A000032 -id:A000032 - cp's OEIS frontend

A338736 a(n) = L(L(n)) mod n, where L = Lucas numbers = A000032.

0, 0, 1, 1, 4, 0, 3, 7, 7, 4, 10, 3, 9, 10, 7, 15, 12, 0, 10, 9, 7, 4, 22, 3, 1, 4, 7, 1, 4, 18, 30, 31, 7, 4, 29, 15, 1, 34, 34, 39, 35, 24, 29, 29, 7, 4, 46, 3, 1, 4, 7, 29, 29, 0, 21, 55, 7, 54, 35, 3, 45, 4, 7, 63, 64, 36, 2, 29, 7, 4, 6, 3, 43, 4, 7, 29

Offset: 1

Alois P. Heinz, Nov 05 2020

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000032, A005371, A263112, A338638, A338889.

b:= proc(n) local r, M, p; r, M, p:=
      <<1|0>, <0|1>>, <<0|1>, <1|1>>, n;
      do if irem(p, 2, 'p')=1 then r:=
        `if`(nargs=1, r.M, r.M mod args[2]) fi;
         if p=0 then break fi; M:=
        `if`(nargs=1, M.M, M.M mod args[2])
      od; (r.<<2, 1>>)[1$2]
    end:
a:= n-> (f-> b(f, n) mod n)(b(n)):
seq(a(n), n=1..80);

a(n) = A005371(n) mod n.

A338889 a(n) = L(L(L(n))) mod L(L(n)), where L = Lucas numbers = A000032.

Original entry on oeis.org

1, 0, 3, 1, 1, 1, 0, 1, 1, 29, 7, 1, 19679776435706023589554718882448088434898811874077010905231927243854, 1, 7

Offset: 0

Alois P. Heinz, Nov 14 2020

nonn

a(21) = 2992285359..7163788371 has 5090 decimal digits.

Alois P. Heinz, Table of n, a(n) for n = 0..20

Cf. A000032, A005371, A262361, A274996, A338638, A338736.

b:= proc(n) local r, M, p; r, M, p:=
      <<1|0>, <0|1>>, <<0|1>, <1|1>>, n;
      do if irem(p, 2, 'p')=1 then r:=
        `if`(nargs=1, r.M, r.M mod args[2]) fi;
         if p=0 then break fi; M:=
        `if`(nargs=1, M.M, M.M mod args[2])
      od; (r.<<2, 1>>)[1$2]
    end:
a:= n-> (h-> b(h$2) mod h)(b(b(n))):
seq(a(n), n=0..15);

a(n) = A262361(n) mod A005371(n).

A339517 Odd composite integers m such that A000032(2*m-J(m,5)) == J(m,5) (mod m), where J(m,5) is the Jacobi symbol.

Original entry on oeis.org

323, 377, 1001, 1183, 1729, 1891, 3827, 4181, 5777, 6601, 6721, 8149, 8841, 10877, 11663, 13201, 13981, 15251, 17119, 17711, 18407, 19043, 23407, 25877, 26011, 27323, 30889, 34561, 34943, 35207, 39203, 40501, 41041

Offset: 1

Ovidiu Bagdasar, Dec 07 2020

nonn

The generalized Pell-Lucas sequences of integer parameters (a,b) defined by V(m+2)=a*V(m+1)-b*V(m) and V(0)=2, V(1)=a, satisfy V(k*p-J(p,D)) == V(k-1)*J(p,D) (mod p) whenever p is prime, k is a positive integer, b=-1 and D=a^2+4.
The composite integers m with the property V(k*m-J(m,D)) == V(k-1)*J(m,D) (mod m) are called generalized Pell-Lucas pseudoprimes of level k- and parameter a.
Here b=-1, a=1, D=5 and k=2, while V(m) recovers A000032(m) (Lucas numbers).

D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.
D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021).
D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted).

Dorin Andrica and Ovidiu Bagdasar, On Generalized Lucas Pseudoprimality of Level k, Mathematics (2021) Vol. 9, 838.
Dorin Andrica, Vlad Crişan, and Fawzi Al-Thukair, On Fibonacci and Lucas sequences modulo a prime and primality testing, Arab Journal of Mathematical Sciences, 24(1), 9-15 (2018).

Cf. A000032, A071904, A339125 (a=1, b=-1, k=1).

