two consecutive numbers in an additive series plus an offset is equal to the larger numbers n value plus one to the size of the series.

Figured out the relationship between two consecutive numbers in a series and the next larger series.  It has to do with multiples of the triangular numbers.

https://github.com/BuckRogers1965/Examples/blob/master/Math/additiveSeries/additiveSeries-Relationships.c

As you can see in the following example program:

 #include <stdio.h>


 int main () {


   int i, j, n, x;

   for (x=1; x<20; x++) {
     printf("\tFor additive series of size %d\n", x);
     printf("\tn\tadd\tseries\tformula\t\tnext\tf(n-1)\tf(n)\toffset\n");
     for (i = 1, j = 0, n =1 ; n<10; n++, i+=x ){
       j+=i;
       printf("\t%d\t%d\t%d\t%d\t\t%d\t%d\t%d\t%d\n", n, i, j, (x*n*n-(x-2)*n)/2,
       (x*(n-1)*(n-1)-(x-2)*(n-1))/2 + (x*n*n-(x-2)*n)/2 +(-x*n*n +3*x*n +n*n -3*n -2*x +2)/2,
       (x*(n-1)*(n-1)-(x-2)*(n-1))/2, (x*n*n-(x-2)*n)/2, (-x*n*n +3*x*n +n*n -3*n -2*x +2)/2 );
     }
     printf("\n\n");
   }

 }




 For additive series of size 1
 n add series formula  next f(n-1) f(n) offset
 1 1 1 1  1 0 1 0
 2 2 3 3  4 1 3 0
 3 3 6 6  9 3 6 0
 4 4 10 10  16 6 10 0
 5 5 15 15  25 10 15 0
 6 6 21 21  36 15 21 0
 7 7 28 28  49 21 28 0
 8 8 36 36  64 28 36 0
 9 9 45 45  81 36 45 0


 For additive series of size 2
 n add series formula  next f(n-1) f(n) offset
 1 1 1 1  1 0 1 0
 2 3 4 4  5 1 4 0
 3 5 9 9  12 4 9 -1
 4 7 16 16  22 9 16 -3
 5 9 25 25  35 16 25 -6
 6 11 36 36  51 25 36 -10
 7 13 49 49  70 36 49 -15
 8 15 64 64  92 49 64 -21
 9 17 81 81  117 64 81 -28


 For additive series of size 3
 n add series formula  next f(n-1) f(n) offset
 1 1 1 1  1 0 1 0
 2 4 5 5  6 1 5 0
 3 7 12 12  15 5 12 -2
 4 10 22 22  28 12 22 -6
 5 13 35 35  45 22 35 -12
 6 16 51 51  66 35 51 -20
 7 19 70 70  91 51 70 -30
 8 22 92 92  120 70 92 -42
 9 25 117 117  153 92 117 -56


 For additive series of size 4
 n add series formula  next f(n-1) f(n) offset
 1 1 1 1  1 0 1 0
 2 5 6 6  7 1 6 0
 3 9 15 15  18 6 15 -3
 4 13 28 28  34 15 28 -9
 5 17 45 45  55 28 45 -18
 6 21 66 66  81 45 66 -30
 7 25 91 91  112 66 91 -45
 8 29 120 120  148 91 120 -63
 9 33 153 153  189 120 153 -84


 For additive series of size 5
 n add series formula  next f(n-1) f(n) offset
 1 1 1 1  1 0 1 0
 2 6 7 7  8 1 7 0
 3 11 18 18  21 7 18 -4
 4 16 34 34  40 18 34 -12
 5 21 55 55  65 34 55 -24
 6 26 81 81  96 55 81 -40
 7 31 112 112  133 81 112 -60
 8 36 148 148  176 112 148 -84
 9 41 189 189  225 148 189 -112


 For additive series of size 6
 n add series formula  next f(n-1) f(n) offset
 1 1 1 1  1 0 1 0
 2 7 8 8  9 1 8 0
 3 13 21 21  24 8 21 -5
 4 19 40 40  46 21 40 -15
 5 25 65 65  75 40 65 -30
 6 31 96 96  111 65 96 -50
 7 37 133 133  154 96 133 -75
 8 43 176 176  204 133 176 -105
 9 49 225 225  261 176 225 -140


 For additive series of size 7
 n add series formula  next f(n-1) f(n) offset
 1 1 1 1  1 0 1 0
 2 8 9 9  10 1 9 0
 3 15 24 24  27 9 24 -6
 4 22 46 46  52 24 46 -18
 5 29 75 75  85 46 75 -36
 6 36 111 111  126 75 111 -60
 7 43 154 154  175 111 154 -90
 8 50 204 204  232 154 204 -126
 9 57 261 261  297 204 261 -168


 For additive series of size 8
 n add series formula  next f(n-1) f(n) offset
 1 1 1 1  1 0 1 0
 2 9 10 10  11 1 10 0
 3 17 27 27  30 10 27 -7
 4 25 52 52  58 27 52 -21
 5 33 85 85  95 52 85 -42
 6 41 126 126  141 85 126 -70
 7 49 175 175  196 126 175 -105
 8 57 232 232  260 175 232 -147
 9 65 297 297  333 232 297 -196


