A New NERDY Number!

5/21/2015:

You may remember that my daughter (Peggy Sue - though most people just call her "Doctor") is a big fan of the number 13.  So I have prepared a new NERDY number for her.

The new sequence is called the "Pegganacci Sequence".  It starts off like a Tridecanacci Sequence but each new term is the sum of the the prevous 13 terms, each multiplied by one of the numbers from 1 to 13.  Mathematicians would explain it this way:

The new NERDY number is:

999999999999899999999999979999999999996 999999999999599999999999949999999999993 999999999999299999999999919999999999990 999999999998999999999999889999999999987 
9999999999987.

The the decimal expansion of the inverse of the number shown above is called the "Pegg-anacci Sequence".  It starts off like a Tridecanacci Sequence but each new term is the sum of the the previous 13 terms, each multiplied by one of the numbers from 1 to 13.  Mathematicians would explain it this way: 

This is a Fibonacci like sequence where a(0) = a(1) = a(2) = a(3) = … = a(11) = a(12) = 0, a(13) = 1, and when n>13 
then a(n) = a(n – 1) + 2*a(n – 2) + 3*a(n – 3) + 
4*a(n – 4) + 5*a(n – 5) + 6*a(n – 6) + 7*a(n – 7) + 
8*a(n – 8) + 9*a(n – 9) + 10*a(n – 10) + 11*a(n – 11) + 12*a(n – 12) + 13*a(n – 13).  Each term in the sequence is written in 13 digit chunks, which are accurate to the 31^st non-zero term.

The decimal expansion of the inverse is:

                                 1                                    
999999999999899999999999979999999999996 999999999999599999999999949999999999993 999999999999299999999999919999999999990 999999999998999999999999889999999999987 
9999999999987 = 

0.
0000000000000
0000000000000
0000000000000
0000000000000
0000000000000
0000000000000
0000000000000
0000000000000
0000000000000
0000000000000
0000000000000
0000000000000
0000000000001
0000000000001
0000000000003
0000000000008
0000000000021
0000000000055
0000000000144
0000000000377
0000000000987
0000000002584
0000000006765
0000000017711
0000000046368
0000000121393
0000000317797
0000000831997
0000002178165
0000005702425
0000014928920
0000039083838
0000102321293
0000267876635
0000701299695
0001835999105
0004806636502
0012583750392
0032944195765
0086247740185
0225796153741
0591133204441
1547583781233 …

The new NERDY Number is a 169 digit number (THIRTEEN squared).  It produces a sequence where each new term is the sum of the THIRTEEN prior terms, each multiplied by a number from one to THIRTEEN, and written as THIRTEEN digit numbers.  It produces 31 accurate non-zero terms (and "31" is "13" written backwards).

Make sure you stick this one away in your notes on "13-ish Trivia". 


David 
Free-Range Mathematologist