A084057 -id:A084057 - cp's OEIS frontend

A269992 Decimal expansion of Sum_{n>=1} 2^(1-n)/L(n), where L = A000032 (Lucas numbers).

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

Offset: 1

Clark Kimberling, Mar 12 2016

			1.2552211343298486031402667274403360...

Cf. A000032, A084057, A087131, A269991.

x = N[Sum[2^(1 - n)/LucasL[n], {n, 1, 500}], 100]
RealDigits[x][[1]]

L(n) = real((2 + quadgen(5)) * quadgen(5)^n); \\ A000032
suminf(n=1, 2^(1-n)/L(n)) \\ Michel Marcus, Nov 17 2020

Equals Sum_{n>=1} 1/A084057(n) = 2 * Sum_{n>=1} 1/A087131(n). - Amiram Eldar, Feb 01 2021

A160444 Expansion of g.f.: x^2(1 + x - x^2)/(1 - 2x^2 - 2*x^4).

Original entry on oeis.org

0, 1, 1, 1, 2, 4, 6, 10, 16, 28, 44, 76, 120, 208, 328, 568, 896, 1552, 2448, 4240, 6688, 11584, 18272, 31648, 49920, 86464, 136384, 236224, 372608, 645376, 1017984, 1763200, 2781184, 4817152, 7598336, 13160704, 20759040, 35955712, 56714752

Offset: 1

Willibald Limbrunner (w.limbrunner(AT)gmx.de), May 14 2009

This sequence is the case k=3 of a family of sequences with recurrences a(2*n+1) = a(2*n) + a(2*n-1), a(2*n+2) = k*a(2*n-1) + a(2*n), a(1)=0, a(2)=1. Values of k, for k >= 0, are given by A057979 (k=0), A158780 (k=1), A002965 (k=2), this sequence (k=3). See "Family of sequences for k" link for other connected sequences.
It seems that the ratio of two successive numbers with even, or two successive numbers with odd, indices approaches sqrt(k) for these sequences as n-> infinity.
This algorithm can be found in a historical figure named "Villardsche Figur" of the 13th century. There you can see a geometrical interpretation.

G. C. Greubel, Table of n, a(n) for n = 1..1000
W. Beinert, Villardscher Teilungskanon, Lexikon der Typographie
W. Limbrunner, Das Quadrat, ein Wunder der Geometrie. (in German)
Willibald Limbrunner, Family of sequences for k
M-T. Zenner, Villard de Honnecourt and Euclidean Geoometry, Nexus Network Journal 4 (2002) 65-78.
Index entries for linear recurrences with constant coefficients, signature (0,2,0,2).

Cf. A002532, A002533, A002534, A002535, A002605, A002965, A003665.
Cf. A003683, A015518, A015519, A026150, A046717, A057979, A063727.
Cf. A083098, A083099, A083100, A084057, A158780.

I:=[0,1,1,1]; [n le 4 select I[n] else 2*(Self(n-2) +Self(n-4)): n in [1..40]]; // G. C. Greubel, Feb 18 2023

LinearRecurrence[{0,2,0,2}, {0,1,1,1}, 40] (* G. C. Greubel, Feb 18 2023 *)

@CachedFunction
def a(n): # a = A160444
    if (n<5): return ((n+1)//3)
    else: return 2*(a(n-2) + a(n-4))
[a(n) for n in range(1, 41)] # G. C. Greubel, Feb 18 2023

a(n) = 2*a(n-2) + 2*a(n-4).
a(2*n+1) = A002605(n).
a(2*n) = A026150(n-1).

Edited by R. J. Mathar, May 14 2009

A163305 Numerators of fractions in the approximation of the square root of 5 satisfying: a(n)= (a(n-1)+ c)/(a(n-1)+1); with c=5 and a(1)=0. Also product of the powers of two and five times the Fibonacci numbers.

Original entry on oeis.org

0, 5, 10, 40, 120, 400, 1280, 4160, 13440, 43520, 140800, 455680, 1474560, 4771840, 15441920, 49971200, 161710080, 523304960, 1693450240, 5480120320, 17734041600, 57388564480, 185713295360, 600980848640, 1944814878720

Offset: 1

Mark Dols, Jul 24 2009

For denominators see: A084057 (= product of Lucas numbers (excluding first number (2)), and powers of 2).

