A124297 -id:A124297 - cp's OEIS frontend

A001604 Odd-indexed terms of A124297.

11, 31, 151, 911, 5951, 40051, 272611, 1863551, 12760031, 87424711, 599129311, 4106261531, 28144128251, 192901135711, 1322159893351, 9062207833151, 62113268013311, 425730597768451, 2918000731816531, 20000274041790911, 137083916295800111, 939587136717207031, 6440026032054760351

Offset: 0

N. J. A. Sloane

Old name: Related to factors of Fibonacci numbers.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

John Cerkan, Table of n, a(n) for n = 0..1187
Dov Jarden, Recurring Sequences, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See p. 20.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (11, -33, 33, -11, 1).

Cf. A001603, A124296, A124297.

A001604:=-(11-90*z+173*z**2-90*z**3+11*z**4)/(z-1)/(z**2-3*z+1)/(z**2-7*z+1); # conjectured (correctly) by Simon Plouffe in his 1992 dissertation

5 #^2 + 5 # + 1 &@ Fibonacci@ # & /@ Range[1, 45, 2] (* Michael De Vlieger, Apr 03 2017 *)

G.f.: -(11-90*x+173*x^2-90*x^3+11*x^4)/((x-1)*(x^2-3*x+1)*(x^2-7*x+1)). [After Simon Plouffe]

a(n) = (5+sqrt(5))/2*((3+sqrt(5))/2)^n+(5-sqrt(5))/2*((3-sqrt(5))/2)^n+(3+sqrt(5))/2*((7+3*sqrt(5))/2)^n+(3-sqrt(5))/2*((7-3*sqrt(5))/2)^n+3. [Tim Monahan, Aug 15 2011]

Entry revised by Michel Marcus and N. J. A. Sloane, Jun 06 2015

A133320 Numbers k such that both A124296(k) = 5F(k)^2 - 5F(k) + 1 and A124297(k) = 5F(k)^2 + 5F(k) + 1 are prime, where F(k) = Fibonacci(k).

Original entry on oeis.org

3, 4, 5, 10, 40

Offset: 1

Alexander Adamchuk, Oct 18 2007

Cf. A124297 (5*F(n)^2 + 5*F(n) + 1, where F(n) = Fibonacci(n)).
Cf. A124296 (5*F(n)^2 - 5*F(n) + 1, where F(n) = Fibonacci(n)).
Cf. A000032, A000045, A121171, A001946.

Do[ F=Fibonacci[n]; f=5*F^2-5*F+1; g=5*F^2+5*F+1; If[ PrimeQ[f], If[ PrimeQ[g], Print[ {n,f,g} ] ] ], {n,1,1000} ]

A001946 a(n) = 11*a(n-1) + a(n-2).

Original entry on oeis.org

2, 11, 123, 1364, 15127, 167761, 1860498, 20633239, 228826127, 2537720636, 28143753123, 312119004989, 3461452808002, 38388099893011, 425730551631123, 4721424167835364, 52361396397820127, 580696784543856761, 6440026026380244498, 71420983074726546239

Offset: 0

N. J. A. Sloane, Simon Plouffe

For odd n there is the Aurifeuillian factorization a(n) = Lucas[5n] = Lucas[n]*A[n]*B[n] = A000032[n]*A124296[n]*A124297[n], where A[n] = A124296[n] = 5*F(n)^2 - 5*F(n) + 1 and B[n] = A124297[n] = 5*F(n)^2 + 5*F(n) + 1, where F(n) = Fibonacci[n]. The largest prime divisors of a(n) for n>0 are listed in A121171[n] = {11, 41, 31, 2161, 151, 2521, 911, ...}. - Alexander Adamchuk, Oct 25 2006

For more information about this type of recurrence follow the Khovanova link and see A086902 and A054413. - Johannes W. Meijer, Jun 12 2010

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 139.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)
Index entries for linear recurrences with constant coefficients, signature (11, 1).

Cf. A000032, A000045, A121171, A124296, A124297.

[ Lucas(5*n) : n in [0..100]]; // Vincenzo Librandi, Apr 14 2011

A001946:=(-2+11*z)/(-1+11*z+z**2); # Conjectured by Simon Plouffe in his 1992 dissertation

Table[Fibonacci[5n-1]+Fibonacci[5n+1],{n,0,30}] (* Alexander Adamchuk, Oct 25 2006 *)
LinearRecurrence[{11,1},{2,11},20] (* Harvey P. Dale, Jan 25 2024 *)

