A001588 -id:A001588 - cp's OEIS frontend

A382608 Long leg of the unique primitive Pythagorean triple whose inradius is A000045(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

4, 4, 12, 24, 60, 144, 364, 924, 2380, 6160, 16020, 41760, 109044, 285012, 745420, 1950312, 5104012, 13359280, 34969884, 91543980, 239651724, 627394464, 1642504612, 4300075584, 11257651300, 29472763684, 77160454284, 202008299064, 528863957340, 1384582787280

Offset: 1

Miguel-Ángel Pérez García-Ortega, Mar 31 2025

			Triangle begins:
  n=1:      3,    4,    5;
  n=2:      3,    4,    5;
  n=3:      5,   12,   13;
where this sequence is the middle column.

Miguel-Ángel Pérez García-Ortega, El Libro de las Ternas Pitagóricas

Cf. A000045 (inradius), A001588 (short leg), A382609 (semiperimeter), A382610 (area).

a=Table[Fibonacci[n],{n,0,16}];Apply[Join,Map[{2#+1,2#^2+2#,2#^2+2#+1}&,a]]

a(n) = 2*F(n)*(F(n) + 1) where F(n) = A000045(n).

A124067 Numbers k such that 2*F(k) + 1 is a prime, where F = A000045.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 11, 13, 19, 21, 22, 24, 27, 40, 44, 51, 62, 92, 139, 208, 224, 293, 307, 421, 509, 741, 861, 966, 989, 1046, 1100, 1102, 1109, 1182, 1547, 1766, 1813, 2048, 2720, 2726, 6012, 6790, 7132, 8301, 8699, 10062, 11102, 15827, 23918, 26747, 29389, 37229, 38211, 68726

Offset: 1

Giovanni Teofilatto, Dec 12 2006

nonn

Cf. A000045, A001588, A124081 (associated primes).

[n: n in [1..1200] | IsPrime(2*Fibonacci(n)+1)]; // Vincenzo Librandi, Aug 13 2013

Select[Range[10000], PrimeQ[2 Fibonacci[#] + 1]&] (* Vincenzo Librandi, Aug 13 2013 *)

isok(n) = isprime(2*fibonacci(n)+1) \\ Michel Marcus, Jun 03 2013

for(n=1,10^9,if(ispseudoprime(2*fibonacci(n)+1),print1(n,", "))); \\ Joerg Arndt, Aug 13 2013

Inserted a(1)=1 and extended by Michel Marcus, Jun 03 2013
More terms from Vincenzo Librandi, Aug 13 2013
a(48) from Jorge Coveiro, Sep 05 2022
a(49)-a(55) from Michael S. Branicky, Jun 17 2023

A256007 Numbers k satisfying |k + 1 - 2F| <= 1 for some positive Fibonacci number F.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 14, 15, 16, 24, 25, 26, 40, 41, 42, 66, 67, 68, 108, 109, 110, 176, 177, 178, 286, 287, 288, 464, 465, 466, 752, 753, 754, 1218, 1219, 1220, 1972, 1973, 1974, 3192, 3193, 3194, 5166, 5167, 5168, 8360, 8361, 8362, 13528, 13529

Offset: 0

Clark Kimberling, May 07 2015

For r > 0, define f(n) = floor(n*r) if n is odd and f(n) = floor(n/r) if n is even. Let S(r,n) be the set {n, f(n), f(f(n)), ...} of iterates of f starting with n. Conjecture: if r = (1 + sqrt(5))/2, then S(r,n) is bounded if and only if n is in this sequence.

			F(1) = F(2) contributes {0,1,2}; F(3) contributes {1,2,3}.

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A000045, A001588, A019274.

u = Table[Fibonacci[k], {k, 2, 30}]; Union[2 u - 2, 2 u - 1, 2 u]

Conjectures from Colin Barker, May 24 2015: (Start)
  a(n) = 2*a(n-3)-a(n-9) for n>12.
  G.f.: -x*(x^11+x^10+x^9+2*x^8+x^7-x^4-2*x^3-3*x^2-2*x-1) / ((x-1)*(x^2+x+1)*(x^6+x^3-1)).
(End)

A217762 Square array T, read by antidiagonals: T(n,k) = F(n) + 2*F(k) where F(n) is the n-th Fibonacci number.

Original entry on oeis.org

0, 2, 1, 2, 3, 1, 4, 3, 3, 2, 6, 5, 3, 4, 3, 10, 7, 5, 4, 5, 5, 16, 11, 7, 6, 5, 7, 8, 26, 17, 11, 8, 7, 7, 10, 13, 42, 27, 17, 12, 9, 9, 10, 15, 21, 68, 43, 27, 18, 13, 11, 12, 15, 23, 34, 110, 69, 43, 28, 19, 15, 14, 17, 23, 36, 55, 178, 111, 69, 44, 29, 21

Offset: 0

Philippe Deléham, Apr 07 2013

			Square array begins:
...0....2....2....4....6...10...16...26...42...
...1....3....3....5....7...11...17...27...43...
...1....3....3....5....7...11...17...27...43...
...2....4....4....6....8...12...18...28...44...
...3....5....5....7....9...13...19...29...45...
...5....7....7....9...11...15...21...31...47...
...8...10...10...12...14...18...24...34...50...
..13...15...15...17...19...23...29...39...55...
..21...23...23...25...27...31...37...47...63...
..34...36...36...38...40...44...50...60...76...
..55...57...57...59...61...65...71...81...97...
..89...91...91...93...95...99..105..115..131...
.144..146..146..148..150..154..160..170..186...
...

Cf. A000045

T(n,0) = A000045(n).
T(1,k) = A001588(k).
T(n,1) = T(n,2) = A157725(n).
T(n,3) = A157727(n).
T(n,n)= A022086(n) = 3*A000045(n).
T(n+1,n) = A000032(n+1) = A000204(n+1).
T(n+2,n) = A000285(n).
T(n+3,n) = A013655(n+1) = A001060(n+1).
T(n+4,n) = A021120(n).
T(n+5,n) = A022088(n+2) = 5*A000045(n+2).
T(n+6,n) = A022097(n+2).
T(n+7,n) = A022122(n+2).
T(n+8,n) = 3*A013655(n+2).
T(n+9,n) = A097657(n+2).
T(n+10,n) = A022118(n+4).
T(n,n+1) = A000045(n+3).
T(n,n+2) = A013655(n+1) = A001060(n+1).
T(n,n+3) = A000032(n+3).
T(n,n+4) = A022095(n+2).
T(n,n+5) = A022120(n+2).
T(n,n+6) = A022136(n+2).
T(n,n+7) = A022098(n+4).
T(n,n+8) = A022380(n+4).
T(n,n+9) = A206419(n+6).
Sum(T(n-k,k), 0<=k<=n) = 3*A000071(n+2).

cp's OEIS Frontend

A382608 Long leg of the unique primitive Pythagorean triple whose inradius is A000045(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

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A124067 Numbers k such that 2*F(k) + 1 is a prime, where F = A000045.

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A256007 Numbers k satisfying |k + 1 - 2F| <= 1 for some positive Fibonacci number F.

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A217762 Square array T, read by antidiagonals: T(n,k) = F(n) + 2*F(k) where F(n) is the n-th Fibonacci number.

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