perplexus.info :: Just Math : Formula Formulation

Formula Formulation (Posted on 2014-03-03)

A function f(N) is defined as:
f(1)=sod(1²), f(2) = sod(12²)
f(3) = sod(121²), f(4) = sod(1212²), .... and, so on

Derive the general formula for f(N) in terms of N.

*** sod(x) denotes the sum of the digits of base ten expansion of x.

No Solution Yet Submitted by K Sengupta

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first few

| Comment 3 of 4 |

5   S=""
10   for N=1 to 40
20      S=S+mid("21",N@2+1,1)
30      B=val(S)
40      P=B^2
50      Ps=cutspc(str(P))
60      T=0
70      for I=1 to len(Ps):T=T+val(mid(Ps,I,1)):next
80      print N,S,tab(48);T;sqrt(T)
90   next

finds

n                                              f(n)    sqrt(f(n))  
7      1212121                                  55  7.4161984870956629486
8      12121212                                 72  8.4852813742385702927
9      121212121                                79  8.88819441731558885
10     1212121212                               90  9.4868329805051379959
11     12121212121                              94  9.695359714832658028
12     121212121212                             108  10.3923048454132637611
13     1212121212121                            118  10.8627804912002157238
14     12121212121212                           126  11.2249721603218241567
15     121212121212121                          133  11.5325625946707958893
16     1212121212121212                         144  12.0
17     12121212121212121                        157  12.5299640861416677884
18     121212121212121212                       162  12.7279220613578554391
19     1212121212121212121                      163  12.7671453348037046616
20     12121212121212121212                     171  13.0766968306220206566
21     121212121212121212121                    178  13.3416640641263337124
22     1212121212121212121212                   198  14.0712472794702886636
23     12121212121212121212121                  193  13.8924439894498045083
24     121212121212121212121212                 207  14.3874945699381586247
25     1212121212121212121212121                208  14.4222051018559571724
26     12121212121212121212121212               234  15.29705854077835449
27     121212121212121212121212121              232  15.2315462117278165712
28     1212121212121212121212121212             252  15.8745078663875435429
29     12121212121212121212121212121            247  15.7162336455017109802
30     121212121212121212121212121212           270  16.4316767251549834036
31     1212121212121212121212121212121          271  16.4620776331543279476
32     12121212121212121212121212121212         288  16.9705627484771405855
33     121212121212121212121212121212121        286  16.9115345252877628981
34     1212121212121212121212121212121212       306  17.4928556845359014125
35     12121212121212121212121212121212121      310  17.6068168616590091457
36     121212121212121212121212121212121212     324  18.0
37     1212121212121212121212121212121212121    325  18.0277563773199464655
38     12121212121212121212121212121212121212   342  18.4932420089069293507
39     121212121212121212121212121212121212121  349  18.6815416922694043484
40     1212121212121212121212121212121212121212 360  18.9736659610102759919

Posted by Charlie on 2014-03-03 17:22:45

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