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

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

Offset

0,2

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.

References

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

Links

Robert Israel, Table of n, a(n) for n = 0..866

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... .

Example

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.

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 *)

Crossrefs

Cf. A001107, A244647, A250550, A250551.

Sequence in context: A336027 A166811 A089906 * A287317 A084943 A055055

Adjacent sequences: A294512 A294513 A294514 * A294516 A294517 A294518

Keyword

nonn,frac,easy

Author

Wolfdieter Lang, Nov 02 2017

Status

approved