A120785 -id:A120785 - cp's OEIS frontend

A121503 Numerators of partial sums of a series for sqrt(2) + sqrt(3) involving Catalan numbers.

13, 203, 1615, 51595, 412529, 6599099, 52788535, 3378355987, 27026481101, 432421205841, 3459361042977, 110699432952143, 885595037556565, 14169517557800915, 113356129507566775, 14509583941597490435

Offset: 0

Wolfdieter Lang, Aug 16 2006

The corresponding denominators are 4*A120785(n).
Sqrt(2)+sqrt(3) = (4*sin(Pi/4) + 6*tan(Pi/6))/2 = 3.146264370 (maple10, 10 digits). This is the arithmetic mean of the areas of an 8-gon (octagon), resp. 6-gon (hexagon) inscribed, resp. circumscribed in a unit circle.
Popper (see the reference) argues that Plato knew about the sum of sqrt(2)+sqrt(3). This sum approximates Pi with a relative error of 0.15%. The two right triangles, one with side lengths (1,1/2,sqrt(3)/2) and the other with side lengths (sqrt(2),1,1) are used in Plato's Timaios [53d] to build four of the five regular polyhedra (Platonic solids).
The Taylor series for sqrt(2) = sqrt(1+1) and sqrt(3) = 3*sqrt(1-2/3) are used here. Therefore lim_{n->oo} r(n) = sqrt(2)+sqrt(3), with rationals r(n) defined below.

			Rationals r(n): [13/4, 203/64, 1615/512, 51595/16384, 412529/131072, 6599099/2097152, 52788535/16777216,...].

K. R. Popper, Die Welt des Parmenides, Piper, 2001, 2005. Ch. 8: Platon und die Geometrie (1950), pp. 326-337. English: The World of Parmenides, Routledge, London, New York, 1998.

Wolfdieter Lang, Rationals r(n), limit.

Cf. A000108, A120785.

a(n) = numerator(4 - sum(k=0, n, binomial(2*k,k)/(k+1)*(1+2^(k+1))/16^k)/4); \\ Michel Marcus, Sep 20 2023

a(n) = numerator(r(n)) with r(n) := 4-(Sum_{k=0..n} C(k)*(1+2^(k+1))/16^k)/4, with C(k) = A000108(k) (Catalan numbers).

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

Wolfdieter Lang, Aug 16 2006

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.

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

W. Lang: Rationals r(n), limit.

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

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

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

Wolfdieter Lang, Aug 16 2006

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.

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

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

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.

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

Wolfdieter Lang, Jul 20 2006

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.

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

Wolfdieter Lang, Rationals r(n) and limit.

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

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.

A120784 Numerators of partial sums of Catalan numbers scaled by powers of 1/16.

Original entry on oeis.org

1, 17, 137, 4389, 35119, 561925, 4495433, 287708141, 2301665843, 36826655919, 294613251551, 9427624079025, 75420992684203, 1206735883132973, 9653887065398089, 1235697544380650237

Offset: 0

Wolfdieter Lang, Jul 20 2006

Denominators are given under A120785.

From the expansion of sqrt(3)/2 = sqrt(1-1/4) = 1-(1/8)*sum(C(k)/16^k,k=0..infinity) one has r:=limit(r(n),n to infinity)= 4*(2 - sqrt(3)) = 1.071796769..., with the partial sums r(n) defined below.

			Rationals r(n): [1, 17/16, 137/128, 4389/4096, 35119/32768, 561925/524288, 4495433/4194304, 287708141/268435456,...].

W. Lang: Rationals r(n) and limit.

a(n)=numerator(r(n)), with the rationals r(n):=sum(C(k)/16^k,k=0..n) with C(k):=A000108(k) (Catalan numbers). Rationals r(n) are taken in lowest terms.

A121504 Numerators of partial sums of a series used for the series of sqrt(2) + sqrt(3) involving Catalan numbers.

Original entry on oeis.org

3, 53, 433, 13941, 111759, 1789509, 14320329, 916611309, 7333257267, 117334608047, 938685468127, 30038055403185, 240304869286059, 3844880951681069, 30759058568289097, 3937160132112061181

Offset: 0

Wolfdieter Lang, Aug 16 2006

The corresponding denominators are A120785(n).
The limit of this series is 4*(4-(sqrt(2)+sqrt(3))) = 3.414942518 (maple10, 10 digits).
See A121503 for the geometric interpretation of sqrt(2)+sqrt(3) and a Popper reference.

			Rationals r(n): [3, 53/16, 433/128, 13941/4096, 111759/32768,
1789509/524288, 14320329/4194304, 916611309/268435456,...].

G. C. Greubel, Table of n, a(n) for n = 0..830
W. Lang, Rationals r(n), limit.

A121503/(4*A120785) are the partial sums of a series for sqrt(2)+sqrt(3).

[Numerator( (&+[Binomial(2*k,k)*(1 + 2^(k+1))/(16^k*(k+1)): k in [0..n]]) ): n in [0..30]]; // G. C. Greubel, Sep 27 2018

Table[Numerator[Sum[CatalanNumber[k]*(1 + 2^(k + 1))/16^k, {k, 0, n}]], {n, 0, 50}] (* G. C. Greubel, Sep 27 2018 *)

for(n=0, 30, print1(numerator(sum(k=0,n, binomial(2*k,k)*(1 + 2^(k+1))/(16^k*(k+1)))), ", ")) \\ G. C. Greubel, Sep 27 2018

a(n) = numerator(r(n)) with r(n):= sum(C(k)*(1+2^(k+1))/16^k,k=0..n), n>=0, with C(k)=A000108(k) (Catalan numbers).

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A121503 Numerators of partial sums of a series for sqrt(2) + sqrt(3) involving Catalan numbers.

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

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

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

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A120784 Numerators of partial sums of Catalan numbers scaled by powers of 1/16.

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A121504 Numerators of partial sums of a series used for the series of sqrt(2) + sqrt(3) involving Catalan numbers.

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