 For additive series of size 9
 n add series formula  next f(n-1) f(n) offset
 1 1 1 1  1 0 1 0
 2 10 11 11  12 1 11 0
 3 19 30 30  33 11 30 -8
 4 28 58 58  64 30 58 -24
 5 37 95 95  105 58 95 -48
 6 46 141 141  156 95 141 -80
 7 55 196 196  217 141 196 -120
 8 64 260 260  288 196 260 -168
 9 73 333 333  369 260 333 -224


 For additive series of size 10
 n add series formula  next f(n-1) f(n) offset
 1 1 1 1  1 0 1 0
 2 11 12 12  13 1 12 0
 3 21 33 33  36 12 33 -9
 4 31 64 64  70 33 64 -27
 5 41 105 105  115 64 105 -54
 6 51 156 156  171 105 156 -90
 7 61 217 217  238 156 217 -135
 8 71 288 288  316 217 288 -189
 9 81 369 369  405 288 369 -252


 For additive series of size 11
 n add series formula  next f(n-1) f(n) offset
 1 1 1 1  1 0 1 0
 2 12 13 13  14 1 13 0
 3 23 36 36  39 13 36 -10
 4 34 70 70  76 36 70 -30
 5 45 115 115  125 70 115 -60
 6 56 171 171  186 115 171 -100
 7 67 238 238  259 171 238 -150
 8 78 316 316  344 238 316 -210
 9 89 405 405  441 316 405 -280


 For additive series of size 12
 n add series formula  next f(n-1) f(n) offset
 1 1 1 1  1 0 1 0
 2 13 14 14  15 1 14 0
 3 25 39 39  42 14 39 -11
 4 37 76 76  82 39 76 -33
 5 49 125 125  135 76 125 -66
 6 61 186 186  201 125 186 -110
 7 73 259 259  280 186 259 -165
 8 85 344 344  372 259 344 -231
 9 97 441 441  477 344 441 -308


 For additive series of size 13
 n add series formula  next f(n-1) f(n) offset
 1 1 1 1  1 0 1 0
 2 14 15 15  16 1 15 0
 3 27 42 42  45 15 42 -12
 4 40 82 82  88 42 82 -36
 5 53 135 135  145 82 135 -72
 6 66 201 201  216 135 201 -120
 7 79 280 280  301 201 280 -180
 8 92 372 372  400 280 372 -252
 9 105 477 477  513 372 477 -336


 For additive series of size 14
 n add series formula  next f(n-1) f(n) offset
 1 1 1 1  1 0 1 0
 2 15 16 16  17 1 16 0
 3 29 45 45  48 16 45 -13
 4 43 88 88  94 45 88 -39
 5 57 145 145  155 88 145 -78
 6 71 216 216  231 145 216 -130
 7 85 301 301  322 216 301 -195
 8 99 400 400  428 301 400 -273
 9 113 513 513  549 400 513 -364


 For additive series of size 15
 n add series formula  next f(n-1) f(n) offset
 1 1 1 1  1 0 1 0
 2 16 17 17  18 1 17 0
 3 31 48 48  51 17 48 -14
 4 46 94 94  100 48 94 -42
 5 61 155 155  165 94 155 -84
 6 76 231 231  246 155 231 -140
 7 91 322 322  343 231 322 -210
 8 106 428 428  456 322 428 -294
 9 121 549 549  585 428 549 -392


 For additive series of size 16
 n add series formula  next f(n-1) f(n) offset
 1 1 1 1  1 0 1 0
 2 17 18 18  19 1 18 0
 3 33 51 51  54 18 51 -15
 4 49 100 100  106 51 100 -45
 5 65 165 165  175 100 165 -90
 6 81 246 246  261 165 246 -150
 7 97 343 343  364 246 343 -225
 8 113 456 456  484 343 456 -315
 9 129 585 585  621 456 585 -420


 For additive series of size 17
 n add series formula  next f(n-1) f(n) offset
 1 1 1 1  1 0 1 0
 2 18 19 19  20 1 19 0
 3 35 54 54  57 19 54 -16
 4 52 106 106  112 54 106 -48
 5 69 175 175  185 106 175 -96
 6 86 261 261  276 175 261 -160
 7 103 364 364  385 261 364 -240
 8 120 484 484  512 364 484 -336
 9 137 621 621  657 484 621 -448


 For additive series of size 18
 n add series formula  next f(n-1) f(n) offset
 1 1 1 1  1 0 1 0
 2 19 20 20  21 1 20 0
 3 37 57 57  60 20 57 -17
 4 55 112 112  118 57 112 -51
 5 73 185 185  195 112 185 -102
 6 91 276 276  291 185 276 -170
 7 109 385 385  406 276 385 -255
 8 127 512 512  540 385 512 -357
 9 145 657 657  693 512 657 -476


 For additive series of size 19
 n add series formula  next f(n-1) f(n) offset
 1 1 1 1  1 0 1 0
 2 20 21 21  22 1 21 0
 3 39 60 60  63 21 60 -18
 4 58 118 118  124 60 118 -54
 5 77 195 195  205 118 195 -108
 6 96 291 291  306 195 291 -180
 7 115 406 406  427 291 406 -270
 8 134 540 540  568 406 540 -378
 9 153 693 693  729 540 693 -504