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[ January 16, 2005 ]
Sumas Palindromizadas con Factores de Corrección
Números Naturales del 1 al 100
Hugo Sánchez (email)
This is a new batch of palindromes !
Patrones Palindrómicos Infinitos con Factores de Corrección,
aplicados a la Sucesión de Números Naturales de 1 al 100
(1 al 9 y luego 10 al 100).
Sumatorias Generalizadas de la Secuencia de los Naturales.
Explicaciones Preliminares:
Parte I. Secuencia de Naturales con un Dígito.
| a) | 9
S
i=1 | (in)  |
n = 2 : 11 + 22 + 33 + ... + 99 = 495
n = 3 : 111 + 222 + 333 + ... + 999 = 4995
n = 4 : 1111 + 2222 + 3333 + ... + 9999 = 49995
•••
n = 20 : 11...11 + 22...22 + 33...33 + ... + 99...99 =
4_9999999999999999999_5
b) El patrón es claro : 4(9n-1)5 si se le resta 1,
se Palindromiza el resultado, así :
9
S
i=1 | (in) – 1 | = 4(9n-1)4 , n > 1 |
Parte II. Secuencia de los Naturales con dos Dígitos.
| a) | 99
S
i=10 | (in) |
n = 1 : 10 + 11 + 12 + 13 + ... + 98 + 99 = 4905
n = 2 : 1010 + 1111 + 1212 + 1313 + ... + 9898 + 9999 = 495405
n = 3 : 101010 + 111111 + 121212 + ... + 989898 + 999999 = 49545405
•••
n = 20 : 10...10 + 11...11 + 12...12 + ... + 98...98 + 99...99 =
49_54545454545454545454545454545454545454_05 = 49(5419)05
b) El patrón es claro : 49(54n-1)05. Notar que
| n = 2 : | 495405 + 535
10 | = 49594 = 49(542-2)594 |
| n = 3 : | 49545405 + 535
10 | = 4954594 = 49(543-2)594 |
| n = 20 : | 49(5419)05 + 535
10 | = 49(5420-2)594 |
Esto evidencia que se Palindromiza la Sumatoria así :
99
S (in) + 535
i=10
10 |
= 49(54n-2)594 , con n > 1 |
Parte III. Los Patrones con tres Dígitos (100 a 999)
en adelante (4, 5, 6, ...)
no son Palindromizables con Factores de Corrección.
Caso I : Sumatorias Generalizadas Unidigitales
| 3. | 4
S
i=1 | (in) + 1 | = 1n+1 , n ⩾ 1 |
| 4. | 5
S
i=1 | (in) – 4 | = 1(6n-1)1 , n ⩾ 2 |
| 5. | 6
S
i=1 | (in) + 1 | = 2(3n-1)2 , n ⩾ 1 |
| 6. | 7
S
i=1 | (in) + 5 | = 3(1n-1)3 , n ⩾ 1 |
| 7. | 8
S
i=1 | (in) – 3 | = 3(9n-1)3 , n ⩾ 1 |
| 8. | 9
S
i=1 | (in) – 1 | = 4(6n-1)4 , n ⩾ 1 |
| 9. | 9
S
i=2 | (in) | = 4(8n-1)4 , n ⩾ 2 |
| 10. | 8
S
i=2 | (in) – 2 | = 3(8n-1)3 , n ⩾ 1 |
| 11. | 7
S
i=2 | (in) – 5 | = 2(9n-1)2 , n ⩾ 1 |
| 12. | 6
S
i=2 | (in) + 2 | = 2n+1 , n ⩾ 1 |
| 13. | 5
S
i=2 | (in) – 3 | = 1(5n-1)1 , n ⩾ 1 |
| 14. | 4
S
i=2 | (in) | = 9n , n ⩾ 1 |
| 15. | 3
S
i=2 | (in) | = 5n , n ⩾ 1 |
| 16. | 9
S
i=3 | (in) + 2 | = 4(6n-1)4 , n ⩾ 1 |
| 17. | 8
S
