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.

A120787 Denominators of partial sums of Catalan numbers scaled by powers of 1/20.

Original entry on oeis.org

1, 20, 200, 1600, 80000, 1600000, 16000000, 1280000000, 2560000000, 256000000000, 2560000000000, 102400000000000, 1024000000000000, 4096000000000000, 40960000000000000, 6553600000000000000
Offset: 0

Views

Author

Wolfdieter Lang, Jul 20 2006

Keywords

Comments

Numerators are given under A120786.

Formula

a(n)=denominator(r(n)), with the rationals r(n):=sum(C(k)/20^k,k=0..n) with C(k):=A000108(k) (Catalan numbers).

A121002 Numerators of partial sums of Catalan numbers scaled by powers of 1/5.

Original entry on oeis.org

1, 6, 32, 33, 839, 4237, 21317, 107014, 4292, 2687362, 13453606, 67326816, 336842092, 336990672, 1685488248, 8429380209, 42153972579, 210795791853, 210814897401, 5270725887663, 26354942262399
Offset: 0

Views

Author

Wolfdieter Lang, Aug 16 2006

Keywords

Comments

Denominators are given under A121003.
This is the first member (p=0) of the second p-family of partial sums of normalized scaled Catalan series CsnII(p):=sum(C(k)/((5^k)*F(2*p+1)^(2*k)),k=0..infinity) with limit F(2*p+1)*(L(2*p+2) - L(2*p+1)*phi) = F(2*p+1)*sqrt(5)/phi^(2*p+1), with C(n)=A000108(n) (Catalan), F(n)= A000045(n) (Fibonacci), L(n) = A000032(n) (Lucas) and phi:=(1+sqrt(5))/2 (golden section).
The partial sums of the above mentioned second p-family are rII(p;n):=sum(C(k)/((5^k)*F(2*p+1)^(2*k)),k=0..n), n>=0, for p=0,1,...
For more details on this p-family and the other three ones see the W. Lang link under A120996.
The limit lim_{n->infinity} r(n) = (3 - phi) = (2*phi-1)/phi = 1.38196601125010 (maple10, 15 digits). This is the square of the dimensionless pentagon side length.

Examples

			Rationals r(n): [1, 6/5, 32/25, 33/25, 839/625, 4237/3125,
21317/15625, 107014/78125, 4292/3125, 2687362/1953125,...].

Links

Crossrefs

Cf. A120786 (numerators, second member p=1).

Formula

a(n)=numerator(r(n)) with r(n) := rII(p=0,n) = sum(C(k)/5^k,k=0..n) and C(k):=A000108(k) (Catalan). The rationals r(n) are given in lowest terms.

A121004 Numerators of partial sums of Catalan numbers scaled by powers of 1/(5*5^2)=1/125.

Original entry on oeis.org

1, 126, 15752, 393801, 246125639, 30765704917, 3845713114757, 480714139345054, 12017853483626636, 7511158427266652362, 938894803408331562046, 117361850426041445314536, 14670231303255180664525012
Offset: 0

Views

Author

Wolfdieter Lang, Aug 16 2006

Keywords

Comments

Denominators are given under A121005.
This is the third member (p=2) of the second p-family of partial sums of normalized scaled Catalan series CsnII(p):=sum(C(k)/((5^k)*F(2*p+1)^(2*k)),k=0..infinity) with limit F(2*p+1)*(L(2*p+2) - L(2*p+1)*phi) = F(2*p+1)*sqrt(5)/phi^(2*p+1), with C(n)=A000108(n) (Catalan), F(n)= A000045(n) (Fibonacci), L(n) = A000032(n) (Lucas) and phi:=(1+sqrt(5))/2 (golden section).
The partial sums of the above mentioned second p-family are rII(p;n):=sum(C(k)/((5^k)*F(2*p+1)^(2*k)),k=0..n), n>=0, for p=0,1,...
For more details on this p-family and the other three ones see the W. Lang link under A120996.

Examples

			Rationals r(n): [1, 126/125, 15752/15625, 393801/390625,
246125639/244140625, 30765704917/30517578125,...].

Links

Crossrefs

The second member (p=2) is A120786/A120787.

Programs

  • Maple
    The value of the series is lim_{n->infinity}(r(n) := rII(2;n)) = 5*(18 - 11*phi) = 5*sqrt(5)/phi^5 = 1.0081306187560 (maple10, 15 digits).

Formula

a(n)=numerator(r(n)) with r(n) := rII(p=2,n) = sum(C(k)/5^(3*k),k=0..n) and C(k):=A000108(k) (Catalan). The rationals r(n) are given in lowest terms.
Showing 1-3 of 3 results.

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