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A103532 Number of divisors of 240^n. 4
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 (list; graph; refs; listen; history; text; internal format)
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

LINKS

Table of n, a(n) for n=0..35.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

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

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

Adjacent sequences:  A103529 A103530 A103531 * A103533 A103534 A103535

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

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Last modified February 20 12:12 EST 2018. Contains 299387 sequences. (Running on oeis4.)