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,4).

Cf. A000032, A000079, A084057, A000045.

LinearRecurrence[{2,4},{0,5},30] (* Harvey P. Dale, Mar 01 2016 *)

a(n)=5*fibonacci(n-1)*2^(n-2) \\ Franklin T. Adams-Watters, Aug 06 2009

From Colin Barker, Jun 20 2012: (Start)
a(n) = 2*a(n-1) + 4*a(n-2).
G.f.: 5*x^2/(1-2*x-4*x^2). (End)
a(n) = 5*A063727(n-1). - R. J. Mathar, Mar 08 2021

More terms from Franklin T. Adams-Watters, Aug 06 2009

A162962 a(n) = 5*a(n-2) for n > 2; a(1) = 1, a(2) = 5.

Original entry on oeis.org

1, 5, 5, 25, 25, 125, 125, 625, 625, 3125, 3125, 15625, 15625, 78125, 78125, 390625, 390625, 1953125, 1953125, 9765625, 9765625, 48828125, 48828125, 244140625, 244140625, 1220703125, 1220703125, 6103515625, 6103515625, 30517578125

Offset: 1

Klaus Brockhaus, Jul 19 2009

Apparently a(n) = A074872(n+1), a(n) = A056451(n-1) for n > 1.

Binomial transform is A084057 without initial 1, second binomial transform is A048876, third binomial transform is A082762, fourth binomial transform is A162769, fifth binomial transform is A093145 without initial 0.

Index entries for linear recurrences with constant coefficients, signature (0,5).

Cf. A000351 (powers of 5), A074872 (powers of 5 repeated), A056451 (5^floor((n+1)/2)), A084057, A048876, A082762, A162769, A093145.

[ n le 2 select 4*n-3 else 5*Self(n-2): n in [1..30] ];

LinearRecurrence[{0,5},{1,5},30] (* Harvey P. Dale, Mar 18 2023 *)

a(n) = 5^((1/4)*(2*n-1+(-1)^n)).

G.f.: x*(1+5*x)/(1-5*x^2).

A247584 a(n) = 5a(n-1) - 10a(n-2) + 10a(n-3) - 5a(n-4) + 3*a(n-5) with a(0) = a(1) = a(2) = a(3) = a(4) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 13, 43, 113, 253, 509, 969, 1849, 3719, 8009, 18027, 40897, 91257, 198697, 423777, 894081, 1886011, 4007301, 8594411, 18560081, 40181493, 86872293, 187197193, 402060793, 861827743, 1846685729, 3960390059, 8504658049, 18283290609, 39325827729

Offset: 0

Alexander Samokrutov, Sep 20 2014

a(n)/a(n-1) tends to 2.1486... = 1 + 2^(1/5), the real root of the polynomial x^5 - 5*x^4 + 10*x^3 - 10*x^2 + 5*x - 3.
If x^5 = 2 and n >= 0, then there are unique integers a, b, c, d, g  such that (1 + x)^n = a + b*x + c*x^2 + d*x^3 + g*x^4. The coefficient a is a(n) (from A052102). - Alexander Samokrutov, Jul 11 2015
If x=a(n), y=a(n+1), z=a(n+2), s=a(n+3), t=a(n+4) then x, y, z, s, t satisfies Diophantine equation (see link). - Alexander Samokrutov, Jul 11 2015

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..48 from Alexander Samokrutov)
Alexander Samokrutov, Diophantine equation
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,3).

The following sequences belong to the same family: A000129, A001333, A002532, A002533, A002605, A015518, A015519, A026150, A046717, A052101, A052102, A052103, A063727, A083098, A083099, A083100, A084057, A093406, A247344.

Cf. A005531.

