Bernoulli numbers: Difference between revisions
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return A[0] # (which is Bn) |
return A[0] # (which is Bn) |
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for i in range(61) |
bn = [(i, bernoulli(i)) for i in range(61)] |
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bn = [(i, b) for i,b in bn if b] |
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b = bernoulli(i) |
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width = max(len(str(b.numerator)) for i,b in bn) |
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if b: |
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for i,b in bn: |
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print('B(%2i) = %*i/%i' % (i, width, b.numerator, b.denominator</lang> |
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{{out}} |
{{out}} |
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<pre>B(0) = |
<pre>B( 0) = 1/1 |
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B( 1) = 1/2 |
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B(1) = Fraction(1, 2) |
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B( 2) = 1/6 |
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B(2) = Fraction(1, 6) |
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B( 4) = -1/30 |
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B(4) = Fraction(-1, 30) |
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B( 6) = 1/42 |
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B(6) = Fraction(1, 42) |
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B( 8) = -1/30 |
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B(8) = Fraction(-1, 30) |
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B(10) = 5/66 |
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B(10) = Fraction(5, 66) |
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B(12) = |
B(12) = -691/2730 |
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B(14) = 7/6 |
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B(14) = Fraction(7, 6) |
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B(16) = |
B(16) = -3617/510 |
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B(18) = 43867/798 |
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B(18) = Fraction(43867, 798) |
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B(20) = |
B(20) = -174611/330 |
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B(22) = 854513/138 |
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B(22) = Fraction(854513, 138) |
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B(24) = |
B(24) = -236364091/2730 |
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B(26) = 8553103/6 |
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B(26) = Fraction(8553103, 6) |
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B(28) = |
B(28) = -23749461029/870 |
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B(30) = 8615841276005/14322 |
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B(30) = Fraction(8615841276005, 14322) |
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B(32) = |
B(32) = -7709321041217/510 |
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B(34) = 2577687858367/6 |
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B(34) = Fraction(2577687858367, 6) |
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B(36) = |
B(36) = -26315271553053477373/1919190 |
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B(38) = 2929993913841559/6 |
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B(38) = Fraction(2929993913841559, 6) |
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B(40) = |
B(40) = -261082718496449122051/13530 |
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B(42) = 1520097643918070802691/1806 |
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B(42) = Fraction(1520097643918070802691, 1806) |
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B(44) = |
B(44) = -27833269579301024235023/690 |
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B(46) = 596451111593912163277961/282 |
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B(46) = Fraction(596451111593912163277961, 282) |
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B(48) = |
B(48) = -5609403368997817686249127547/46410 |
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B(50) = 495057205241079648212477525/66 |
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B(50) = Fraction(495057205241079648212477525, 66) |
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B(52) = |
B(52) = -801165718135489957347924991853/1590 |
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B(54) = 29149963634884862421418123812691/798 |
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B(54) = Fraction(29149963634884862421418123812691, 798) |
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B(56) = |
B(56) = -2479392929313226753685415739663229/870 |
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B(58) = 84483613348880041862046775994036021/354 |
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B(58) = Fraction(84483613348880041862046775994036021, 354) |
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B(60) = |
B(60) = -1215233140483755572040304994079820246041491/56786730</pre> |
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=={{header|REXX}}== |
=={{header|REXX}}== |
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Revision as of 19:23, 11 March 2014
Bernoulli numbers
Bernoulli numbers are used in some series expansions of serval functions (trigonometric, hyperbolic, gamma, etc.), and are extremely important in number theory and analysis.
There are two definitions of Bernoulli numbers; this task will be using the modern usage (as per the National Institute of Standards and Technology convention), and are expressed as Bn.
task requirements
- show the Bernoulli numbers B0 through B60.
- express the numbers as fractions (most are improper fractions).
- fractions should be reduced.
- suppress all 0 (zero) values suppressed.
- index each number in some way so that it can be discerned which number is being displayed.
- (extra credit) align the solidi [/] if used.
(Other than B1, all odd Bernoulli have a value of 0 (zero).
see also
- Sequence A027641 Numerator of Bernoulli number B_n on The On-Line Encyclopedia of Integer Sequences.
- Sequence A027642 Denominator of Bernoulli number B_n on The On-Line Encyclopedia of Integer Sequences.
- Entry Bernoulli number on The Eric Weisstein's World of Mathematics.
- Wiki entry Bernoulli number.
