A322159 -id:A322159 - cp's OEIS frontend

A296182 Decimal expansion of (2 + phi)/2, with the golden section phi from A001622.

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

Offset: 1

Wolfdieter Lang, Jan 08 2018

In a regular pentagon this is the distance between a vertex and the midpoint of the opposite side in units of the radius of the circumscribing circle.

			1.809016994374947424102293417182819058860154589902881431067724311352630231409451...

Index entries for algebraic numbers, degree 2.

Cf. A001622, A019863, A081003, A322159.

RealDigits[(5 + Sqrt[5])/4, 10, 111][[1]] (* Robert G. Wilson v, Jan 14 2018 *)
RealDigits[(2+GoldenRatio)/2,10,120][[1]] (* Harvey P. Dale, Aug 10 2025 *)

(5 + sqrt(5))/4 \\ Michel Marcus, Jan 08 2018

Equals (2 + phi)/2 = (5 + sqrt(5))/4 = (2*phi - 1)*phi/2 = with phi from A001622.
Equals 1 + A019863.
From Amiram Eldar, Nov 28 2024: (Start)
Equals 1/A322159.
Equals Product_{k>=0} (1 + 1/A081003(k)). (End)

A214293 a(n) = 1 if n is a square, -1 if n is five times a square.

Original entry on oeis.org

1, 0, 0, 1, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0

Offset: 1

Michael Somos, Jul 10 2012

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = q + q^4 - q^5 + q^9 + q^16 - q^20 + q^25 + q^36 - q^45 + q^49 + q^64 + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000
Shaun Cooper and Michael Hirschhorn, On some infinite product identities, Rocky Mountain J. Math., 31 (2001) 131-139. see p. 134 Theorem 4.
Michael Somos, Introduction to Ramanujan theta functions.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A214284, A214295, A214960, A244612, A245485, A322159.

Basis( ModularForms( Gamma1(20), 1/2), 65) [2]; /* Michael Somos, Jul 01 2014 */

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] - EllipticTheta[ 3, 0, q^5]) / 2,  {q, 0, n}];
a[ n_] := If[ n < 0, 0, Boole[ OddQ [ Length @ Divisors @ n]] - Boole[ OddQ [ Length @ Divisors [5 n]]]];

PARI
```
{a(n) = issquare(n) - issquare(5*n)};
```

{a(n) = if( n<1, 0, direuler( p=2, n, if( p==5, 1 - X, 1) / (1 - X^2 ))[n])};

Expansion of (phi(q) - phi(q^5)) / 2 in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Sep 24 2013
Expansion of q * f(q^3, q^7) * f(-q^4, -q^16) / f(-q^8, -q^12) in powers of q where f() is Ramanujan's two-variable theta function.
Expansion of q * f(x, x^9) * f(-q, -q^4) / f(-q^2, -q^3) in powers of q where f() is Ramanujan's two-variable theta function. - Michael Somos, Sep 24 2013
Euler transform of period 20 sequence [ 0, 0, 1, -1, 0, -1, 1, 1, 0, -1, 0, 1, 1, -1, 0, -1, 1, 0, 0, -1, ...].
Multiplicative with a(5^e) = (-1)^e, a(p^e) = 1 if e even, 0 otherwise.
G.f.: (theta_3(q) - theta_3(q^5)) / 2 = Sum_{k>0} x^(k^2) - x^(5*k^2).
Dirichlet g.f.: zeta(2*s) * (1 - 5^-s).
a(4*n + 2) = a(4*n + 3) = 0. a(4*n) = a(n). a(5*n) = -a(n).
a(4*n) = A214293(n). a(4*n+1) = A214960(n). - Michael Somos, Sep 24 2013
Sum_{k=1..n} a(k) ~ c*sqrt(n), where c = 1 - 1/sqrt(5) = 0.5527864... (A322159). - Amiram Eldar, Oct 24 2023

A081010 a(n) = Fibonacci(4n+1) + 2, or Fibonacci(2n-1)*Lucas(2n+2).

Original entry on oeis.org

3, 7, 36, 235, 1599, 10948, 75027, 514231, 3524580, 24157819, 165580143, 1134903172, 7778742051, 53316291175, 365435296164, 2504730781963, 17167680177567, 117669030460996, 806515533049395, 5527939700884759, 37889062373143908, 259695496911122587

Offset: 0

R. K. Guy, Mar 01 2003

Hugh C. Williams, Edouard Lucas and Primality Testing, John Wiley and Sons, 1998, p. 75.

Nathaniel Johnston, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (8,-8,1).

Cf. A000045 (Fibonacci numbers), A000032 (Lucas numbers), A001519, A001906, A322159.