a(n) = Lucas(5n) = Fibonacci(5n-1) + Fibonacci(5n+1). - Alexander Adamchuk, Oct 25 2006
a(n) = ((11 + 5*sqrt(5))/2)^n + ((11 - 5*sqrt(5))/2)^n. - Tanya Khovanova, Feb 06 2007
Contribution from Johannes W. Meijer, Jun 12 2010: (Start)
a(2n+1) = 11*A097842(n), a(2n) = A065705(n).
a(3n+1) = A041226(5n), a(3n+2) = A041226(5n+3), a(3n+3) = 2* A041226(5n+4).
Limit(a(n+k)/a(k), k=infinity) = (A001946(n) + A049666(n)*sqrt(125))/2.
Limit(A001946(n)/A049666(n), n=infinity) = sqrt(125). (End)
From Peter Bala, Mar 22 2015: (Start)
a(n) = Fibonacci(10*n)/Fibonacci(5*n) for n >= 1.
a(n) = ( Fibonacci(5*n + 2*k) - F(5*n - 2*k) )/Fibonacci(2*k) for nonzero integer k.
a(n) = ( Fibonacci(5*n + 2*k + 1) + F(5*n - 2*k - 1) )/Fibonacci(2*k + 1) for arbitrary integer k.
a(n) = Sum_{k = 0..2*n} binomial(2*n,k)*Lucas(n + k). (End)
a(n) = [x^n] ( (1 + 11*x + sqrt(1 + 22*x + 125*x^2))/2 )^n for n >= 1. - Peter Bala, Jun 26 2015
E.g.f.: 2*cosh(5*sqrt(5)*x/2)*(cosh(11*x/2) + sinh(11*x/2)). - Stefano Spezia, Jan 18 2025

A124296 a(n) = 5F(n)^2 - 5F(n) + 1, where F(n) = Fibonacci(n).

Original entry on oeis.org

1, 1, 1, 11, 31, 101, 281, 781, 2101, 5611, 14851, 39161, 102961, 270281, 708761, 1857451, 4865911, 12744061, 33372361, 87382901, 228792301, 599019851, 1568309051, 4105974961, 10749725281, 28143378001, 73680695281, 192899171531

Offset: 0

Alexander Adamchuk, Oct 25 2006

11 = Lucas(5) divides a(3+10k), a(7+10k), and a(8+10k). The last digit of a(n) is 1, so a(n) mod 10 = 1. For odd n there exists the so-called Aurifeuillian factorization A001946(n) = Lucas(5n) = Lucas(n)*A(n)*B(n) = A000032(n)*A124296(n)*A124297(n), where A(n) = A124296(n) = 5*F(n)^2 - 5*F(n) + 1 and B(n) = A124297(n) = 5*F(n)^2 + 5*F(n) + 1, where F(n) = Fibonacci(n).

John Cerkan, Table of n, a(n) for n = 0..2375
Eric Weisstein's World of Mathematics, Aurifeuillean Factorization
Index entries for linear recurrences with constant coefficients, signature (4,-2,-6,4,2,-1).

Cf. A000032, A000045, A121171, A001946, A124297.

Bisections: A001604, A156094.

Table[5*Fibonacci[n]^2-5*Fibonacci[n]+1,{n,0,50}]
5#^2-5#+1&/@Fibonacci[Range[0,30]] (* Harvey P. Dale, Nov 29 2011 *)

a(n)=subst(5*t*(t-1)+1, t, fibonacci(n)) \\ Charles R Greathouse IV, Jan 03 2013

a(n) = 5*Fibonacci(n)^2 - 5*Fibonacci(n) + 1.

G.f.: -(x^5+9*x^4-15*x^3+x^2+3*x-1) / ((x-1)*(x+1)*(x^2-3*x+1)*(x^2+x-1)). [Colin Barker, Jan 03 2013]

A156094 5 F(2n) (F(2n) - 1) + 1 where F(n) denotes the n-th Fibonacci number.

Original entry on oeis.org

1, 1, 31, 281, 2101, 14851, 102961, 708761, 4865911, 33372361, 228792301, 1568309051, 10749725281, 73680695281, 505017569551, 3461448647801, 23725139605861, 162614572159411, 1114576979567761, 7639424583421961, 52361395886149351

Offset: 0

Stuart Clary, Feb 04 2009

Natural bilateral extension (brackets mark index 0): ..., 15401, 2311, 361, 61, 11, [1], 1, 31, 281, 2101, 14851, ... This is A156095-reversed followed by A156094, without repeating the central 1. That is, A156094(-n) = A156095(n).

Cf. A124296, A124297, A001603, A001604, A156095

a[n_Integer] := 5 Fibonacci[2n] (Fibonacci[2n] - 1) + 1
5(#*(#-1))&/@Fibonacci[Range[0,40,2]]+1 (* Harvey P. Dale, Jan 06 2013 *)

Let F(n) be the Fibonacci number A000045(n) and let L(n) be the Lucas number A000032(n).
Alternate formula: a(n) = L(4n) - 5 F(2n) - 1.
Recurrence: a(n) - 10 a(n-1) + 23 a(n-2) - 10 a(n-3) + a(n-4) = -5.
Recurrence: a(n) - 11 a(n-1) + 33 a(n-2) - 33 a(n-3) + 11 a(n-4) - a(n-5) = 0.
G.f.: A(x) = (1 - 10 x + 53 x^2 - 60 x^3 + 11 x^4)/(1 - 11 x + 33 x^2 - 33 x^3 + 11 x^4 - x^5) = (1 - 10 x + 53 x^2 - 60 x^3 + 11 x^4)/((1 - x) (1 - 7 x + x^2) (1 - 3 x + x^2)).
a(n)=A056854(n)-5*A001906(n)-1. - R. J. Mathar, Feb 23 2009
a(n)=((2*sqrt(5))/2)*(((3-sqrt(5))/2)^n-((3+sqrt(5))/2)^n)+((7+3*sqrt(5))/2)^n+((7-3*sqrt(5))/2)^n-1. - Tim Monahan, Aug 15 2011

A001603 Odd-indexed terms of A124296.

