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 1-4 of 4 results.

A244647 Decimal expansion of the sum of the reciprocals of the decagonal numbers (A001107).

Original entry on oeis.org

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

Views

Author

Robert G. Wilson v, Jul 03 2014

Keywords

Comments

For the partial sums of the reciprocals of the (positive) decagonal numbers see A250551(n+1)/A294515(n), n >= 0. - Wolfdieter Lang, Nov 07 2017

Examples

			1.216745956158244182494339352004760382108361700922772890949837441544696356350....

Links

Crossrefs

Cf. A001107, A000038, A013661, A244639, A016627, A244645, A244646, A244648, A244649, A250551(n+1)/A294515(n).

Programs

  • Mathematica
    RealDigits[ Log[2] + Pi/6, 10, 111][[1]] (* or *)
RealDigits[ Sum[1/(4n^2 - 3n), {n, 1 , Infinity}], 10, 111][[1]]
  • PARI
    log(2)+Pi/6 \\ Charles R Greathouse IV, Feb 08 2023

Formula

Sum_{n>0} 1/(4n^2 - 3n) = log(2) + Pi/6, (A002162 + A019673).

A294515 Denominators of partial sums of the reciprocals of the decagonal numbers.

Original entry on oeis.org

1, 10, 270, 7020, 119340, 835380, 4176900, 242260200, 888287400, 32866633800, 1347531985800, 4042595957400, 28298171701800, 1499803100195400, 28496258903712600, 3476543586252937200, 3476543586252937200, 26653500827939185200, 1945705560439560519600, 1945705560439560519600, 52534050131868134029200
Offset: 0

Views

Author

Wolfdieter Lang, Nov 02 2017

Keywords

Comments

The corresponding numerators are given by A250551(n+1), n >= 0.
The positive decagonal numbers are A001107(k+1) = (k + 1)*(4*k + 1), k >= 0.
For the general case V(m,r;n) = Sum_{k=0..n} 1/((k + 1)*(m*k + r)) = (1/(m - r))*Sum_{k=0..n} (m/(m*k + r) - 1/(k+1)), for r = 1, ..., m-1 and m = 2, 3, ..., and their limits see a comment in A294512. Here [m,r] = [4,1].
The limit of the series is V(4,1) = lim_{n -> oo} V(4,1;n) = log(2) + Pi/6 = 1.216745956158244182... given in A244647.

Examples

			The rationals V(4,1;n), n >= 0, begin: 1, 11/10, 307/270, 8117/7020, 139393/119340, 982381/835380, 4935773/4176900, 287319059/242260200, 1056494083/888287400, 39179109811/32866633800, ...
V(4,1;10^4) = 1.216720959 (Maple, 10 digits) to be compared with 1.216745956 from V(4,1) from A244647.

References

  • Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, pp. 189 - 193.

Links

Crossrefs

Cf. A001107, A244647, A250550, A250551.

Programs

  • Maple
    map(denom,ListTools:-PartialSums([seq(1/((k+1)*(4*k+1)),k=0..50)])); # Robert Israel, Nov 08 2017
  • Mathematica
    Denominator@ Accumulate@ Array[1/PolygonalNumber[10, #] &, 23] (* Michael De Vlieger, Nov 02 2017 *)

Formula

a(n) = denominator(V(4,1;n)) with V(4,1;n) = Sum_{k=0..n} 1/((k + 1)*(4*k + 1)) = Sum_{k=0..n} 1/A001107(n+1) = (1/3)*Sum_{k=0..n} (4/(4*k + 1) - 1/(k+1)).
a(n) = A250550(n+1)/(n+1) [conjecture].
In the Koecher reference v_4(1) = (3/4)*V(4,1) = (3/4)*log(2) + Pi/8 = 0.91255946711868313687... .

A250550 Numerator of the harmonic mean of the first n 10-gonal numbers.

Original entry on oeis.org

1, 20, 810, 28080, 596700, 5012280, 29238300, 1938081600, 7994586600, 328666338000, 14822851843800, 48511151488800, 367876232123400, 20997243402735600, 427443883555689000, 55624697380046995200, 59101240966299932400, 479763014902905333600
Offset: 1

Views

Author

Colin Barker, Nov 25 2014

Keywords

Examples

			a(3) = 810 because the first 3 10-gonal numbers are [1,10,27], and 3/(1/1+1/10+1/27) = 810/307.

Links

Crossrefs

Cf. A001107 (10-gonal numbers), A250551 (denominators).

Programs

  • Mathematica
    Module[{nn=20,pn},pns=PolygonalNumber[10,Range[nn]];Table[HarmonicMean[ Take[ pns,n]],{n,nn}]]//Numerator (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 09 2017 *)
  • PARI
    harmonicmean(v) = #v / sum(k=1, #v, 1/v[k])
s=vector(30); for(n=1, #s, s[n]=numerator(harmonicmean(vector(n, k, (8*k^2-6*k)/2)))); s

A294516 Numerators of the partial sums of the reciprocals of (k+1)*(4*k+3) = A033991(k+1), for k >= 0.

Original entry on oeis.org

1, 17, 67, 2087, 40577, 315967, 8627249, 539432053, 543008461, 7096662277, 306487877071, 14457409539227, 246534893826499, 49437672710843, 14617658229054773, 29294219493288391, 1966205309547985477, 139821581165897995307, 700098935135639210887, 55378426713778630607653, 4601722042202662057443599, 12144567347216934480292961
Offset: 0

Views

Author

Wolfdieter Lang, Nov 07 2017

Keywords

Comments

The corresponding numerators are given in A294517.
For the general case V(m,r;n) = Sum_{k=0..n} 1/((k + 1)*(m*k + r)) = (1/(m - r))*Sum_{k=0..n} (m/(m*k + r) - 1/(k+1)), for r = 1, ..., m-1 and m = 2, 3, ..., and their limits see a comment in A294512. Here [m,r] = [4,3].
The limit of the series is V(4,3) = lim_{n -> oo} V(4,3;n) = 3*log(2) - Pi/2 = 0.50864521488493930902... given in A294518.

Examples

			The rationals V(4,3;n), n >= 0, begin: 1/3, 17/42, 67/154, 2087/4620, 40577/87780, 315967/672980, 8627249/18170460, 539432053/1126568520, 543008461/1126568520, 7096662277/14645390760, 306487877071/629751802680, ...
V(4,3;10^4) = 0.508620219 (Maple, 10 digits) to be compared with 0.508645215 from V(4,3) given in A294518.

References

  • Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, pp. 189 - 193.

Links

Crossrefs

Cf. A294512, A250551(n+1)/A294515(n) (V(4,1;n)), A294517, A294518.

Programs

  • PARI
    a(n) = numerator(sum(k=0, n, 1/((k + 1)*(4*k + 3)))); \\ Michel Marcus, Nov 15 2017

Formula

a(n) = numerator(V(4,3;n)) with V(4,3;n) = Sum_{k=0..n} 1/((k + 1)*(4*k + 3)) = Sum_{k=0..n} 1/A033991(k+1) = Sum_{k=0..n} (4/(4*k + 3) - 1/(k+1)).
V(4,3;n) = 3*log(2) - Pi/2 + Psi(n+7/4) - Psi(n+2) with the digamma function Psi. Note that Psi(1) - Psi(3/4) = 3*log(2) - Pi/2. - Wolfdieter Lang, Nov 15 2017
Showing 1-4 of 4 results.

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