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.

Previous Showing 11-18 of 18 results.

A121000 Numerators of partial sums of Catalan numbers scaled by powers of 1/18^2 = 1/324.

Original entry on oeis.org

1, 325, 52651, 34117853, 5527092193, 596925956851, 96702005009873, 125325798492795551, 60908338067498638501, 19734301533869558876755, 3196956848486868538038509, 2071628037819490812648983225
Offset: 0

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Author

Wolfdieter Lang, Aug 16 2006

Keywords

Comments

Denominators are given under A121001.
This is the fourth member (p=3) of the first p-family of partial sums of normalized scaled Catalan series CsnI(p):=sum(C(k)/L(2*p)^(2*k),k=0..infinity) with limit L(2*p)*(F(2*p+1) - F(2*p)*phi) = L(2*p)/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 first p-family are rI(p;n):=sum(C(k)/L(2*p)^(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) = 18*(13 - 8* phi) = 18/phi^6 = 1.003105620014 (maple10, 15 digits).

Examples

			Rationals r(n): [1, 325/324, 52651/52488, 34117853/34012224,
5527092193/5509980288, 596925956851/595077871104, ...].

Links

Formula

a(n)=numerator(r(n)) with r(n) := rI(p=3,n) = sum(C(k)/L(6)^(2*k),k=0..n), with Lucas L(6)=18 and C(k):=A000108(k) (Catalan). The rationals r(n) are given in lowest terms.

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.

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.

A120997 Denominators of partial sums of Catalan numbers scaled by powers of 1/9.

Original entry on oeis.org

1, 9, 81, 729, 6561, 19683, 177147, 1594323, 43046721, 387420489, 3486784401, 31381059609, 282429536481, 2541865828329, 2541865828329, 22876792454961, 205891132094649, 5559060566555523, 50031545098999707
Offset: 0

Views

Author

Wolfdieter Lang, Aug 16 2006

Keywords

Comments

Numerators are given under A120996.
See A120996 for details and a W. Lang link found there for the definition of four p-families of such scaled Catalan sums.

Formula

a(n)=denominator(r(n)) with r(n) := rI(p=1,n) = sum(C(k)/L(2)^(2*k),k=0..n), with Lucas L(2)=3 and C(k):=A000108(k) (Catalan).

A121003 Denominators of partial sums of Catalan numbers scaled by powers of 1/5.

Original entry on oeis.org

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

Views

Author

Wolfdieter Lang, Aug 16 2006

Keywords

Comments

Numerators are given under A121002.
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).
For more details on this p-family and the other three ones see the W. Lang links under A120996 and A121002.

Examples

			Rationals r(n): [1, 6/5, 32/25, 33/25, 839/625, 4237/3125,
21317/15625, 107014/78125, 4292/3125, 2687362/1953125,...].
A120787 (denominators, second member p=1).

Formula

a(n)=denominator(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.

A121005 Denominators of partial alternating sums of Catalan numbers scaled by powers of 1/125.

Original entry on oeis.org

1, 125, 15625, 390625, 244140625, 30517578125, 3814697265625, 476837158203125, 11920928955078125, 7450580596923828125, 931322574615478515625, 116415321826934814453125, 14551915228366851806640625
Offset: 0

Views

Author

Wolfdieter Lang, Aug 16 2006

Keywords

Comments

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)*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 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.
Numerators are given under A121004.

Examples

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

Formula

a(n)=denominator(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.

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.
Previous Showing 11-18 of 18 results.

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