# [seqfan] Re: Last two digits added --> new integer rhymes

Andrew Weimholt andrew.weimholt at gmail.com
Sat Feb 15 23:11:40 CET 2014

```If you modify the definition to...

a(1) = 1
a(2) = 2
for n>2
a(n) = smallest unused natural number ending in sum mod 18 of last two
digits of sequence

then the new sequence IS a permutation of the naturals.

The first difference in the two sequences is at a(146)=20 (vs 218)
following a(145)=99.
100 shows up at a(1240) in the new sequence.

Andrew

On Fri, Feb 14, 2014 at 11:16 PM, Jim Nastos <nastos at gmail.com> wrote:

> Eric,
>   It is still a nice sequence. Here are 500 terms.
> J
>
> 1 2 3 5 8 13 4 7 11 12 23 15 6 111 22 14 25 17 18 9 117 28 10 21 33 16 27
> 19 110 31 24 26 38 211 32 35 48 112 43 37 210 41 45 29 311 42 36 39 212 53
> 58 113 34 47 411 52 57 312 63 49 213 44 68 114 55 310 51 46 410 61 67 313
> 54 59 214 65 511 62 78 115 56 611 72 69 215 66 412 73 510 71 88 116 77 314
> 75 512 83 711 82 610 81 79 216 87 315 76 413 64 710 91 810 101 121 93 612
> 103 123 85 513 74 811 92 911 102 122 84 712 133 86 414 95 514 105 125 97
> 316 107 127 89 217 98 317 108 118 99 218 109 119 910 131 94 613 104 124 96
> 415 106 126 128 1010 141 135 138 1011 132 145 129 1111 142 136 139 812 143
> 137 1110 151 146 1210 161 147 1211 152 157 912 153 148 1012 163 149 713 134
> 167 813 144 158 913 154 159 614 155 1310 171 168 714 165 1311 162 178 515
> 156 1411 172 169 615 166 1112 173 1410 181 179 416 177 814 175 1212 183
> 1511 182 1510 191 1610 201 221 193 1312 203 223 185 1013 164 1710 231 174
> 1611 192 1711 202 222 184 1412 233 176 1113 194 1213 204 224 186 914 195
> 1014 205 225 187 715 196 815 206 226 188 516 197 616 207 227 189 417 198
> 517 208 228 1810 241 235 238 1811 232 245 199 318 209 219 1910 251 236 229
> 1911 242 246 2010 261 237 2110 271 248 1512 243 247 2011 252 257 1612 253
> 258 1313 234 267 1413 244 268 1114 255 2210 281 239 1712 263 249 1513 254
> 259 1214 265 2111 262 278 915 256 2211 272 269 1015 266 1812 273 2310 291
> 2410 301 321 283 2311 282 2510 331 264 2610 341 275 1912 293 2012 303 323
> 285 1613 274 2411 292 2511 302 322 284 2112 333 276 1713 294 1813 304 324
> 286 1314 295 1414 305 325 277 1514 335 288 716 287 1115 296 1215 306 326
> 298 617 308 328 2710 351 336 279 816 297 916 307 327 289 717 338 2611 332
> 345 299 418 309 319 2810 361 337 2910 371 348 2212 343 347 2711 342 346
> 3010 381 329 2811 352 357 2312 353 358 1913 334 367 2013 344 368 1614 355
> 3110 391 3210 401 421 363 339 2412 373 3310 431 354 349 2113 364 3410 441
> 365 2911 362 378 1315 356 3011 372 359 1714 375 2512 383 3111 382 3510 451
> 366 2612 393 2712 403 423 385 2213 374 3211 392 3311 402 422 384 2812
>
>
> On Fri, Feb 14, 2014 at 3:20 PM, Eric Angelini <Eric.Angelini at kntv.be
> >wrote:
>
> > > it is not a permutation of the Naturals. For instance, 100 (or any
> > number ending in "100") does not appear in this sequence since the last
> two
> > digits of S will never sum to 0, 00, or 100
> >
> > ... Jim Nastos is right, sorry...
> > Shame on me...
> > My first attempt to produce a permutation
> > failed too; I had tried the sequence
> > where the word "ending", hereunder,
> > was replaced by "beginning".
> > Didn't check enough, though, the
> > "new" definition...
> >
> >
> > > Le 14 févr. 2014 à 20:23, "Eric Angelini" <Eric.Angelini at kntv.be> a
> > écrit :
> > >
> > >
> > > Hello SeqFans,
> > > here is another nice permutation of the Naturals;
> > >
> > > S=1,2,3,5,8,13,4,7,11,12,23,15,6,111,
> > > 22,14,25,17,18,9,117,28,10,21,33,16,
> > > 27,19,110,31,...
> > >
> > > a(1)=1
> > > a(2)=2
> > > a(n) is the smallest integer not yet in S
> > > ending with the sum of the last two digits
> > > of S.
> > >
> > > We see after 13 that S was extended with 4
> > > as 1+3=4 and 4 was not yet in S;
> > >
> > > We see after 4 that S was extended with 7
> > > as 3+4=7 and 7 was not yet in S;
> > >
> > > We see after 7 that S was extended with 11
> > > as 4+7=11 and 11 was not yet in S;
> > >
> > > We see after 11 that S was extended with 12
> > > as 1+1=2 and 2 being already in S was replaced by 12, smallest integer
> > > not yet in S and ending with "2";
> > >
> > > We see after 12 that S was extended with 23
> > > as 1+2=3 and 3 being already in S, as 13, we used the smallest integer
> > > not yet in S and ending with "3";
> > > ...
> > > Best,
> > > É.
> > >
> > >
> > > _______________________________________________
> > >
> > > Seqfan Mailing list - http://list.seqfan.eu/
> >
> > _______________________________________________
> >
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
> _______________________________________________
>
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>
```