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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 1-10 of 18 results. Next

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

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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.

A121009 Denominators of partial alternating sums of Catalan numbers scaled by powers of 1/(5*3^2) = 1/45.

Original entry on oeis.org

1, 45, 2025, 18225, 4100625, 61509375, 2767921875, 124556484375, 672605015625, 756680642578125, 34050628916015625, 1532278301220703125, 68952523554931640625, 620572711994384765625, 3102863559971923828125
Offset: 0

Views

Author

Wolfdieter Lang, Aug 16 2006

Keywords

Comments

Numerators are given under A121008.
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 and A121008.

Examples

			Rationals r(n): [1, 44/45, 1982/2025, 17837/18225, 4013339/4100625,
60200071/61509375, 2709003239/2767921875,...].

Crossrefs

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

Formula

a(n)=denominator(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.

A121012 Numerators of partial alternating sums of Catalan numbers scaled by powers of 1/(11^2) = 1/121.

Original entry on oeis.org

1, 120, 14522, 1757157, 212616011, 25726537289, 282991910191, 34242021133072, 4143284557101842, 501337431409322440, 667280121205808184436, 80740894665902790257970, 9769648254574237621422382
Offset: 0

Views

Author

Wolfdieter Lang, Aug 16 2006

Keywords

Comments

Denominators are given under A121013.
This is the second member (p=2) of the fourth (normalized) p-family of partial sums of normalized scaled Catalan series CsnIV(p):=sum(((-1)^k)*C(k)/L(2*p+1)^(2*k),k=0..infinity) with limit L(2*p+1)*(-F(2*p+2) + F(2*p+1)*phi) = L(2*p+1)/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 fourth p-family are rIV(p;n):=sum(((-1)^k)*C(k)/L(2*p+1)^(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, 120/121, 14522/14641, 1757157/1771561,
212616011/214358881, 25726537289/25937424601,...].

Links

Crossrefs

The first member is A120794/A120785. The third member is A121498/A121499.

Programs

  • Maple
    The limit lim_{n->infinity} (r(n) := rIV(2;n)) = 11*(-8 + 5*phi) = 11/phi^5 = 0.9918693812443 (maple10, 10 digits).

Formula

a(n)=numerator(r(n)) with r(n) := rIV(p=2,n) = sum(((-1)^k)*C(k)/L(2*2+1)^(2*k), k=0..n), with L(5)=11 and C(k):=A000108(k) (Catalan). The rationals r(n) are given in lowest terms.

A121013 Denominators of partial alternating sums of Catalan numbers scaled by powers of 1/(11^2) = 1/121.

Original entry on oeis.org

1, 121, 14641, 1771561, 214358881, 25937424601, 285311670611, 34522712143931, 4177248169415651, 505447028499293771, 672749994932560009201, 81402749386839761113321, 9849732675807611094711841
Offset: 0

Views

Author

Wolfdieter Lang, Aug 16 2006

Keywords

Comments

This is the second member (p=2) of the fourth (normalized) p-family of partial sums of the normalized scaled Catalan series CsnIV(p):=sum(((-1)^k)*C(k)/L(2*p+1)^(2*k),k=0..infinity) with limit L(2*p+1)*(-F(2*p+2) + F(2*p+1)*phi) = L(2*p+1)/phi^(2*p+1), where 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 fourth p-family are rIV(p;n):=sum(((-1)^k)*C(k)/L(2*p+1)^(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 A121012.

Examples

			Rationals r(n): [1, 120/121, 14522/14641, 1757157/1771561, 212616011/214358881, 25726537289/25937424601,...].

Crossrefs

The first member is A120794/A120785. The third member is A121498/A121499.

Formula

a(n)=denominator(r(n)) with r(n) := rIV(p=2,n) = sum(((-1)^k)*C(k)/L(2*2+1)^(2*k), k=0..n), with L(5)=11 and C(k):=A000108(k) (Catalan). The rationals r(n) are given in lowest terms.

A120786 Numerators of partial sums of Catalan numbers scaled by powers of 1/20.

Original entry on oeis.org

1, 21, 211, 1689, 84457, 1689161, 16891643, 1351331869, 2702663881, 270266390531, 2702663909509, 108106556409753, 1081065564149533, 4324262256635277, 43242622566419631, 6918819610629079929
Offset: 0

Views

Author

Wolfdieter Lang, Jul 20 2006

Keywords

Comments

Denominators are given under A120787.
From the expansion of 2*sqrt(5)/5 = sqrt(1-1/5) = 1-(1/10)*Sum_{k>=0} C(k)/20^k one has r := lim_{n->oo} r(n) = 2*(5 - 2*sqrt(5)) = 2*(7 - 4*phi) = 1.055728090..., where phi := (1+sqrt(5))/2 (golden section) and the partial sums r(n) are defined below.
This is the second member (p=1) in the second p-family of partial sums of the normalized scaled Catalan series CsnII(p) := Sum_{k>=0} C(k)/((5^k)*F(2*p+1)^(2*k)) with limit F(2*p+1)*(L(2*p+2) - L(2*p+1)*phi), 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_{k=0..n} C(k)/((5^k)*F(2*p+1)^(2*k)), n>=0, for p=0,1,...
For more details about this p-family and the other three ones see the W. Lang link under A120996.

Examples

			Rationals r(n): [1, 21/20, 211/200, 1689/1600, 84457/80000, 1689161/1600000, 16891643/16000000, 1351331869/1280000000,...].

Links

Formula

a(n) = numerator(r(n)), with the rationals r(n) := Sum_{k=0..n} C(k)/20^k with C(k) := A000108(k) (Catalan numbers). Rationals r(n) are taken in lowest terms.

