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URL: https://oeis.org/A103532

⇱ A103532 - OEIS

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A103532
Number of divisors of 240^n.
8
1, 20, 81, 208, 425, 756, 1225, 1856, 2673, 3700, 4961, 6480, 8281, 10388, 12825, 15616, 18785, 22356, 26353, 30800, 35721, 41140, 47081, 53568, 60625, 68276, 76545, 85456, 95033, 105300, 116281, 128000, 140481, 153748, 167825, 182736
OFFSET
0,2
COMMENTS
Geometric interpretation: Take a simple cubical grid of size (2n+1). Number the coordinates along each axis from 1 to (2n+1). Select only the cells that have at least two odd coordinates, and discard the rest. The number of selected cells is a(n). - Arun Giridhar, Mar 27 2015
FORMULA
From R. J. Mathar and Stefan Steinerberger, Aug 31 2008: (Start)
a(n) = (4*n+1)*(n+1)^2.
G.f.: (1+16x+7x^2)/(1-x)^4.
Inverse binomial transform: 1, 19, 42, 24, 0 (0 continued). (End)
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>3. - Harvey P. Dale, Jan 21 2013
a(n) = (n+1)*A001107(n+1), where A001107 are the partial sums of A017007. - J. M. Bergot, Jul 08 2013
a(n) = Sum_{i=0..n} (n+1)*(8*i+1). [Bruno Berselli, Sep 08 2015]
Sum_{n>=0} 1/a(n) = 2*Pi/9 - Pi^2/18 + 4*log(2)/3 = 1.07401658592825... . - Vaclav Kotesovec, Oct 04 2016
E.g.f.: exp(x)*(1 + 19*x + 21*x^2 + 4*x^3). - Stefano Spezia, Jan 31 2025
Sum_{n>=0} (-1)^n/a(n) = 2*sqrt(2)*Pi/9 - Pi^2/36 - (4/9)*(log(2) + sqrt(2)*log(sqrt(2)-1)). - Amiram Eldar, Aug 15 2025
EXAMPLE
a(2) = 81 because 240^2 has 81 divisors.
a(2) = 81 because a 5 X 5 X 5 grid has 81 cells with at least two odd coordinates each, coordinate numbering starting at 1.
MAPLE
A103532 := proc(n) (4*n+1)*(n+1)^2 ; end proc: # R. J. Mathar, Aug 31 2008
MATHEMATICA
Table[(4 n + 1) (n + 1)^2, {n, 0, 40}] (* Stefan Steinerberger, Aug 31 2008 *)
DivisorSigma[0, 240^Range[0, 40]] (* or *) LinearRecurrence[{4, -6, 4, -1}, {1, 20, 81, 208}, 40] (* Harvey P. Dale, Jan 21 2013 *)
PROG
(Magma) [(4*n+1)*(n+1)^2: n in [0..45]]; // Vincenzo Librandi, Feb 10 2016
CROSSREFS
Cf. similar sequences, with the formula (k*n-k+2)*n^2/2, listed in A262000.
Sequence in context: A211463 A244449 A041776 * A041778 A219473 A256869
KEYWORD
nonn,easy
AUTHOR
J. Lowell, Aug 30 2008
EXTENSIONS
More terms from Stefan Steinerberger and R. J. Mathar, Aug 31 2008
Example corrected by Harvey P. Dale, Jan 21 2013
STATUS
approved