i=3 | (in) | = 3(6n-1)3 , n ⩾ 1 |
| 18. | 7
S
i=3 | (in) – 3 | = 2(7n-1)2 , n ⩾ 1 |
| 19. | 6
S
i=3 | (in) – 7 | = 1(9n-1)1 , n ⩾ 1 |
| 20. | 5
S
i=3 | (in) – 1 | = 1(3n-1)1 , n ⩾ 1 |
| 21. | 4
S
i=3 | (in) | = 7n , n ⩾ 1 |
| 22. | 9
S
i=4 | (in) + 5 | = 4(3n-1)4 , n ⩾ 2 |
| 23. | 8
S
i=4 | (in) + 3 | = 3n+1 , n ⩾ 2 |
| 24. | 7
S
i=4 | (in) | = 2(4n-1)2 , n ⩾ 2 |
| 25. | 6
S
i=4 | (in) – 4 | = 1(6n-1)1 , n ⩾ 2 |
| 26. | 5
S
i=4 | (in) | = 9n , n ⩾ 1 |
| 27. | 9
S
i=5 | (in) – 2 | = 3(8n-1)3 , n ⩾ 2 |
| 28. | 8
S
i=5 | (in) – 4 | = 2(8n-1)2 , n ⩾ 2 |
| 29. | 7
S
i=5 | (in) – 7 | = 1(9n-1)1 , n ⩾ 2 |
| 30. | 6
S
i=5 | (in) | = 1(2n-1)1 , n ⩾ 2 |
| 31. | 9
S
i=6 | (in) + 3 | = 3n+1 , n ⩾ 2 |
| 32. | 8
S
i=6 | (in) + 1 | = 2(3n-1)2 , n ⩾ 2 |
| 33. | 7
S
i=6 | (in) – 2 | = 1(4n-1)1 , n ⩾ 2 |
| 34. | 9
S
i=7 | (in) – 2 | = 2(6n-1)2 , n ⩾ 2 |
| 35. | 8
S
i=7 | (in) – 4 | = 1(6n-1)1 , n ⩾ 2 |
| 36. | 9
S
i=8 | (in) – 6 | = 1(8n-1)1 , n ⩾ 2 |
Caso II : Sumatorias Generalizadas Bidigitales
1 ) | 99
S (in) + 535
i=10
10 |
= 49(54n-2)594 , n ⩾ 2 |
2 ) | 98
S (in) + 434
i=10
10 |
= 48(54n-2)584 , n ⩾ 2 |
3 ) | 97
S (in) + 232
i=10
10 |
= 47(55n-2)574 , n ⩾ 2 |
4 ) | 96
S (in) – 71
i=10
10 |
= 46(57n-2)564 , n ⩾ 2 |
5 ) | 95
S (in) + 525
i=10
10 |
= 45(60n-2)654 , n ⩾ 2 |
6 ) | 94
S (in) + 20
i=10
10 |
= 44(64n-2)644 , n ⩾ 2 |
7 ) | 93
S (in) – 586
i=10
10 |
= 43(69n-2)634 , n ⩾ 2 |
8 ) | 92
S (in) – 293
i=10
10 |
= 42(75n-2)724 , n ⩾ 2 |
9 ) | 91
S (in) – 101
i=10
10 |
= 41(82n-2)814 , n ⩾ 2 |
10 ) | 90
S (in) – 10
i=10
10 |
= 40(90n-2)904 , n ⩾ 2 |
| 11 ) | 89
S
i=10 | (in) + 33 = 3(92.n)3 , n ⩾ 2 |
12 ) | 88
S (in) + 959
i=10
10 |
= 39(10n-2)193 , n ⩾ 2 [primes possible] |
13 ) | 87
S (in) + 747
i=10
10 |
= 38(21n-2)283 , n ⩾ 2 [primes possible] |
| 14 ) | 86
S
i=10 | (in) + 77 = 37(32.(n-1))73 , n ⩾ 2 [primes possible] |
15 ) | 85
S (in) + 20
i=10
10 |
a ) = 36(46n-2)463 , n ⩾ 2 or
b ) = 3(64n-1)63 , n ⩾ 1 [primes possible] |
16 ) | 84
S (in) + 505
i=10
10 |
= 35(60n-2)653 , n ⩾ 2 [primes possible] |
17 ) | 83
S (in) – 201
i=10
10 |
= 34(75n-2)743 , n ⩾ 2 [primes possible] |
18 ) | 82
S (in) + 172
i=10
10 |
= 33(91n-2)933 , n ⩾ 2 [primes possible] |
19 ) | 81
S (in) – 546
i=10
10 |
= 33(09n-2)033 , n ⩾ 2 |
20 ) | 80
S (in) – 465
i=10
10 |
= 32(27n-2)223 , n ⩾ 2 [primes possible] |
21 ) | 79
S (in) – 485
i=10
10 |
= 31(46n-2)413 , n ⩾ 2 [primes possible] |
| 22 ) | 78
S
i=10 | (in) – 33 = 30(62.(n-1))03 , n ⩾ 1 |