Python
<lang python>from fractions import Fraction as Fr
def bernoulli(n):
A = [0] * (n+1)
for m in range(n+1):
A[m] = Fr(1, m+1)
for j in range(m, 0, -1):
A[j-1] = j*(A[j-1] - A[j])
return A[0] # (which is Bn)
bn = [(i, bernoulli(i)) for i in range(61)] bn = [(i, b) for i,b in bn if b] width = max(len(str(b.numerator)) for i,b in bn) for i,b in bn:
print('B(%2i) = %*i/%i' % (i, width, b.numerator, b.denominator</lang>
- Output:
B( 0) = 1/1 B( 1) = 1/2 B( 2) = 1/6 B( 4) = -1/30 B( 6) = 1/42 B( 8) = -1/30 B(10) = 5/66 B(12) = -691/2730 B(14) = 7/6 B(16) = -3617/510 B(18) = 43867/798 B(20) = -174611/330 B(22) = 854513/138 B(24) = -236364091/2730 B(26) = 8553103/6 B(28) = -23749461029/870 B(30) = 8615841276005/14322 B(32) = -7709321041217/510 B(34) = 2577687858367/6 B(36) = -26315271553053477373/1919190 B(38) = 2929993913841559/6 B(40) = -261082718496449122051/13530 B(42) = 1520097643918070802691/1806 B(44) = -27833269579301024235023/690 B(46) = 596451111593912163277961/282 B(48) = -5609403368997817686249127547/46410 B(50) = 495057205241079648212477525/66 B(52) = -801165718135489957347924991853/1590 B(54) = 29149963634884862421418123812691/798 B(56) = -2479392929313226753685415739663229/870 B(58) = 84483613348880041862046775994036021/354 B(60) = -1215233140483755572040304994079820246041491/56786730
REXX
<lang rexx>/*REXX program calculates a number of Bernoulli numbers (as fractions). */ parse arg N .; if N== then N=60 /*get N. If ¬ given, use default*/ w=max(length(N),4); Nw=N+N%5 /*used for aligning the output. */ say 'B(n)' center('Bernoulli number expressed as a fraction', max(78-w,Nw)) say copies('─',w) copies('─', max(78-w,Nw+2*w))
do #=0 to N /*process numbers from 0 ──► N. */
b=bern(#); if b==0 then iterate /*calculate Bernoulli#, skip if 0*/
indent=max(0, nW-pos('/', b)) /*calculate alignment indentation*/
say right(#,w) left(,indent) b /*display the indented Bernoulli#*/
end /* [↑] align the Bernoulli number*/
exit /*stick a fork in it, we're done.*/ /*──────────────────────────────────BERN subroutine─────────────────────*/ bern: parse arg x /*obtain the subroutine argument.*/ if x==0 then return '1/1' /*handle the special case of zero*/ if x==1 then return '-1/2' /* " " " " " one.*/ if x//2 then return 0 /* " " " " " odds*/
/* [↓] process all #s up to X, */
do j=2 to x by 2; jp=j+1; d=j+j /* & set some shortcut vars.*/
if d>digits() then numeric digits d /*increase precision if needed. */
sn=1-j /*set the numerator. */
sd=2 /* " " denominator. */
do k=2 to j-1 by 2 /*calculate a SN/SD sequence. */
parse var @.k bn '/' ad /*get a previously calculated fra*/
an=comb(jp,k)*bn /*use COMBination for next term. */
lcm=lcm(sd,ad) /*use Least Common Denominator. */
sn=lcm%sd*sn; sd=lcm /*calculate current numerator. */
an=lcm%ad*an; ad=lcm /* " next " */
sn=sn+an /* " current " */
end /*k*/ /* [↑] calculate SN/SD sequence.*/
sn=-sn /*adjust the sign for numerator. */
sd=sd*jp /*calculate the denomitator. */
if sn\==1 then do /*reduce the fraction if possible*/
_=gcd(sn,sd) /*get Greatest Common Denominator*/
sn=sn%_; sd=sd%_ /*reduce numerator & denominator.*/
end /* [↑] done with the reduction.*/
@.j=sn'/'sd /*save the result for next round.*/
end /*j*/ /* [↑] done with calculating B#.*/
return sn'/'sd /*──────────────────────────────────COMB subroutine─────────────────────*/ comb: procedure; parse arg x,y; if x==y then return 1
if x-y<y then y=x-y; z=perm(x,y); do j=2 to y; z=z/j;end; return z
/*──────────────────────────────────GCD subroutine──────────────────────*/ gcd: procedure; $=; do i=1 for arg(); $=$ arg(i); end; parse var $ x z .
if x=0 then x=z; x=abs(x); do j=2 to words($); y=abs(word($,j))
if y=0 then iterate; do until _==0; _=x//y; x=y; y=_; end; end
return x
/*──────────────────────────────────LCM subroutine──────────────────────*/ lcm: procedure; $=; do j=1 for arg(); $=$ arg(j); end; x=abs(word($,1))
do k=2 to words($); !=abs(word($,k)); if !=0 then return 0
x=x * ! / gcd(x,!); end
return x
/*──────────────────────────────────PERM subroutine─────────────────────*/ perm: procedure; parse arg x,y; z=1; do j=x-y+1 to x; z=z*j; end; return z</lang> output when using the default input:
B(n) Bernoulli number expressed as a fraction ──── ──────────────────────────────────────────────────────────────────────────────── 0 1/1 1 -1/2 2 1/6 4 -1/30 6 1/42 8 -1/30 10 5/66 12 -691/2730 14 7/6 16 -3617/510 18 43867/798 20 -174611/330 22 854513/138 24 -236364091/2730 26 8553103/6 28 -23749461029/870 30 8615841276005/14322 32 -7709321041217/510 34 2577687858367/6 36 -26315271553053477373/1919190 38 2929993913841559/6 40 -261082718496449122051/13530 42 1520097643918070802691/1806 44 -27833269579301024235023/690 46 596451111593912163277961/282 48 -5609403368997817686249127547/46410 50 495057205241079648212477525/66 52 -801165718135489957347924991853/1590 54 29149963634884862421418123812691/798 56 -2479392929313226753685415739663229/870 58 84483613348880041862046775994036021/354 60 -1215233140483755572040304994079820246041491/56786730