cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A276987 Decimal expansion of (phi-1)_inf = (1/phi)_inf, where (q)_inf is the q-Pochhammer symbol, phi = (1+sqrt(5))/2 is the golden ratio.

Original entry on oeis.org

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

Views

Author

Vladimir Reshetnikov, Sep 24 2016

Keywords

Comments

(1/phi)_inf appears as a coefficient in asymptotics of A274983, A274985.

Examples

			0.1208019218617061294237231569887920563...

Links

Crossrefs

Cf. A274983, A274985, A062073, A227681.

Programs

  • Mathematica
    RealDigits[QPochhammer[1/GoldenRatio], 10, 100][[1]]

Formula

(1/phi)inf = Product{k > 0} (1 - 1/phi^k).

A274985 a(n) = ([n]phi! - [n]{1-phi}!)/sqrt(5), where [n]_q! is the q-factorial, phi = (1+sqrt(5))/2.

Original entry on oeis.org

0, 0, 1, 6, 58, 948, 25992, 1179016, 87713040, 10646068080, 2101395344400, 673242645670320, 349671381118477440, 294206779308703578240, 400822226102433353285760, 883965927408694948620295680, 3155212287401150653204012531200
Offset: 0

Views

Author

Vladimir Reshetnikov, Sep 23 2016

Keywords

Examples

			For n = 3, [3]_phi! = 1060 + 474*sqrt(5), so A274983(5) = 2*1060 = 2120 and a(5) = 2*474 = 948.

Links

Crossrefs

Cf. A274983, A005329, A275706, A276474, A276688, A276987.

Programs

  • Mathematica
    Round@Table[(QFactorial[n, GoldenRatio] - QFactorial[n, 1 - GoldenRatio])/Sqrt[5], {n, 0, 20}] (* Round is equivalent to FullSimplify here, but is much faster *)

Formula

[n]_phi! = (A274983(n) + a(n)*sqrt(5))/2.
[n]_{1-phi}! = (A274983(n) - a(n)*sqrt(5))/2.
a(n) ~ c * phi^(n*(n+3)/2) / sqrt(5), where c = QPochhammer(phi-1) = A276987 = 0.1208019218617061294237231569887920563043992516794... . - Vaclav Kotesovec, Sep 24 2016

A276990 a(n) = (phi; phi)_n + (1-phi; 1-phi)_n, where (q; q)_n is the q-Pochhammer symbol, phi = (1+sqrt(5))/2 is the golden ratio.

Original entry on oeis.org

2, 1, 2, -2, 20, -190, 3240, -90800, 4174920, -313173840, 38204662320, -7564715117520, 2428250059593600, -1262694691720176000, 1063187432567808662400, -1449125250052431355430400, 3196769645011428154428883200, -11412468527893653264760022630400
Offset: 0

Views

Author

Vladimir Reshetnikov, Sep 24 2016

Keywords

Links

Crossrefs

Cf. A274983, A276474, A276987, A276991.

Programs

  • Mathematica
    Round@Table[QPochhammer[GoldenRatio, GoldenRatio, n] + QPochhammer[1 - GoldenRatio, 1 - GoldenRatio, n], {n, 0, 20}] (* Round is equivalent to FullSimplify here, but is much faster *)

Formula

(phi; phi)_n = (a(n) + A276991(n)*sqrt(5))/2.
(1-phi; 1-phi)_n = (a(n) - A276991(n)*sqrt(5))/2.
a(n) ~ c * (-1)^n * phi^(n*(n+1)/2), where c = (1/phi)_inf = A276987 = 0.1208019218617061294237231569887920563043992516794...

A276991 a(n) = ((phi; phi)_n - (1-phi; 1-phi)_n)/sqrt(5), where (q; q)_n is the q-Pochhammer symbol, phi = (1+sqrt(5))/2 is the golden ratio.

Original entry on oeis.org

0, -1, 0, -2, 8, -86, 1448, -40608, 1867080, -140055600, 17085644400, -3383043446640, 1085946439923840, -564694233102890880, 475471874409018791040, -648068513405723438730240, 1429638846930684965104992000, -5103811083889432701541321459200
Offset: 0

Views

Author

Vladimir Reshetnikov, Sep 24 2016

Keywords

Links

Crossrefs

Cf. A274983, A276474, A276987, A276990.

Programs

  • Mathematica
    Round@Table[(QPochhammer[GoldenRatio, GoldenRatio, n] - QPochhammer[1 - GoldenRatio, 1 - GoldenRatio, n])/Sqrt[5], {n, 0, 20}] (* Round is equivalent to FullSimplify here, but is much faster *)

Formula

(phi; phi)_n = (A276990(n) + a(n)*sqrt(5))/2.
(1-phi; 1-phi)_n = (A276990(n) - a(n)*sqrt(5))/2.
a(n) ~ c * (-1)^n * phi^(n*(n+1)/2) / sqrt(5), where c = (1/phi)_inf = A276987 = 0.1208019218617061294237231569887920563043992516794...
Showing 1-4 of 4 results.

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