List([0..30], n-> Fibonacci(4*n+1)+2); # G. C. Greubel, Jul 14 2019

[Fibonacci(4*n+1) +2: n in [0..30]]; // Vincenzo Librandi, Apr 15 2011

with(combinat) for n from 0 to 30 do printf(`%d,`,fibonacci(4*n+1)+2) od # James Sellers, Mar 03 2003

Fibonacci[4*Range[0,30]+1]+2 (* G. C. Greubel, Jul 14 2019 *)

vector(30, n, n--; fibonacci(4*n+1)+2) \\ G. C. Greubel, Jul 14 2019

[fibonacci(4*n+1)+2 for n in (0..30)] # G. C. Greubel, Jul 14 2019

a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3).
a(n) = 2 + (A001906(n+1)^2 + A001519(n)^2)/2. - Creighton Dement, Aug 15 2004
G.f.: (3-17*x+4*x^2)/((1-x)*(1-7*x+x^2)). - Colin Barker, Jun 24 2012
Product_{n>=0} (1 - 1/a(n)) = 1 - 1/sqrt(5) = A322159. - Amiram Eldar, Nov 28 2024

More terms from James Sellers, Mar 03 2003

A081013 a(n) = Fibonacci(4n+3) - 2, or Fibonacci(2n)Lucas(2n+3).

Original entry on oeis.org

0, 11, 87, 608, 4179, 28655, 196416, 1346267, 9227463, 63245984, 433494435, 2971215071, 20365011072, 139583862443, 956722026039, 6557470319840, 44945570212851, 308061521170127, 2111485077978048, 14472334024676219

Offset: 0

R. K. Guy, Mar 01 2003

Hugh C. Williams, Edouard Lucas and Primality Testing, John Wiley and Sons, 1998, p. 75.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-8,1).

Cf. A000045 (Fibonacci numbers), A000032 (Lucas numbers), A322159.

List([0..40], n-> Fibonacci(4*n+3) -2); # G. C. Greubel, Jul 14 2019

[Fibonacci(4*n+3)-2: n in [0..40]]; // Vincenzo Librandi, Apr 20 2011

[Fibonacci(4*n+3)-2: n in [0..40]]; // G. C. Greubel, Jul 14 2019

with(combinat) for n from 0 to 40 do printf(`%d,`,fibonacci(4*n+3)-2) od # James Sellers, Mar 03 2003

LinearRecurrence[{8,-8,1},{0,11,87},40] (* Harvey P. Dale, Dec 05 2013 *)
Table[Fibonacci[2n] LucasL[2n+3], {n,1,40}] (* Rigoberto Florez, Apr 20 2019 *)
Table[Sum[Binomial[2n-1+i, 2n-1-i], {i, 1, 2n-1}]-1, {n, 1, 40}] (* Rigoberto Florez, Apr 20 2019 *)

my(x='x+O('x^40)); concat([0], Vec(x*(11-x)/((1-x)*(1-7*x+x^2)))) \\ G. C. Greubel, Dec 24 2017

vector(40, n, n--; fibonacci(4*n+3)-2) \\ G. C. Greubel, Jul 14 2019

[fibonacci(4*n+3)-2 for n in (0..40)] # G. C. Greubel, Jul 14 2019

a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3).
From R. J. Mathar, Sep 03 2010: (Start)
G.f.: x*(11 - x)/((1-x)*(1-7*x+x^2)).
a(n) = A033891(n) - 2.
a(n+1) - a(n) = A056914(n+1), n>0. (End)
a(n) = 7*a(n-1) - a(n-2) + 10, n>=2. - R. J. Mathar, Nov 07 2015
From Rigoberto Florez, Apr 20 2019: (Start)
a(n) = Sum_{i=0..2n} F(i)*L(i+2), F(i) = A000045(i) and L(i) = A000032(i).
a(n) = (Sum_{i=1..2n-1} binomial(2n-1+i,2n-1-i)) - 1. (End)
Product_{n>=1} (1 + 1/a(n)) = 2*(1-1/sqrt(5)) = 2*A322159. - Amiram Eldar, Nov 28 2024

cp's OEIS Frontend

A296182 Decimal expansion of (2 + phi)/2, with the golden section phi from A001622.

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A214293 a(n) = 1 if n is a square, -1 if n is five times a square.

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A081010 a(n) = Fibonacci(4n+1) + 2, or Fibonacci(2n-1)*Lucas(2n+2).

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A081013 a(n) = Fibonacci(4*n+3) - 2, or Fibonacci(2*n)*Lucas(2*n+3).

Views

Author

Keywords

References

Links

Crossrefs

Programs

GAP

Magma

Magma

Maple

Mathematica

PARI

PARI

Sage

Formula

A081013 a(n) = Fibonacci(4n+3) - 2, or Fibonacci(2n)Lucas(2n+3).