Original entry on oeis.org

1, 11, 101, 781, 5611, 39161, 270281, 1857451, 12744061, 87382901, 599019851, 4105974961, 28143378001, 192899171531, 1322154751061, 9062194370461, 62113232767531, 425730505493801, 2918000490238361, 20000273409331051, 137083914639998701, 939587132382262661

Offset: 0

N. J. A. Sloane

Old name: Related to factors of Fibonacci numbers.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

John Cerkan, Table of n, a(n) for n = 0..1187
Dov Jarden, Recurring Sequences, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See p. 20.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (11, -33, 33, -11, 1).

Cf. A001604, A124296, A124297.

A001603:=-(1+13*z**2+z**4)/(z-1)/(z**2-3*z+1)/(z**2-7*z+1); # conjectured (correctly) by Simon Plouffe in his 1992 dissertation

5 #^2 - 5 # + 1 &@ Fibonacci@ # & /@ Range[1, 43, 2] (* Michael De Vlieger, Apr 03 2017 *)

G.f.: -(1+13*x^2+x^4)/((x-1)*(x^2-3*x+1)*(x^2-7*x+1)). [After Simon Plouffe]

Entry revised by Michel Marcus and N. J. A. Sloane, Jun 06 2015

A156095 5 F(2n) (F(2n) + 1) + 1 where F(n) denotes the n-th Fibonacci number.

Original entry on oeis.org

1, 11, 61, 361, 2311, 15401, 104401, 712531, 4875781, 33398201, 228859951, 1568486161, 10750188961, 73681909211, 505020747661, 3461456968201, 23725161388951, 162614629188281, 1114577128871281, 7639424974303651, 52361396909490901

Offset: 0

Stuart Clary, Feb 04 2009

Natural bilateral extension (brackets mark index 0): ..., 14851, 2101, 281, 31, 1, [1], 11, 61, 361, 2311, 15401, ... This is A156095-reversed followed by A156095, without repeating the central 1. That is, A156095(-n) = A156094(n).

Cf. A124296, A124297, A001603, A001604, A156094.

a[n_Integer] := 5 Fibonacci[2n] (Fibonacci[2n] + 1) + 1

Let F(n) be the Fibonacci number A000045(n) and let L(n) be the Lucas number A000032(n).
Alternate formula: a(n) = L(4n) + 5 F(2n) - 1.
Recurrence: a(n) - 10 a(n-1) + 23 a(n-2) - 10 a(n-3) + a(n-4) = -5.
Recurrence: a(n) - 11 a(n-1) + 33 a(n-2) - 33 a(n-3) + 11 a(n-4) - a(n-5) = 0.
G.f.: A(x) = (1 - 27 x^2 + 20 x^3 + x^4)/(1 - 11 x + 33 x^2 - 33 x^3 + 11 x^4 - x^5) = (1 - 27 x^2 + 20 x^3 + x^4)/((1 - x) (1 - 7 x + x^2) (1 - 3 x + x^2)).
a(n)=((2*sqrt(5))/2)*(((3+sqrt(5))/2)^n-((3-sqrt(5))/2)^n)+((7+3*sqrt(5))/2)^n+((7-3*sqrt(5))/2)^n-1. - Tim Monahan, Aug 15 2011

cp's OEIS Frontend

A001604 Odd-indexed terms of A124297.

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A133320 Numbers k such that both A124296(k) = 5*F(k)^2 - 5*F(k) + 1 and A124297(k) = 5*F(k)^2 + 5*F(k) + 1 are prime, where F(k) = Fibonacci(k).

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A001946 a(n) = 11*a(n-1) + a(n-2).

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A124296 a(n) = 5*F(n)^2 - 5*F(n) + 1, where F(n) = Fibonacci(n).

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A156094 5 F(2n) (F(2n) - 1) + 1 where F(n) denotes the n-th Fibonacci number.

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A001603 Odd-indexed terms of A124296.

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Maple

Mathematica

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Extensions

A156095 5 F(2n) (F(2n) + 1) + 1 where F(n) denotes the n-th Fibonacci number.

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Author

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Comments

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Mathematica

Formula

A133320 Numbers k such that both A124296(k) = 5F(k)^2 - 5F(k) + 1 and A124297(k) = 5F(k)^2 + 5F(k) + 1 are prime, where F(k) = Fibonacci(k).

A124296 a(n) = 5F(n)^2 - 5F(n) + 1, where F(n) = Fibonacci(n).