926, 109801, 4769326, 230701876, 10716675201, 505618944676, 23714405408926, 1114769987764201, 52357935173823126, 2459933168462154076, 115560463558534156801, 5428954301161174383676, 255043991670277234750326

Offset: 1

Artur Jasinski, Dec 20 2008

nonn

All numbers in this sequence are:
congruent to 1 mod 100 (iff n is congruent to 0 mod 3),
congruent to 26 mod 100 (iff n is congruent to 2 or 4 mod 6),
congruent to 76 mod 100 (iff n is congruent to 1 or 5 mod 6).

G. C. Greubel, Table of n, a(n) for n = 1..595
Index entries for linear recurrences with constant coefficients, signature (34,714,-4641,-12376,12376,4641,-714,-34,1).

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

Cf. A153179, A153180. - R. J. Mathar, Oct 22 2010

[Lucas(9*n)/Lucas(n): n in [0..30]]; // G. C. Greubel, Dec 21 2017

Table[LucasL[9*n]/LucasL[n], {n, 1, 50}]
LinearRecurrence[{34,714,-4641,-12376,12376,4641,-714,-34,1},{76,1926,109801,4769326,230701876,10716675201,505618944676,23714405408926,1114769987764201},20] (* Harvey P. Dale, Aug 12 2012 *)

{lucas(n) = fibonacci(n+1) + fibonacci(n-1)};
for(n=0,30, print1( lucas(9*n)/lucas(n), ", ")) \\ G. C. Greubel, Dec 21 2017

From R. J. Mathar, Oct 22 2010: (Start)
a(n) = 34*a(n-1) +714*a(n-2) -4641*a(n-3) -12376*a(n-4) +12376*a(n-5) +4641*a(n-6) -714*a(n-7) -34*a(n-8) +a(n-9).
G.f.: -x*(76-658*x-9947*x^2+13644*x^3+26020*x^4-5306*x^5-1372*x^6+42*x^7 +x^8) / ((x-1)*(x^2+18*x+1)*(x^2-47*x+1)*(x^2+3*x+1)*(x^2-7*x+1)).
a(n) = 1-(-1)^n*A087215(n) -(-1)^n*A005248(n) +A056854(n) +A087265(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]

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.

cp's OEIS Frontend

A157696 Define k(n) to be the sequence of integers such that k(n)F(n)=F(2n)(Fibonacci sequence) (A000204); in turn define g(n) to be the sequence of integers such that g(n)k(n)=k(3n)(A110391); finally a(n) is the sequence of integers such that a(n)g(n)=g(5n).

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A159583 Values of A110391(5n)/A110391(n).

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A153173 a(n) = L(5*n)/L(n) where L(n) = Lucas number A000204(n).

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

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A153177 a(n) = L(9*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

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

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

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