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 1-3 of 3 results.

A218490 Decimal expansion of Lucas factorial constant.

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Oct 30 2012

Keywords

Comments

The Lucas factorial constant is associated with the Lucas factorial A135407.

Examples

			1.35787840761210570138743970976060718557860586529567870449687825438407191103...

Links

Crossrefs

Cf. A062073, A135407, A070825, A003266, A000032, A000045, A186269, A001622.

Programs

  • Mathematica
    RealDigits[QPochhammer[-1, -1/GoldenRatio^2], 10, 105][[1]] (* slightly modified by Robert G. Wilson v, Dec 21 2017 *)
  • PARI
    prodinf(j=0, 1 + ((sqrt(5) - 3)/2)^j) \\ Iain Fox, Dec 21 2017

Formula

Equals exp( Sum_{k>=1} 1/(k*(((3-sqrt(5))/2)^k-(-1)^k)) ). - Vaclav Kotesovec, Jun 08 2013
Equals Product_{k=0..infinity} (1 + (-1)^k/phi^(2*k)). - G. C. Greubel, Dec 23 2017
Equals lim_{n->oo} A135407(n)/phi^(n*(n+1)/2), where phi is the golden ratio (A001622). - Amiram Eldar, Jan 23 2022

A186270 a(n)=Product{k=0..n, A003665(k)}.

Original entry on oeis.org

1, 1, 10, 280, 38080, 18887680, 39286374400, 319319651123200, 10504339243348787200, 1374135642457914946355200, 721146385161913763847208960000, 1511615130036671973985522422906880000, 12683442560532981918553467630898150113280000, 425533759542581882449393472981756918078982062080000
Offset: 0

Views

Author

Paul Barry, Feb 16 2011

Keywords

Comments

a(n) is the determinant of the symmetric matrix (if(j<=floor((i+j)/2), 2^j*J(j+1),
2^i*J(i+1)))_{0<=i,j<=n}, where J(n)=A001045(n).

Examples

			a(3)=280 since det[1, 1, 1, 1; 1, 2, 2, 2; 1, 2, 12, 12; 1, 2, 12, 40]=280.

Crossrefs

Cf. A186269.

Programs

  • Mathematica
    Table[Product[4^k/2+(-2)^k/2,{k,0,n}],{n,0,15}] (* Vaclav Kotesovec, Jul 11 2015 *)

Formula

a(n)=Product{k=0..n, 4^k/2+(-2)^k/2}=Product{k=0..n, sum{j=0..floor(k/2), binomial(n,2k)*9^k}}.
a(n) ~ c * 2^(n^2 - 1), where c = 2*QPochhammer(1/2, -1/2) = 1.1373978925308570119099534741488893085817049027787180586386880920367... . - Vaclav Kotesovec, Jul 11 2015, updated Mar 18 2024

A186271 a(n)=Product{k=0..n, A001333(k)}.

Original entry on oeis.org

1, 1, 3, 21, 357, 14637, 1449063, 346326057, 199830134889, 278363377900377, 936136039878967851, 7600488507777339982269, 148977175240943640992454669, 7049748909576694035403947391749, 805384464676770256686653161875581007
Offset: 0

Views

Author

Paul Barry, Feb 16 2011

Keywords

Comments

a(n) is the determinant of the symmetric matrix (if(j<=floor((i+j)/2), Pell(j+1),
Pell(i+1)))_{0<=i,j<=n}, where Pell(n)=A000129(n).

Examples

			a(3)=21 since det[1, 1, 1, 1; 1, 2, 2, 2; 1, 2, 5, 5; 1, 2, 5, 12]=21.

Crossrefs

Cf. A186269.

Programs

  • Mathematica
    Table[Product[Sum[Binomial[k,2*j]*2^j,{j,0,Floor[k/2]}],{k,0,n}],{n,0,15}] (* Vaclav Kotesovec, Jul 11 2015 *)
Table[FullSimplify[Product[((1+Sqrt[2])^k + (1-Sqrt[2])^k)/2, {k, 0, n}]], {n, 0, 15}] (* Vaclav Kotesovec, Jul 11 2015 *)

Formula

a(n)=Product{k=0..n, sum{j=0..floor(k/2), binomial(k,2j)*2^j}}.
a(n) ~ c * (1+sqrt(2))^(n*(n+1)/2) / 2^(n+1), where c = 1.6982679851338713863950411843311686297311132648098280324748781109134... . - Vaclav Kotesovec, Jul 11 2015
Showing 1-3 of 3 results.

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