A208899 -id:A208899

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A124397

Numerators of partial sums of a series for sqrt(5)/3.

+10
2

1, 3, 21, 17, 99, 2223, 12039, 56763, 59337, 286961, 7358781, 36088473, 183146521, 181066401, 36534213, 4535753121, 22798981683, 113528187171, 113891192583, 568042152363, 14228623114839, 71035463999307, 355598139789279

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Denominators are given by A124398.

The alternating sums over central binomial coefficients scaled by powers of 5, r(n) = Sum_{k=0..n} (-1)^k*binomial(2*k,k)/5^k, have the limit s = lim_{n-> infinity} r(n) = sqrt(5)/3. From the expansion of 1/sqrt(1+x) for x=4/5.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

Wolfdieter Lang, Rationals and more.

FORMULA

a(n) = numerator(r(n)) with the rationals r(n) = Sum_{k=0..n} (-1)^k * binomial(2*k,k)/5^k in lowest terms.

r(n) = Sum_{k=0..n} (-1)^k*((2*k-1)!!/((2*k)!!)*(4/5)^k, n>=0, with the double factorials A001147 and A000165.

EXAMPLE

a(3)=17 because r(3) = 1 - 2/5 + 6/25 - 4/25 = 17/25 = a(3)/A124398(3).

MAPLE

seq(numer(add((-1)^k*binomial(2*k, k)/5^k, k = 0..n)), n = 0..20); # G. C. Greubel, Dec 25 2019

MATHEMATICA

Table[Numerator[Sum[(-1)^k*(k+1)*CatalanNumber[k]/5^k, {k, 0, n}]], {n, 0, 20}] (* G. C. Greubel, Dec 25 2019 *)

PROG

(PARI) a(n) = numerator(sum(k=0, n, ((-1)^k)*binomial(2*k, k)/5^k)); \\ Michel Marcus, Aug 11 2019

(Magma) [Numerator(&+[(-1)^k*(k+1)*Catalan(k)/5^k: k in [0..n]]): n in [0..20]]; // G. C. Greubel, Dec 25 2019

(Sage) [numerator(sum((-1)^k*(k+1)*catalan_number(k)/5^k for k in (0..n))) for n in (0..20)] # G. C. Greubel, Dec 25 2019

(GAP) List([0..20], n-> NumeratorRat(Sum([0..n], k-> (-1)^k*Binomial(2*k, k)/5^k)) ); # G. C. Greubel, Dec 25 2019

CROSSREFS

Cf. A123747/A123748 partial sums for a series for sqrt(5).

Cf. A123749/A124396 partial sums for a series for 3/sqrt(5).

Cf. A124398 (denominators), A208899 (sqrt(5)/3).

KEYWORD

nonn,frac,easy

AUTHOR

Wolfdieter Lang, Nov 10 2006

STATUS

approved

A124398

Denominators of partial sums of a series for sqrt(5)/3.

+10
2

1, 5, 25, 25, 125, 3125, 15625, 78125, 78125, 390625, 9765625, 48828125, 244140625, 244140625, 48828125, 6103515625, 30517578125, 152587890625, 152587890625, 762939453125, 19073486328125, 95367431640625, 476837158203125

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Denominators of alternating sums over central binomial coefficients scaled by powers of 5.

Numerators are given by A124397.

For the rationals r(n) see the W. Lang link under A124397.

r(n) is not 1/3 times the rational sequence A123747/A123748 which converges to sqrt(5).

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

FORMULA

a(n) = denominator(r(n)) with the rationals r(n) = Sum_{k=0..n} (-1)^k * binomial(2*k,k)/5^k, in lowest terms.

r(n) = Sum_{k=0..n} (-1)^k*((2*k-1)!!/((2*k)!!)*(4/5)^k, n>=0, with the double factorials A001147 and A000165.

EXAMPLE

a(3) = 25 because r(3)= 1 - 2/5 + 6/25 - 4/25 = 17/25 = A124397(3)/a(3).

MAPLE

seq(denom(add((-1)^k*binomial(2*k, k)/5^k, k = 0..n)), n = 0..20); # G. C. Greubel, Dec 25 2019

MATHEMATICA

Table[Denominator[Sum[(-1)^k*(k+1)*CatalanNumber[k]/5^k, {k, 0, n}]], {n, 0, 20}] (* G. C. Greubel, Dec 25 2019 *)

PROG

(PARI) a(n) = denominator(sum(k=0, n, ((-1)^k)*binomial(2*k, k)/5^k)); \\ Michel Marcus, Aug 11 2019

(Magma) [Denominator(&+[(-1)^k*(k+1)*Catalan(k)/5^k: k in [0..n]]): n in [0..20]]; // G. C. Greubel, Dec 25 2019

(Sage) [denominator(sum((-1)^k*(k+1)*catalan_number(k)/5^k for k in (0..n))) for n in (0..20)] # G. C. Greubel, Dec 25 2019

(GAP) List([0..20], n-> DenominatorRat(Sum([0..n], k-> (-1)^k*Binomial(2*k, k)/5^k)) ); # G. C. Greubel, Dec 25 2019

CROSSREFS

Cf. A124397 (numerators), A208899 (sqrt(5)/3).

KEYWORD

nonn,frac,easy

AUTHOR

Wolfdieter Lang, Nov 10 2006

STATUS

approved

A378388

Decimal expansion of the surface area of a tetrakis hexahedron with unit shorter edge length.

+10
0

1, 1, 9, 2, 5, 6, 9, 5, 8, 7, 9, 9, 9, 8, 8, 7, 8, 3, 8, 0, 8, 4, 8, 9, 2, 6, 2, 3, 3, 2, 3, 3, 4, 7, 3, 2, 5, 5, 6, 8, 3, 2, 9, 7, 9, 1, 7, 9, 2, 8, 1, 3, 7, 1, 9, 6, 1, 1, 1, 4, 5, 1, 9, 7, 5, 5, 2, 2, 7, 7, 8, 2, 7, 0, 0, 6, 8, 2, 9, 2, 7, 9, 6, 8, 7, 6, 8, 7, 6, 8

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

2,3

COMMENTS

The tetrakis hexahedron is the dual polyhedron of the truncated octahedron.

LINKS

Paolo Xausa, Table of n, a(n) for n = 2..10000

Eric Weisstein's World of Mathematics, Tetrakis Hexahedron.

FORMULA

Equals (16/3)*sqrt(5) = (16/3)*A002163 = 16*A208899.

EXAMPLE

11.925695879998878380848926233233473255683297917928...

MATHEMATICA

First[RealDigits[16*Sqrt[5]/3, 10, 100]] (* or *)

First[RealDigits[PolyhedronData["TetrakisHexahedron", "SurfaceArea"], 10, 100]]

CROSSREFS

Cf. A374359 (volume - 1), A010532 (inradius*10), A179587 (midradius + 1), A378389 (dihedral angle).

Cf. A377341 (surface area of a truncated octahedron with unit edge).

Cf. A002163, A208899.

KEYWORD

nonn,cons,easy,new

AUTHOR

Paolo Xausa, Nov 27 2024

STATUS

approved

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