Pythagorean / Fibonacci Progressions

Pythagorean triples provide integer solutions to the equation:

a^2+b^2=c^2

Consider the equations:

a^2+b^2+c^2...+m^2=n^2

It is easily possible to formulate many series that provide integer solutions for these progressions, the simplest of which are shown below:

Consider a^2+b^2+c^2+d^2=e^2,

let a = an odd integer greater than 1
let b = a+1
let c = 2a

Then d = ((c*(b+c))/2
and e = d+1

Then, consider a^2+b^2+c^2+d^2+e^2=f^2

a, b, and c are as above
now let d = b+c (the beginning of an off-set Fibonacci series)
then e = (d*(c+d))/2
and f = e+1

Then, consider a^2+b^2+c^2+d^2+e^2+f^2=g^2

a, b, c, and d are as above
now let e = c+d (the second member of the off-set Fibonnacci series)
then f = (e*(d+e))/2
and g = f+1

etc.

Some examples:

a b c d e f g h i j k l m n o p q r s t u
1 3 4 6 30 31
2 3 4 6 10 80 81 2.66666666666667
3 3 4 6 10 16 208 209 2.6
4 3 4 6 10 16 26 546 547 2.625
5 3 4 6 10 16 26 42 1428 1429 2.61538461538462
6 3 4 6 10 16 26 42 68 3740 3741 2.61904761904762
7 3 4 6 10 16 26 42 68 110 9790 9791 2.61764705882353
8 3 4 6 10 16 26 42 68 110 178 25632 25633 2.61818181818182
9 3 4 6 10 16 26 42 68 110 178 288 67104 67105 2.61797752808989
10 3 4 6 10 16 26 42 68 110 178 288 466 175682 175683 2.61805555555556
11 3 4 6 10 16 26 42 68 110 178 288 466 754 459940 459941 2.61802575107296
12 3 4 6 10 16 26 42 68 110 178 288 466 754 1220 1204140 1204141 2.61803713527852
13 3 4 6 10 16 26 42 68 110 178 288 466 754 1220 1974 3152478 3152479 2.61803278688525
14 3 4 6 10 16 26 42 68 110 178 288 466 754 1220 1974 3194 8253296 8253297 2.61803444782168
15 3 4 6 10 16 26 42 68 110 178 288 466 754 1220 1974 3194 5168 21607408 21607409 2.61803381340013
16 3 4 6 10 16 26 42 68 110 178 288 466 754 1220 1974 3194 5168 8362 56568930 56568931 2.61803405572755
17 3 4 6 10 16 26 42 68 110 178 288 466 754 1220 1974 3194 5168 8362 13530 148099380 148099381 2.61803396316671


a b c d e f g h i j k l m n o p q r s t u
1 5 6 10 80 81
2 5 6 10 16 208 209 2.6
3 5 6 10 16 26 546 547 2.625
4 5 6 10 16 26 42 1428 1429 2.61538461538462
5 5 6 10 16 26 42 68 3740 3741 2.61904761904762
6 5 6 10 16 26 42 68 110 9790 9791 2.61764705882353
7 5 6 10 16 26 42 68 110 178 25632 25633 2.61818181818182
8 5 6 10 16 26 42 68 110 178 288 67104 67105 2.61797752808989
9 5 6 10 16 26 42 68 110 178 288 466 175682 175683 2.61805555555556
10 5 6 10 16 26 42 68 110 178 288 466 754 459940 459941 2.61802575107296
11 5 6 10 16 26 42 68 110 178 288 466 754 1220 1204140 1204141 2.61803713527852
12 5 6 10 16 26 42 68 110 178 288 466 754 1220 1974 3152478 3152479 2.61803278688525
13 5 6 10 16 26 42 68 110 178 288 466 754 1220 1974 3194 8253296 8253297 2.61803444782168
14 5 6 10 16 26 42 68 110 178 288 466 754 1220 1974 3194 5168 21607408 21607409 2.61803381340013
15 5 6 10 16 26 42 68 110 178 288 466 754 1220 1974 3194 5168 8362 56568930 56568931 2.61803405572755
16 5 6 10 16 26 42 68 110 178 288 466 754 1220 1974 3194 5168 8362 13530 148099380 148099381 2.61803396316671
17 5 6 10 16 26 42 68 110 178 288 466 754 1220 1974 3194 5168 8362 13530 21892 387729212 387729213 2.6180339985218


a b c d e f g h i j k l m n o p q r s t u
1 7 8 14 154 155
2 7 8 14 22 396 397 2.57142857142857
3 7 8 14 22 36 1044 1045 2.63636363636364
4 7 8 14 22 36 58 2726 2727 2.61111111111111
5 7 8 14 22 36 58 94 7144 7145 2.62068965517241
6 7 8 14 22 36 58 94 152 18696 18697 2.61702127659575
7 7 8 14 22 36 58 94 152 246 48954 48955 2.61842105263158
8 7 8 14 22 36 58 94 152 246 398 128156 128157 2.61788617886179
9 7 8 14 22 36 58 94 152 246 398 644 335524 335525 2.61809045226131
10 7 8 14 22 36 58 94 152 246 398 644 1042 878406 878407 2.61801242236025
11 7 8 14 22 36 58 94 152 246 398 644 1042 1686 2299704 2299705 2.61804222648752
12 7 8 14 22 36 58 94 152 246 398 644 1042 1686 2728 6020696 6020697 2.61803084223013
13 7 8 14 22 36 58 94 152 246 398 644 1042 1686 2728 4414 15762394 15762395 2.61803519061584
14 7 8 14 22 36 58 94 152 246 398 644 1042 1686 2728 4414 7142 41266476 41266477 2.6180335296783
15 7 8 14 22 36 58 94 152 246 398 644 1042 1686 2728 4414 7142 11556 108037044 108037045 2.61803416409969
16 7 8 14 22 36 58 94 152 246 398 644 1042 1686 2728 4414 7142 11556 18698 282844646 282844647 2.61803392177224
17 7 8 14 22 36 58 94 152 246 398 644 1042 1686 2728 4414 7142 11556 18698 30254 740496904 740496905 2.61803401433308