23 ) | 77
S (in) + 162
i=10
10 |
= 29(87n-2)892 , n ⩾ 2 |
24 ) | 76
S (in) + 939
i=10
10 |
= 29(10n-2)192 , n ⩾ 2 |
25 ) | 75
S (in) + 515
i=10
10 |
= 28(33n-2)382 , n ⩾ 2 |
26 ) | 74
S (in) – 10
i=10
10 |
= 27(57n-2)572 , n ⩾ 2 |
27 ) | 73
S (in) + 364
i=10
10 |
= 26(82n-2)862 , n ⩾ 2 |
28 ) | 72
S (in) – 264
i=10
10 |
= 26(09n-2)062 , n ⩾ 2 |
29 ) | 71
S (in) – 91
i=10
10 |
= 25(36n-2)352 , n ⩾ 2 |
30 ) | 70
S (in) – 20
i=10
10 |
= 24(64n-2)642 , n ⩾ 2 |
31 ) | 69
S (in) – 50
i=10
10 |
= 23(93n-2)932 , n ⩾ 2 |
32 ) | 68
S (in) – 81
i=10
10 |
= 23(24n-2)232 , n ⩾ 2 |
33a ) | 67
S (in) – 313
i=10
10 |
= 22(55n-2)522 , n ⩾ 2 |
| 33b ) | 67
S
i=10 | (in) – 11 = 22(52.(n-1))22 , n ⩾ 1 |
34 ) | 66
S (in) – 646
i=10
10 |
= 21(87n-2)812 , n ⩾ 2 |
35 ) | 65
S (in) + 20
i=10
10 |
= 21(21n-2)212 , n ⩾ 2 |
36a ) | 64
S (in) – 515
i=10
10 |
= 20(55n-2)502 , n ⩾ 2 |
| 36b ) | 64
S
i=10 | (in) – 33 = 20(52.(n-1))02 , n ⩾ 1 |
37 ) | 63
S (in) + 839
i=10
10 |
= 19(90n-2)991 , n ⩾ 2 [primes possible] |
38 ) | 62
S (in) + 202
i=10
10 |
= 19(27n-2)291 , n ⩾ 2 [primes possible] |
39 ) | 61
S (in) + 364
i=10
10 |
= 18(64n-2)681 , n ⩾ 2 [primes possible] |
40 ) | 60
S (in) + 525
i=10
10 |
= 18(03n-2)081 , n ⩾ 2 [primes possible] |
41 ) | 59
S (in) + 525
i=10
10 |
= 17(42n-2)471 , n ⩾ 2 [primes possible] |
42 ) | 58
S (in) + 344
i=10
10 |
= 16(82n-2)861 , n ⩾ 2 [primes possible] |
43 ) | 57
S (in) + 202
i=10
10 |
= 16(24n-2)261 , n ⩾ 2 [primes possible] |
44a ) | 56
S (in) – 141
i=10
10 |
= 15(66n-2)651 , n ⩾ 2 [primes possible] |
| 44b ) | 56
S
i=10 | (in) = 15(62.(n-1))51 , n ⩾ 1 [primes possible] |
45 ) | 55
S (in) + 515
i=10
10 |
= 15(10n-2)151 , n ⩾ 2 [primes possible] |
46 ) | 54
S (in) – 30
i=10
10 |
= 14(54n-2)541 , n ⩾ 2 [primes possible] |
47 ) | 53
S (in) – 676
i=10
10 |
= 13(99n-2)931 , n ⩾ 2 [primes possible] |
48 ) | 52
S (in) – 323
i=10
10 |
= 13(46n-2)431 , n ⩾ 2 [primes possible] |
49 ) | 51
S (in) – 171
i=10
10 |
= 12(93n-2)921 , n ⩾ 2 [primes possible] |
50 ) | 50
S (in) – 20
i=10
10 |
= 12(42n-2)421 , n ⩾ 2 [primes possible] |
51 ) | 49
S (in) – 70
i=10
10 |
= 11(91n-2)911 , n ⩾ 2 [primes possible] |
52 ) | 48
S (in) – 121
i=10
10 |
= 11(42n-2)411 , n ⩾ 2 [primes possible] |
53 ) | 47
S (in) – 373
i=10
10 |
= 10(93n-2)901 , n ⩾ 2 [primes possible] |
54 ) | 46
S (in) – 626
i=10
10 |
= 10(46n-2)401 , n ⩾ 2 [primes possible] |
55 ) | 45
S (in)
i=10
10 |
= 92.n |
