A259623 -id:A259623 - cp's OEIS frontend

A259624 Strictly increasing list of F - 1, F, and F + 1, where F = A000045, the Fibonacci numbers.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 14, 20, 21, 22, 33, 34, 35, 54, 55, 56, 88, 89, 90, 143, 144, 145, 232, 233, 234, 376, 377, 378, 609, 610, 611, 986, 987, 988, 1596, 1597, 1598, 2583, 2584, 2585, 4180, 4181, 4182, 6764, 6765, 6766, 10945, 10946, 10947

Offset: 1

Clark Kimberling and Peter J. C. Moses, Jul 01 2015

Clark Kimberling, Table of n, a(n) for n = 1..2000
Index entries for linear recurrences with constant coefficients, signature (-1, -1, 1, 1, 1, 1, 1, 1).

Cf. A000045, A258085, A259623.

Union[# - 1, #, # + 1] &[Fibonacci[Range[50]]]
CoefficientList[Series[-((x (1+x) (1+x+x^2+x^4) (1+x+2 x^2+x^3+x^4+x^5))/((1+x+x^2) (-1+x^3+x^6))),{x,0,60}],x] (* or *) LinearRecurrence[{-1,-1,1,1,1,1,1,1},{0,1,2,3,4,5,6,7,8,9,12,13},60] (* Harvey P. Dale, Nov 21 2024 *)

G.f.: -((x (1 + x) (1 + x + x^2 + x^4) (1 + x + 2 x^2 + x^3 + x^4 + x^5))/((1 + x + x^2) (-1 + x^3 + x^6))).

A259626 List of numbers L and L + 1, where L = A000032, the Lucas numbers, sorted into increasing order and duplicates removed.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 11, 12, 18, 19, 29, 30, 47, 48, 76, 77, 123, 124, 199, 200, 322, 323, 521, 522, 843, 844, 1364, 1365, 2207, 2208, 3571, 3572, 5778, 5779, 9349, 9350, 15127, 15128, 24476, 24477, 39603, 39604, 64079, 64080, 103682, 103683, 167761, 167762

Offset: 1

Clark Kimberling and Peter J. C. Moses, Jul 01 2015

Clark Kimberling, Table of n, a(n) for n = 1..2000
Index entries for linear recurrences with constant coefficients, signature (0, 2, 0, 0, 0, -1).

Cf. A000032, A258085, A259623, A259624, A259625, A259627.

Mathematica
```
Union[#, # + 1] &[LucasL[Range[50]]]
```

G.f.: -((-1 - 2 x - x^2 + x^4 + x^5 + x^6 + x^7 + x^8)/((-1 + x) (1 + x) (-1 + x^2 + x^4)))

A259627 List of numbers L - 1, L, and L+1, where L = A000032, the Lucas numbers, sorted into increasing order and duplicates removed.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 17, 18, 19, 28, 29, 30, 46, 47, 48, 75, 76, 77, 122, 123, 124, 198, 199, 200, 321, 322, 323, 520, 521, 522, 842, 843, 844, 1363, 1364, 1365, 2206, 2207, 2208, 3570, 3571, 3572, 5777, 5778, 5779, 9348, 9349, 9350, 15126

Offset: 1

Clark Kimberling and Peter J. C. Moses, Jul 01 2015

Clark Kimberling, Table of n, a(n) for n = 1..2000
Index entries for linear recurrences with constant coefficients, signature (-1, -1, 1, 1, 1, 1, 1, 1).

Cf. A000032, A258085, A259623-A259626.

