## Monday, January 20, 2014

### 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.

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);
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

```