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 1-5 of 5 results.

A121008 Numerators of partial alternating sums of Catalan numbers scaled by powers of 1/(5*3^2) = 1/45.

Original entry on oeis.org

1, 44, 1982, 17837, 4013339, 60200071, 2709003239, 121905145612, 658287786362, 740573759652388, 33325819184374256, 1499661863296782734, 67484783848355431042, 607363054635198730798, 3036815273175993713422
Offset: 0

Views

Author

Wolfdieter Lang, Aug 16 2006

Keywords

Comments

Denominators are given under A121009.
This is the second member (p=2) of the third p-family of partial sums of normalized scaled Catalan series CsnIII(p):=sum(((-1)^k)*C(k)/((5^k)*F(2*p)^(2*k)),k=0..infinity) with limit F(2*p)*(-L(2*p+1) + L(2*p)*phi) = F(2*p)*sqrt(5)/phi^(2*p), 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 third p-family are rIII(p;n):=sum(((-1)^k)*C(k)/((5^k)*F(2*p)^(2*k)),k=0..n), n>=0, for p=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, 44/45, 1982/2025, 17837/18225, 4013339/4100625,
60200071/61509375, 2709003239/2767921875,...].

Links

Crossrefs

The first member (p=1) is A121006/A121007.

Programs

  • Maple
    The limit lim_{n->infinity}(r(n) := rIII(2;n)) = 3*(-11 + 7*phi) = 3*sqrt(5)/phi^4 = 0.9787137637479 (maple10, 15 digits).

Formula

a(n)=numerator(r(n)) with r(n) := rIII(p=2,n) = sum(((-1)^k)*C(k)/((5^k)*F(2*2)^(2*k)),k=0..n), with F(4)=3 and C(k):=A000108(k) (Catalan). The rationals r(n) are given in lowest terms.

A121006 Numerators of partial alternating sums of Catalan numbers scaled by powers of 1/5.

Original entry on oeis.org

1, 4, 22, 21, 539, 2653, 13397, 66556, 66842, 1666188, 8347736, 41679894, 208607482, 208458902, 1042829398, 5212208021, 26068111639, 130314629237, 26066746957, 3257989916987, 16291262409019
Offset: 0

Views

Author

Wolfdieter Lang, Aug 16 2006

Keywords

Comments

Denominators are given under A121007.
This is the first member (p=1) of the third p-family of partial sums of normalized scaled Catalan series CsnIII(p):=sum(((-1)^k)*C(k)/((5^k)*F(2*p)^(2*k)),k=0..infinity) with limit F(2*p)*(-L(2*p+1) + L(2*p)*phi)= F(2*p)*sqrt(5)/phi^(2*p), 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 third p-family are rIII(p;n):=sum(((-1)^k)*C(k)/((5^k)*F(2*p)^(2*k)),k=0..n), n>=0, for p=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) := rIII(1;n)) = -4 + 3*phi = sqrt(5)/phi^2 = 0.85410196624968...

Examples

			Rationals r(n): 1, 4/5, 22/25, 21/25, 539/625, 2653/3125,
13397/15625, 66556/78125, 66842/78125, 1666188/1953125, 8347736/9765625,...

Links

Crossrefs

The second member (p=2) is A121008/A121009.

Formula

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

A121007 Denominators of partial alternating sums of Catalan numbers scaled by powers of 1/5.

Original entry on oeis.org

1, 5, 25, 25, 625, 3125, 15625, 78125, 78125, 1953125, 9765625, 48828125, 244140625, 244140625, 1220703125, 6103515625, 30517578125, 152587890625, 30517578125, 3814697265625, 19073486328125
Offset: 0

Views

Author

Wolfdieter Lang, Aug 16 2006

Keywords

Comments

Numerators are given under A121006. See A121006 for further comments.

Examples

			Rationals r(n): 1, 4/5, 22/25, 21/25, 539/625, 2653/3125, 13397/15625, 66556/78125, 66842/78125, 1666188/1953125, 8347736/9765625,...

Crossrefs

The second member (p=2) is A121008/A121009.

Formula

a(n)=denominator(r(n)) with r(n) := rIII(p=1,n) = sum(((-1)^k)*C(k)/5^k,k=0..n) and C(k):=A000108(k) (Catalan). The rationals r(n) are given in lowest terms.

A121010 Numerators of partial alternating sums of Catalan numbers scaled by powers of 1/(5*8^2) = 1/320.

Original entry on oeis.org

1, 319, 51041, 6533247, 5226597607, 1672511234219, 267601797475073, 342530300768093011, 2192193924915795299, 17537551399326362389569, 2806008223892217982335239, 1795845263291019508694523567
Offset: 0

Views

Author

Wolfdieter Lang, Aug 16 2006

Keywords

Comments

Denominators are given under A121011.
This is the third member (p=3) of the third p-family of partial sums of normalized scaled Catalan series CsnIII(p):=sum(((-1)^k)*C(k)/((5^k)*F(2*p)^(2*k)),k=0..infinity) with limit F(2*p)*(-L(2*p+1) + L(2*p)*phi) = F(2*p)*sqrt(5)/phi^(2*p), 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 third p-family are rIII(p;n):=sum(((-1)^k)*C(k)/((5^k)*F(2*p)^(2*k)),k=0..n), n>=0, for p=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, 319/320, 51041/51200, 6533247/6553600,
5226597607/5242880000, 1672511234219/1677721600000,...].

Links

Crossrefs

The second member is A121008/A121009.

Programs

  • Maple
    The limit lim_{n->infinity} (r(n) := rIII(3;n)) = 8*(-29 + 18*phi) = 8*sqrt(5)/phi^6 = 0.9968943824 (maple10, 10 digits).

Formula

a(n)=numerator(r(n)) with r(n) := rIII(p=3,n) = sum(((-1)^k)*C(k)/((5^k)*F(2*3)^(2*k)),k=0..n), with F(6)=8 and C(k):=A000108(k) (Catalan). The rationals r(n) are given in lowest terms.

A121011 Denominators of partial alternating sums of Catalan numbers scaled by powers of 1/(5*8^2) = 1/320.

Original entry on oeis.org

1, 320, 51200, 6553600, 5242880000, 1677721600000, 268435456000000, 343597383680000000, 2199023255552000000, 17592186044416000000000, 2814749767106560000000000, 1801439850948198400000000000
Offset: 0

Views

Author

Wolfdieter Lang, Aug 16 2006

Keywords

Comments

Numerators are given under A121010.
This is the third member (p=3) of the third p-family of partial sums of normalized scaled Catalan series CsnIII(p):=sum(((-1)^k)*C(k)/((5^k)*F(2*p)^(2*k)),k=0..infinity) with limit F(2*p)*(-L(2*p+1) + L(2*p)*phi) = F(2*p)*sqrt(5)/phi^(2*p), 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 third p-family are rIII(p;n):=sum(((-1)^k)*C(k)/((5^k)*F(2*p)^(2*k)),k=0..n), n>=0, for p=1,...
For more details on this p-family and the other three ones see the W. Lang links under A120996 and A121010.

Examples

			Rationals r(n): [1, 319/320, 51041/51200, 6533247/6553600,
5226597607/5242880000, 1672511234219/1677721600000,...].

Crossrefs

The second member is A121008/A121009.

Formula

a(n)=denominator(r(n)) with r(n) := rIII(p=3,n) = sum(((-1)^k)*C(k)/((5^k)*F(2*3)^(2*k)),k=0..n), with F(6)=8 and C(k):=A000108(k) (Catalan). The rationals r(n) are given in lowest terms.
Showing 1-5 of 5 results.

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