# [seqfan] Re: Sum of all valid substrings (VS)

Hugo van der Sanden human at google.com
Sat Jan 17 11:19:44 CET 2009

```I think the description should include "unique substrings" as a phrase (or
"set of"): I would have expected a(3) = 13 = 11 + 1 + 1, from the
description alone.

Hugo

2009/1/16 Maximilian Hasler <maximilian.hasler at gmail.com>

> The following has been sent to the OEIS:
>
> %I A154770
> %S A154770
> 10,11,12,15,21,24,30,33,36,45,54,63,72,81,90,99,108,127,176,283,407,
> %T A154770
> 458,578,733,849,1003,1117,1381,2044,2318,2953,4397,5435,6557,7964,9989,
> %U A154770
> 12279,16572,26426,36970,49970,67738,84716,100181,111599,139413,209606
> %N A154770 a(n+1) = A154771(a(n)) = sum of all "valid substrings" of
> a(n); a(1)=10 (least nontrivial choice).
> %C A154770 "Valid substrings" means all numbers that appear as
> substring of n (written in decimal system). Starting values < 10 would
> yield a constant sequence.
> %H A154770 E. Angelini, <a
> href="http://list.seqfan.eu/pipermail/seqfan/2009-January/000536.html">Sum
> of all valid substrings (VS)</a>, SeqFan list, Jan 16 2009
> %e A154770 a(1) = 10 has { 0, 1, 10 } as substrings,
> %e A154770 a(2) = 0+1+10 = 11 has { 1, 11 } as substrings,
> %e A154770 a(3) = 1+11 = 12 has { 1, 2, 12 } as substrings,
> %e A154770 a(4) = 1+2+12 = 15 has { 1, 5, 15 } as substrings.
> %o A154770 (PARI) A154770( n=2, a=10 ) = { local(d); while( n--,
>
> a=vecsort(concat(vector(d=#Str(a),i,vector(d,j,a%10^j)+(d--&!a\=10))),,8);a*=vector(#a,i,1)~);
> a }
> %o A154770 print1(a=10);for(i=1,50,print1("," a=A154770(,a)))
> %Y A154770 Cf. A004207, A154771.
> %K A154770 base,easy,nonn
> %O A154770 1,1
> %A A154770 Eric Angelini (eric.angelini(AT)kntv.be) and M. F. Hasler
> (Maximilian.Hasler(AT)gmail.com), Jan 16 2009
>
>
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>

```