Reciprocal of 49 shows Powers of 2 in Decimal Expansion

Theorem
The decimal expansion of the reciprocal of $49$ contains the powers of $2$:
 * $\dfrac 1 {49} = 0 \cdotp \dot 02040 \, 81632 \, 65306 \, 12244 \, 89795 \, 91836 \, 73469 \, 38775 \, 5 \dot 1$

Proof
Performing the calculation using long division:

0.020408163265306122448979591836734693877551 - 49)1.000000000000000000000000000000000000000000     98     --      200      196      ---        400        392        ---          80          49          --          310          294          ---           160           147           ---            130             98             ---             320             294             ---              260              245              ---               150               147               ---                 300                 294                 ---                   60                   49                   --                   110                    98                   ---                    120                     98                    ---                     220                     196                     ---                      240                      196                      ---                       440                       392                       ---                        480                        441                        ---                         390                         343                         ---                          470                          441                          ---                           290                           245                           ---                            450                            441                            ---                              90                              49                              --                              410                              392                              ---                               180                               147                               ---                                330                                294                                ---                                 360                                 343                                 ---                                  170                                  147                                  ---                                   230                                   196                                   ---                                    340                                    294                                    ---                                     460                                     441                                     ---                                      190                                      147                                      ---                                       430                                       392                                       ---                                        380                                        343                                        ---                                         370                                         343                                         ---                                          270                                          245                                          ---                                           250                                           245                                           ---                                             50                                             49                                             --                                              100                                               98                                              ---                                                200                                                ...

Adding up powers of $2$, shifted appropriately to the right: 02 04    08      16        32          64           128             256               512                1024                  2048                    4096                      8192                       16384                         32768                           65536                            131072                              262144                                524288                                 1048576                                   2097152                                     4194304                                       8388608                                        16777216                                          33554432                                            67108864                                             134217728                                               268435456                                                 536870912                                                  1073741824                                                    2147483648                                                      4294967296                                                        8589934592                                                         ...........

020408163265306122448979591836734693877551020408163265305947144192

1 1 1111 11 11212122222112222222322222111

As can be seen, the decimal expansion of $\dfrac 1 {49}$ matches the sum of the shifted powers of $2$ (to the limits of calculation) and the pattern is apparent.