A294515 - OEIS

%I #26 Nov 23 2024 19:36:06

%S 1,10,270,7020,119340,835380,4176900,242260200,888287400,32866633800,

%T 1347531985800,4042595957400,28298171701800,1499803100195400,

%U 28496258903712600,3476543586252937200,3476543586252937200,26653500827939185200,1945705560439560519600,1945705560439560519600,52534050131868134029200

%N Denominators of partial sums of the reciprocals of the decagonal numbers.

%C The corresponding numerators are given by A250551(n+1), n >= 0.

%C The positive decagonal numbers are A001107(k+1) = (k + 1)*(4*k + 1), k >= 0.

%C 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].

%C 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.

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

%H Robert Israel, <a href="/A294515/b294515.txt">Table of n, a(n) for n = 0..866</a>

%F 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)).

%F a(n) = A250550(n+1)/(n+1) [conjecture].

%F In the Koecher reference v_4(1) = (3/4)*V(4,1) = (3/4)*log(2) + Pi/8 = 0.91255946711868313687... .

%e 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, ...

%e V(4,1;10^4) = 1.216720959 (Maple, 10 digits) to be compared with 1.216745956 from V(4,1) from A244647.

%p map(denom,ListTools:-PartialSums([seq(1/((k+1)*(4*k+1)),k=0..50)])); # _Robert Israel_, Nov 08 2017

%t Denominator@ Accumulate@ Array[1/PolygonalNumber[10, #] &, 23] (* _Michael De Vlieger_, Nov 02 2017 *)

%Y Cf. A001107, A244647, A250550, A250551.

%K nonn,frac,easy

%O 0,2

%A _Wolfdieter Lang_, Nov 02 2017