[n le 5 select 1 else 5*Self(n-1) -10*Self(n-2) +10*Self(n-3) -5*Self(n-4) +3*Self(n-5): n in [1..40]]; // Vincenzo Librandi, Jul 11 2015

m:=50; S:=series( (1-x)^4/(1 -5*x +10*x^2 -10*x^3 +5*x^4 -3*x^5), x, m+1):
seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Apr 15 2021

LinearRecurrence[{5,-10,10,-5,3}, {1,1,1,1,1}, 50] (* Vincenzo Librandi, Jul 11 2015 *)

makelist(sum(2^k*binomial(n,5*k), k, 0, floor(n/5)), n, 0, 50); /* Alexander Samokrutov, Jul 11 2015 */

Vec((1-x)^4/(1-5*x+10*x^2-10*x^3+5*x^4-3*x^5) + O(x^100)) \\ Colin Barker, Sep 22 2014

[sum(2^j*binomial(n, 5*j) for j in (0..n//5)) for n in (0..50)] # G. C. Greubel, Apr 15 2021

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + 3*a(n-5).
a(n) = Sum_{k=0...floor(n/5)} (2^k*binomial(n,5*k)). - Alexander Samokrutov, Jul 11 2015
G.f.: (1-x)^4/(1 -5*x +10*x^2 -10*x^3 +5*x^4 -3*x^5). - Colin Barker, Sep 22 2014

A292848 a(n) is the smallest prime of form (1/2)((1 + sqrt(2n))^k + (1 - sqrt(2*n))^k).

Original entry on oeis.org

3, 5, 7, 113, 11, 13, 43, 17, 19, 61, 23, 73, 79, 29, 31, 97, 103, 37, 1241463763, 41, 43, 664973, 47, 2593, 151, 53, 163, 14972833, 59, 61, 4217, 193, 67, 23801, 71, 73, 223, 229, 79, 241, 83, 7561, 61068909859, 89, 271, 277, 283, 97, 10193, 101, 103, 313

Offset: 1

XU Pingya, Sep 24 2017

nonn

When 2n + 1 = p is prime, a(n) = p.
From Robert Israel, Sep 26 2017: (Start)
a(n) is also the first prime in the sequence defined by the recursion x(k+2)=2*x(k+1)+(2*n-1)*x(k) with x(0)=x(1)=1.
a(307), if it exists, has more than 10000 digits.
It appears that x(n*k) is divisible by x(k) if n is odd.  Thus a(n) (if it exists) must be x(k) where k is either a power of 2 or a prime. (End)

			For k = {1, 2, 3, 4}, (1/2)((1 + sqrt(8))^k + (1 - sqrt(8))^k) = {1, 9, 25, 113}. 113 is prime, so a(4) = 113.

Robert Israel, Table of n, a(n) for n = 1..306

Cf. A001333, A026150, A046717, A084057, A002533, A083098, A083100, A003665, A002535, A133294, A090042, A125816, A133343, A133345, A120612, A133356, A125818.

f:= proc(n) local a,b,t;
  a:= 1; b:= 1;
  do
    t:= a; a:= 2*a + (2*n-1)*b;
    if isprime(a) then return a fi;
    b:= t;
  od
end proc:
map(f, [$1..100]); # Robert Israel, Sep 26 2017

f[n_, k_] := ((1 + Sqrt[n])^k + (1 - Sqrt[n])^k)/2;
Table[k = 1; While[! PrimeQ[Expand@f[2n, k]], k++]; Expand@f[2n, k], {n, 52}]

cp's OEIS Frontend

A269992 Decimal expansion of Sum_{n>=1} 2^(1-n)/L(n), where L = A000032 (Lucas numbers).

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A160444 Expansion of g.f.: x^2*(1 + x - x^2)/(1 - 2*x^2 - 2*x^4).

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A163305 Numerators of fractions in the approximation of the square root of 5 satisfying: a(n)= (a(n-1)+ c)/(a(n-1)+1); with c=5 and a(1)=0. Also product of the powers of two and five times the Fibonacci numbers.

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A162962 a(n) = 5*a(n-2) for n > 2; a(1) = 1, a(2) = 5.

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A247584 a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + 3*a(n-5) with a(0) = a(1) = a(2) = a(3) = a(4) = 1.

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A292848 a(n) is the smallest prime of form (1/2)*((1 + sqrt(2*n))^k + (1 - sqrt(2*n))^k).

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Author

Keywords

Comments

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Programs

Maple

Mathematica

A160444 Expansion of g.f.: x^2(1 + x - x^2)/(1 - 2x^2 - 2*x^4).

A247584 a(n) = 5a(n-1) - 10a(n-2) + 10a(n-3) - 5a(n-4) + 3*a(n-5) with a(0) = a(1) = a(2) = a(3) = a(4) = 1.

A292848 a(n) is the smallest prime of form (1/2)((1 + sqrt(2n))^k + (1 - sqrt(2*n))^k).