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A294515
Denominators of partial sums of the reciprocals of the decagonal numbers.
4
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
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
KEYWORD
nonn,frac,easy
AUTHOR
Wolfdieter Lang, Nov 02 2017
STATUS
approved