Cf. A339518 (a=3, b=-1), A339519 (a=5, b=-1), A339520 (a=7, b=-1).

Select[Range[3, 45000, 2], CoprimeQ[#, 5] && CompositeQ[#] && Divisible[LucasL[2*# - JacobiSymbol[#, 5]] - JacobiSymbol[#, 5], #] &]

A347861 a(n) = A000032(n)*A000032(n+1) mod A000032(n+2).

Original entry on oeis.org

2, 3, 5, 6, 5, 24, 5, 71, 5, 194, 5, 516, 5, 1359, 5, 3566, 5, 9344, 5, 24471, 5, 64074, 5, 167756, 5, 439199, 5, 1149846, 5, 3010344, 5, 7881191, 5, 20633234, 5, 54018516, 5, 141422319, 5, 370248446, 5, 969323024, 5, 2537720631, 5, 6643838874, 5, 17393795996, 5, 45537549119, 5, 119218851366, 5

Offset: 0

J. M. Bergot and Robert Israel, Jan 23 2022

The analogous sequence for Fibonacci numbers instead of Lucas numbers is A333599.

			a(3) = A000032(3)*A000032(4) mod A000032(5) = 4*7 mod 11 = 6.

Robert Israel, Table of n, a(n) for n = 0..4760
Index entries for linear recurrences with constant coefficients, signature (-1,3,3,-1,-1).

Cf. A000032, A333599.

L:= n -> combinat:-fibonacci(n-1)+combinat:-fibonacci(n+1):
f:= n -> L(n)*L(n+1) mod L(n+2):
map(f, [$0..40]);

With[{L = LucasL}, Table[Mod[L[n]*L[n + 1], L[n + 2]], {n, 0, 50}]] (* Amiram Eldar, Jan 24 2022 *)

L(n) = fibonacci(n+1)+fibonacci(n-1);
a(n) = L(n)*L(n+1) % L(n+2); \\ Michel Marcus, Jan 24 2022

G.f.: 4*x - 3 - (x + 3)/(2*(x^2 + x - 1)) - (x - 3)/(2*(x^2 - x - 1)) + 5/(x + 1).
a(n) = -a(n-1) + 3*a(n-2) + 3*a(n-3) - a(n-4) - a(n-5) for n >= 7.
a(n) = 5 for even n >= 2.
a(n) = A000032(n+2)-5 for odd n >= 3.

A350902 a(n) = (5F(n)F(n-1)F(2n-1)a(n-1) + F(n-1)L(n)a(n-2))/(L(n-1)F(n)), with a(0) = 1, a(1) = 0, where F(n) = A000045(n) and L(n) = A000032(n).

Original entry on oeis.org

1, 0, 3, 25, 816, 59475, 12031005, 6229446000, 8517168411895, 30387269735449725, 284188952072106783648, 6954889250543118311091775, 445684855849546942072130113089, 74767094861864103592878982016253600, 32838249084789127737424410920015676309123

Offset: 0

Amiram Eldar, Jan 21 2022

Although the recurrence relation involves fractions, all the terms are integers.
The sequence of fractions b(n) = A350903(n)/A350904(n) is defined by the same recurrence relation, but with the initial terms 0 and 1 instead of 1 and 0.
André-Jeannin (1991) used this sequence and the sequence b(n) to prove that s = Sum_{n>=1} 1/F(n) (A079586) is an irrational number.
The sequence of ratios r(n) = b(n)/a(n) rapidly converges to s. For example, abs(r(16)-s) < 10^(-100) and abs(r(49)-s) < 10^(-1000).

Amiram Eldar, Table of n, a(n) for n = 0..69
Richard André-Jeannin, Irrationalité de la somme des inverses de certaines suites récurrentes, Comptes Rendus de l'Académie des Sciences - Series I - Mathematics 308:19 (1989), pp. 539-541.
Richard André-Jeannin, Sequences of Integers Satisfying Recurrence Relations, The Fibonacci Quarterly, Vol. 29, No. 3 (1991), pp. 205-208.

Cf. A000032, A000045, A079586, A350903, A350904.

With[{F = Fibonacci, L = LucasL}, a[0] = 1; a[1] = 0; a[n_] := a[n] = (5*F[n]*F[n - 1]*F[2*n - 1]*a[n - 1] + F[n - 1]*L[n]*a[n - 2])/(L[n - 1]*F[n]); Array[a, 15, 0]]

Limit_{n->oo} A350903(n)/(A350904(n)*a(n)) = A079586 (André-Jeannin, 1991).