= 9(99n-2)999 , n ⩾ 2 |
| 56 ) | 44
S
i=10 | (in) + 14 = 9(54n-1)59 , n ⩾ 1 [primes possible] |
| 57 ) | 43
S
i=10 | (in) + 18 = 9(10n-1)19 , n ⩾ 1 [primes possible] |
| 58 ) | 42
S
i=10 | (in) + 10 = 8(66n-1)68 , n ⩾ 1 |
| 59 ) | 41
S
i=10 | (in) + 12 = 8(24n-1)28 , n ⩾ 1 |
| 60 ) | 40
S
i=10 | (in) + 12 = 7(82n-1)87 , n ⩾ 1 [primes possible] |
| 61 ) | 39
S
i=10 | (in) + 12 = 7(42n-1)47 , n ⩾ 1 [primes possible] |
| 62 ) | 38
S
i=10 | (in) + 11 = 7(03n-1)07 , n ⩾ 1 [primes possible] |
| 63 ) | 37
S
i=10 | (in) + 8 = 6(64n-1)66 , n ⩾ 1 |
| 64 ) | 36
S
i=10 | (in) + 5 = 6(27n-1)26 , n ⩾ 1 |
| 65 ) | 35
S
i=10 | (in) + 10 = 5(90n-1)95 , n ⩾ 1 |
| 66 ) | 34
S
i=10 | (in) + 5 = 52.n+1 = 5(55n-1)55 , n ⩾ 1 |
| 67 ) | 33
S
i=10 | (in) + 9 = 5(21n-1)25 , n ⩾ 1 |
| 68 ) | 32
S
i=10 | (in) + 1 = 4(87n-1)84 , n ⩾ 1 |
| 69 ) | 31
S
i=10 | (in) + 3 = 4(52.n-1)4 , n ⩾ 1 |
| 70 ) | 30
S
i=10 | (in) + 4 = 4(24n) = (42n)4 = 4(24n-1)24 , n ⩾ 1 |
| 71 ) | 29
S
i=10 | (in) + 3 = 3(93n) = (39n)3 = 3(93n-1)93 , n ⩾ 1 |
| 72 ) | 28
S
i=10 | (in) + 2 = 3(64n-1)63 , n ⩾ 1 [primes possible] |
| 73 ) | 27
S
i=10 | (in) = 3(36n-1)33 , n ⩾ 1 |
| 74 ) | 26
S
i=10 | (in) – 3 = 3(09n-1)03 , n ⩾ 1 |
| 75 ) | 25
S
i=10 | (in) + 2 = 2(82n) = (28n)2 , n ⩾ 1 |
| 76 ) | 24
S
i=10 | (in) – 3 = 2(57n-1)52 , n ⩾ 1 |
| 77 ) | 23
S
i=10 | (in) + 1 = 2(33n-1)32 = 2(32.n-1)2 , n ⩾ 1 |
| 78 ) | 22
S
i=10 | (in) + 4 = 2(10n-1)12 , n ⩾ 1 |
| 79 ) | 21
S
i=10 | (in) – 5 = 1(87n-1)81 , n ⩾ 1 [primes possible] |
| 80 ) | 20
S
i=10 | (in) – 4 = 1(62.n-1)1 , n ⩾ 1 [primes possible] |
| 81 ) | 19
S
i=10 | (in) – 4 = 1(46n-1)41 , n ⩾ 1 [primes possible] |
| 82 ) | 18
S
i=10 | (in) – 5 = 1(27n-1)21 , n ⩾ 1 [primes possible] |
| 83 ) | 17
S
i=10 | (in) – 7 = 1(09n-1)01 , n ⩾ 1 [primes possible] |
84a ) | 16
S (in) – 1
i=10
10 |
= (91n-1)9 |
= 9(19n-1)9 , n ⩾ 1 [primes possible] |
| 84b ) | 16
S
i=10 | (in) + 9 = (91n-1)9 , n ⩾ 1 [primes possible] |
85a ) | 15
S (in) – 5
i=10
10 |
= (75n-1)7 |
= 7(57n-1) , n ⩾ 1 [primes possible] |
| 85b ) | 15
S
i=10 | (in) + 7 = (75n-1)7 , n ⩾ 1 [primes possible] |
86 ) | 14
S (in)
i=10
10 |
= (60n-1)6 |
= 6(06n-1) , n ⩾ 1 |
87 ) | 13
S (in) – 6
i=10
10 |
= 4(64n-1) |
= (46n-1)4 , n ⩾ 1 |
| 88 ) | 12
S
i=10 | (in) = 32.n , n ⩾ 1 |
89 ) | 11
S (in) – 1
i=10
10 |
= 2(12n-1) |
= (21n-1)2 , n ⩾ 1 |
 Other Hugo Sánchez WONplates
 won84.htm won106.htm won153.htm
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