Union[# - 1, #, # + 1] &[LucasL[Range[50]]]

G.f.: -((x (1 + x) (1 + x^2) (1 + 2 x + 3 x^2 + 2 x^3 + 2 x^4 + 2 x^5 +

2 x^6))/((1 + x + x^2) (-1 + x^3 + x^6)))

A259625 List of numbers L - 1 and L, where L = A000032, the Lucas numbers, sorted into increasing order and duplicates removed.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 10, 11, 17, 18, 28, 29, 46, 47, 75, 76, 122, 123, 198, 199, 321, 322, 520, 521, 842, 843, 1363, 1364, 2206, 2207, 3570, 3571, 5777, 5778, 9348, 9349, 15126, 15127, 24475, 24476, 39602, 39603, 64078, 64079, 103681, 103682, 167760, 167761

Offset: 1

Clark Kimberling and Peter J. C. Moses, Jul 01 2015

Clark Kimberling, Table of n, a(n) for n = 1..2000
Index entries for linear recurrences with constant coefficients, signature (0, 2, 0, 0, 0, -1).

Cf. A000032, A258085, A259623, A259624, A259626, A259627.

Mathematica
```
Union[#, # - 1] &[LucasL[Range[50]]]
```

G.f.: -((x (-1 - 2 x - x^2 + x^5 + x^6 + x^7))/((-1 + x) (1 + x) (-1 + x^2 + x^4)))

A375274 Decimal expansion of the asymptotic density of the exponentially Fibonacci numbers (A115063).

Original entry on oeis.org

9, 4, 4, 3, 3, 5, 9, 0, 5, 0, 6, 4, 0, 6, 3, 3, 2, 4, 4, 8, 0, 5, 7, 3, 1, 3, 7, 7, 5, 6, 6, 6, 8, 8, 0, 5, 6, 1, 4, 6, 3, 4, 5, 8, 3, 2, 2, 2, 0, 2, 3, 5, 5, 5, 9, 2, 3, 6, 8, 3, 7, 7, 0, 4, 5, 5, 9, 3, 9, 5, 3, 8, 4, 6, 5, 4, 4, 6, 8, 5, 8, 7, 1, 9, 4, 1, 4, 2, 8, 0, 5, 2, 0, 3, 3, 7, 9, 2, 7, 4, 7, 9, 7, 2, 4

Offset: 0

Amiram Eldar, Aug 09 2024

This constant was apparently first calculated by Juan Arias-de-Reyna and Peter J. C. Moses in 2015 (see A115063).

			0.94433590506406332448057313775666880561463458322202...

Cf. A000045, A010056, A115063, A259623.

$MaxExtraPrecision = m = 500; em = 16; f[x_] := Log[(1 - x) * (1 + Sum[x^Fibonacci[e], {e, 2, em}])]; c = Rest[CoefficientList[Series[f[x], {x, 0, m}], x] * Range[0, m]]; RealDigits[Exp[NSum[Indexed[c, k]*PrimeZetaP[k]/k, {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 105][[1]]

c(imax) = prodeulerrat((1-1/p)*(1 + sum(i = 2, imax, 1/p^fibonacci(i))));
f(prec) = {default(realprecision, prec); my(k = 2, c1 = 0, c2 = c(k)); while(c1 != c2, k++; c1 = c2; c2 = c(k)); c1;}
f(120)

Equals Product_{p prime} (1 + Sum_{i>=2} (u(i) - u(i-1))/p^i), where u(i) = A010056(i) is the characteristic function of the Fibonacci numbers (A000045) (first formula at A115063).
Equals Product_{p prime} (1 + Sum_{i>=4} (-1)^(i+1)/p^A259623(i)).
Equals Product_{p prime} ((1 - 1/p) * (1 + Sum_{i>=2} 1/p^Fibonacci(i))).

cp's OEIS Frontend

A259624 Strictly increasing list of F - 1, F, and F + 1, where F = A000045, the Fibonacci numbers.

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A259626 List of numbers L and L + 1, where L = A000032, the Lucas numbers, sorted into increasing order and duplicates removed.

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A259627 List of numbers L - 1, L, and L+1, where L = A000032, the Lucas numbers, sorted into increasing order and duplicates removed.

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A259625 List of numbers L - 1 and L, where L = A000032, the Lucas numbers, sorted into increasing order and duplicates removed.

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A375274 Decimal expansion of the asymptotic density of the exponentially Fibonacci numbers (A115063).

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