A350903 Numerators of the sequence of fractions defined by u(n) = ((5F(n)F(n-1)F(2n-1)u(n-1) + F(n-1)L(n)u(n-2))/(L(n-1)F(n))), with u(0) = 0 and u(1) = 1, where F(n) = A000045(n) and L(n) = A000032(n).

Original entry on oeis.org

0, 1, 10, 84, 8225, 999146, 161691205, 4081394133187, 801267937794945, 451272063930179690869, 955797228958312695758495, 12869303093903467063139191673469, 141131682569461636438244407470674215, 5214528077594695050414454970728001934806021

Offset: 0

Amiram Eldar, Jan 21 2022

See A350902 for details.

			The sequence of fractions begins with 0, 1, 10, 84, 8225/3, 999146/5, 161691205/4, 4081394133187/195, 801267937794945/28, 451272063930179690869/4420, ...

Amiram Eldar, Table of n, a(n) for n = 0..60
Richard André-Jeannin, Sequences of Integers Satisfying Recurrence Relations, The Fibonacci Quarterly, Vol. 29, No. 3 (1991), pp. 205-208.

Cf. A000032, A000045, A079586, A350902, A350904 (denominators).

With[{F = Fibonacci, L = LucasL}, u[0] = 0; u[1] = 1; u[n_] := u[n] = (5*F[n]*F[n - 1]*F[2*n - 1]*u[n - 1] + F[n - 1]*L[n]*u[n - 2])/(L[n - 1]*F[n]); Numerator @ Array[u, 15, 0]]

A350904 Denominators of the sequence of fractions defined by u(n) = ((5F(n)F(n-1)F(2n-1)u(n-1) + F(n-1)L(n)u(n-2))/(L(n-1)F(n))), with u(0) = 0 and u(1) = 1, where F(n) = A000045(n) and L(n) = A000032(n).

Original entry on oeis.org

1, 1, 1, 1, 3, 5, 4, 195, 28, 4420, 1001, 550732, 94248, 20757737, 150585864, 596098336680, 84878386593, 17090110926980520, 1216260982575912, 13296541287045886485, 484071647034823848, 3418959485072391296664264, 19630886922468003512297

Offset: 0

Amiram Eldar, Jan 21 2022

See A350902 for details.

Amiram Eldar, Table of n, a(n) for n = 0..60
Richard André-Jeannin, Sequences of Integers Satisfying Recurrence Relations, The Fibonacci Quarterly, Vol. 29, No. 3 (1991), pp. 205-208.

Cf. A000032, A000045, A079586, A350902, A350903 (numerators).

With[{F = Fibonacci, L = LucasL}, u[0] = 0; u[1] = 1; u[n_] := u[n] = (5*F[n]*F[n - 1]*F[2*n - 1]*u[n - 1] + F[n - 1]*L[n]*u[n - 2])/(L[n - 1]*F[n]); Denominator @ Array[u, 25, 0]]

A355018 Partial sums of F(1) - L(1) + F(2) - L(2) + F(3) - L(3) + ..., where F = A000045 and L = A000032.

Original entry on oeis.org

1, 0, 1, -2, 0, -4, -1, -8, -3, -14, -6, -24, -11, -40, -19, -66, -32, -108, -53, -176, -87, -286, -142, -464, -231, -752, -375, -1218, -608, -1972, -985, -3192, -1595, -5166, -2582, -8360, -4179, -13528, -6763, -21890, -10944, -35420, -17709, -57312, -28655

Offset: 0

Clark Kimberling, Jun 16 2022

The closely related partial sums of L(1) - F(1) + L(2) - F(2) + L(3) - F(3) + .... are given by A355019.

			a(0) = 1
a(1) = 1 - 1 = 0
a(2) = 1 - 1 + 1 = 1
a(3) = 1 - 1 + 1 - 3 = -2.

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

Cf. A000032, A000045, A355019, A355020, A355021.