A120794 Numerators of partial sums of Catalan numbers scaled by powers of -1/16.

Original entry on oeis.org

1, 15, 121, 3867, 30943, 495067, 3960569, 253475987, 2027808611, 32444935345, 259559486959, 8305903553295, 66447228478363, 1063155655468083, 8505245244078969, 1088671391232413187
Offset: 0

Views

Author

Wolfdieter Lang, Jul 20 2006

Keywords

Comments

From the expansion of sqrt(1+1/4) = 1+(1/8)*Sum_{k>=0} C(k)/(-16)^k one has, with the partial sums r(n) are defined below, r := lim_{n->oo} r(n) = 4*(sqrt(5)-2) = 4*(2*phi-3) = 0.944271909...
Denominators coincide with the listed numbers of A120785 but may differ for higher n values.
This is the first member (p=1) of the fourth family of scaled Catalan sums with limits in Q(sqrt(5)). See the W. Lang link under A120996.

Examples

			Rationals r(n): [1, 15/16, 121/128, 3867/4096, 30943/32768, 495067/524288, 3960569/4194304,...].

Links

Crossrefs

The second member (p=2) of this p-family is A121012/A121013.

Formula

a(n)=numerator(r(n)), with the rationals r(n):=Sum_{k=0..n} ((-1)^k)*C(k)/16^k with C(k):=A000108(k) (Catalan numbers). Rationals r(n) are taken 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.

A121498 Numerators of partial alternating sums of Catalan numbers scaled by powers of 1/(29^2) = 1/841.

Original entry on oeis.org

1, 840, 706442, 594117717, 499653000011, 420208173009209, 353395073500744901, 297205256814126461312, 249949620980680353964822, 210207631244752177684410440, 176784617876836581432589196836
Offset: 0

Views

Author

Wolfdieter Lang, Aug 16 2006

Keywords

Comments

Denominators are given under A121499.
This is the third member (p=3) of the fourth (normalized) p-family of partial sums of normalized scaled Catalan series CsnIV(p):=sum(((-1)^k)*C(k)/L(2*p+1)^(2*k),k=0..infinity) with limit L(2*p+1)*(-F(2*p+2) + F(2*p+1)*phi) = L(2*p+1)/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 fourth p-family are rIV(p;n):=sum(((-1)^k)*C(k)/L(2*p+1)^(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, 840/841, 706442/707281, 594117717/594823321,
499653000011/500246412961, 420208173009209/420707233300201,...].

Links

Crossrefs

The second member (p=2) of this p-family is A121012/A121013.

Programs

  • Maple
    The limit lim_{n->infinity}(r(n) := rIV(2;n)) = 29*(-21 + 13*phi) = 29/phi^7 = 0.998813758709 (maple10, 10 digits).

Formula

a(n)=numerator(r(n)) with r(n) := rIV(p=3,n) = sum(((-1)^k)*C(k)/L(2*3+1)^(2*k),k=0..n), with L(7)=29 and C(k):=A000108(k) (Catalan). The rationals r(n) are given in lowest terms.

A120998 Numerators of partial sums of Catalan numbers scaled by powers of 1/7^2 = 1/49.

Original entry on oeis.org

1, 50, 2452, 120153, 841073, 41212583, 14135916101, 692659889378, 33940334580952, 1663076394471510, 81490743329120786, 570435203303853900, 27951324961888870816, 9587304461927883432788, 469777918634466290881052
Offset: 0

Views

Author

Wolfdieter Lang, Aug 16 2006

Keywords

Comments

Denominators are given under A120999.
This is the third member (p=2) 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) = 7*(5 - 3* phi) = 7/phi^4 = 1.0212862362522 (maple10, 15 digits).

Examples

			Rationals r(n): [1, 50/49, 2452/2401, 120153/117649, 841073/823543,
41212583/40353607, 14135916101/13841287201,...].

Links

Formula

a(n)=numerator(r(n)) with r(n) := rI(p=2,n) = sum(C(k)/L(4)^(2*k),k=0..n), with Lucas L(4)=7 and C(k):=A000108(k) (Catalan). The rationals r(n) are given in lowest terms.

A121499 Denominators of partial alternating sums of Catalan numbers scaled by powers of 1/(29^2) = 1/841.

Original entry on oeis.org

1, 841, 707281, 594823321, 500246412961, 420707233300201, 353814783205469041, 297558232675799463481, 250246473680347348787521, 210457284365172120330305161, 176994576151109753197786640401
Offset: 0

Views

Author

Wolfdieter Lang, Aug 16 2006

Keywords

Comments

Numerators are given under A121498.
This is the third member (p=3) of the fourth (normalized) p-family of partial sums of normalized scaled Catalan series CsnIV(p):=sum(((-1)^k)*C(k)/L(2*p+1)^(2*k),k=0..infinity) with limit L(2*p+1)*(-F(2*p+2) + F(2*p+1)*phi) = L(2*p+1)/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 fourth p-family are rIV(p;n):=sum(((-1)^k)*C(k)/L(2*p+1)^(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 A121498.

Examples

			Rationals r(n): [1, 840/841, 706442/707281, 594117717/594823321,
499653000011/500246412961, 420208173009209/420707233300201,...].

Links

Crossrefs

The second member (p=2) of this p-family is A121012/A121013.

Formula

a(n)=denominator(r(n)) with r(n) := rIV(p=3,n) = sum(((-1)^k)*C(k)/L(2*3+1)^(2*k),k=0..n), with L(7)=29 and C(k):=A000108(k) (Catalan). The rationals r(n) are given in lowest terms.
Showing 1-10 of 18 results. Next

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