F:=Fibonacci; [2 - (((n+1) mod 2)*F(Floor((n+2)/2)) + 2*(n mod 2)*F(Floor((n+3)/2))) : n in [0..60]]; // G. C. Greubel, Mar 17 2024

f[n_] := Fibonacci[n]; g[n_] := LucasL[n];
f1[n_] := If[OddQ[n], 2 - 2 f[(n + 3)/2], 2 - f[(n + 2)/2]]
f2 = Table[f1[n], {n, 0, 20}]  (* this sequence *)
g1[n_] := If[OddQ[n], -2 + 2 f[(n + 3)/2], -2 + f[(n + 8)/2]]
g2 = Table[g1[n], {n, 0, 20}]  (* A355019 *)
LinearRecurrence[{1,1,-1,1,-1}, {1,0,1,-2,0}, 61] (* G. C. Greubel, Mar 17 2024 *)

f=fibonacci; [2 - (((n+1)%2)*f(((n+2)//2)) +2*(n%2)*f((n+3)//2)) for n in range(61)] # G. C. Greubel, Mar 17 2024

a(n) = 2 - 2*F((n+3)/2) if n is odd, a(n) = 2 - F((n+2)/2) if n is even, where F = A000045 (Fibonacci numbers).
a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4) - a(n-5) for n >= 5.
G.f.: (1 - x - 2*x^3)/((1 - x)*(1 - x^2 - x^4)).
From G. C. Greubel, Mar 17 2024: (Start)
a(n) = (1/2)*Sum_{j=0..n} ( (1+(-1)^j)*Fibonacci(floor((j+3)/2)) - (1 - (-1)^j)*Lucas(floor((j+1)/2)) ).
a(n) = 2 - (1/2)*( (1+(-1)^n)*Fibonacci(floor((n+2)/2)) + 2*(1-(-1)^n)* Fibonacci(floor((n+3)/2)) ). (End)

A355019 Partial sums of L(1) - F(1) + L(2) - F(2) + L(3) - F(3) + ..., where L = A000032 and F = A000045.

Original entry on oeis.org

1, 0, 3, 2, 6, 4, 11, 8, 19, 14, 32, 24, 53, 40, 87, 66, 142, 108, 231, 176, 375, 286, 608, 464, 985, 752, 1595, 1218, 2582, 1972, 4179, 3192, 6763, 5166, 10944, 8360, 17709, 13528, 28655, 21890, 46366, 35420, 75023, 57312, 121391, 92734, 196416, 150048

Offset: 0

Clark Kimberling, Jun 16 2022

The closely related partial sums of F(1) - L(1) + F(2) - L(2) + F(3) - L(3) + ... are given by A355018.

			a(0) = 1
a(1) = 1 - 1 = 0
a(2) = 1 - 1 + 3 = 3
a(3) = 1 - 1 + 3  - 1 = 2.

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

Cf. A000032, A000045, A355018, A355020, A355021.

F:=Fibonacci; [(((n+1) mod 2)*F(Floor(n/2)+4) + 2*(n mod 2)*F(Floor((n+3)/2))) - 2: n in [0..60]]; // G. C. Greubel, Mar 17 2024

f[n_] := Fibonacci[n]; g[n_] := LucasL[n];
f1[n_] := If[OddQ[n], 2 - 2 f[(n + 3)/2], 2 - f[(n + 2)/2]]
f2 = Table[f1[n], {n, 0, 20}]  (* A355018 *)
g1[n_] := If[OddQ[n], -2 + 2 f[(n + 3)/2], -2 + f[(n + 8)/2]]
g2 = Table[g1[n], {n, 0, 20}]  (* this sequence *)
LinearRecurrence[{1,1,-1,1,-1}, {1,0,3,2,6}, 61] (* G. C. Greubel, Mar 17 2024 *)

f=fibonacci; [(((n+1)%2)*f((n//2)+4) +2*(n%2)*f((n+3)//2)) -2 for n in range(61)] # G. C. Greubel, Mar 17 2024

a(n) = -2 + 2 F((n+3)/2) if n is odd, a(n) = - 2 + F((n+8)/2) if n is even, where F = A000045 (Fibonacci numbers).
a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4) - a(n-5) for n >= 5.
G.f.: (1 - x + 2*x^2)/((1 - x)*(1 - x^2 - x^4)).
From G. C. Greubel, Mar 17 2024: (Start)
a(n) = (1/2)*Sum_{j=0..n} ( (1+(-1)^j)*Lucas(floor(j/2) +1) - (1-(-1)^j) *Fibonacci(floor((j+1)/2)) ).
a(n) = (1/2)*( (1+(-1)^n)*Fibonacci(floor(n/2) +4) + 2*(1-(-1)^n)* Fibonacci(floor((n+3)/2)) ) - 2. (End)

A380823 Semiperimeter of the unique primitive Pythagorean triple whose inradius is A000032(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

Original entry on oeis.org

15, 6, 28, 45, 120, 276, 703, 1770, 4560, 11781, 30628, 79800, 208335, 544446, 1423828, 3725085, 9748320, 25514796, 66787903, 174835650, 457697640, 1198222581, 3136914028, 8212428720, 21500225295, 56288009526, 147363418828, 385801624845, 1010040449160, 2644318093956, 6922911197503

Offset: 0

Miguel-Ángel Pérez García-Ortega, Feb 04 2025

			For n=2, the short leg is A380821(2,1) = 7, the long leg is A380821(2,2) = 24 and the hypotenuse is A380821(2,3) = 25 so the semiperimeter is then a(2) = (7 + 24 + 25)/2 = 28.

Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint 2025.

Miguel-Ángel Pérez García-Ortega, El Libro de las Ternas Pitagóricas.
Index entries for linear recurrences with constant coefficients, signature (4,-2,-6,4,2,-1).

Cf. A380821, A380824, A381721, A000032.

a=Table[LucasL[n],{n,0,30}];Apply[Join,Map[{(#+1)(2#+1)}&,a]]

a(n) = (A380821(n,1) + A380821(n,2) + A380821(n,3))/2.
a(n) = (Lucas(n) + 1)*(2*Lucas(n) + 1).
G.f.: (15 - 54*x + 34*x^2 + 35*x^3 - 28*x^4)/((1 - x)*(1 + x)*(1 - 3*x + x^2)*(1 - x - x^2)). - Stefano Spezia, Mar 08 2025

cp's OEIS Frontend

A338736 a(n) = L(L(n)) mod n, where L = Lucas numbers = A000032.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Formula

A338889 a(n) = L(L(L(n))) mod L(L(n)), where L = Lucas numbers = A000032.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Formula

A339517 Odd composite integers m such that A000032(2*m-J(m,5)) == J(m,5) (mod m), where J(m,5) is the Jacobi symbol.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

A347861 a(n) = A000032(n)*A000032(n+1) mod A000032(n+2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A350902 a(n) = (5*F(n)*F(n-1)*F(2*n-1)*a(n-1) + F(n-1)*L(n)*a(n-2))/(L(n-1)*F(n)), with a(0) = 1, a(1) = 0, where F(n) = A000045(n) and L(n) = A000032(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A350903 Numerators of the sequence of fractions defined by u(n) = ((5*F(n)*F(n-1)*F(2*n-1)*u(n-1) + F(n-1)*L(n)*u(n-2))/(L(n-1)*F(n))), with u(0) = 0 and u(1) = 1, where F(n) = A000045(n) and L(n) = A000032(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A350904 Denominators of the sequence of fractions defined by u(n) = ((5*F(n)*F(n-1)*F(2*n-1)*u(n-1) + F(n-1)*L(n)*u(n-2))/(L(n-1)*F(n))), with u(0) = 0 and u(1) = 1, where F(n) = A000045(n) and L(n) = A000032(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A355018 Partial sums of F(1) - L(1) + F(2) - L(2) + F(3) - L(3) + ..., where F = A000045 and L = A000032.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A350902 a(n) = (5F(n)F(n-1)F(2n-1)a(n-1) + F(n-1)L(n)a(n-2))/(L(n-1)F(n)), with a(0) = 1, a(1) = 0, where F(n) = A000045(n) and L(n) = A000032(n).

A350903 Numerators of the sequence of fractions defined by u(n) = ((5F(n)F(n-1)F(2n-1)u(n-1) + F(n-1)L(n)u(n-2))/(L(n-1)F(n))), with u(0) = 0 and u(1) = 1, where F(n) = A000045(n) and L(n) = A000032(n).

A350904 Denominators of the sequence of fractions defined by u(n) = ((5F(n)F(n-1)F(2n-1)u(n-1) + F(n-1)L(n)u(n-2))/(L(n-1)F(n))), with u(0) = 0 and u(1) = 1, where F(n) = A000045(n) and L(n) = A000032(n).