A "doublely-recursive" sequence

[f.firoozbakht at sci.ui.ac.ir]

Hello,Leroy Quet ,

You did not make any mistake to calculating the first few terms.

for n=0,...,100

  a(n)= :
    1,1,2,4,1,2,4,9,2,4,9,30,15,4,9,30,97,84,84,26,15,127,
    308,30,15,127,898,24,913,97,24,913,308,69,2,4,9,30,2996, 
    4217,308,560,97,69,1040,11,69,868,9,30,2996,7327,14566,
    13618,39,11544,26,1938,913,2875,64695,4292,97,4288,1,2,4,9, 
    30,2996,64721,11852,13618,11852,84,30,2996,211582,13698,3909,
    146861,26,1938,2972,4217,39494,433,676711,481203,146861,223434, 
    2972,2875,676711,11,69,146861,8,26,1938,7327

and {n,a(n)}=:

    {0, 1},{1, 1},{2, 2},{3, 4},{4, 1},{5, 2},{6, 4},{7, 9},{8, 2},{9,4},
    {10, 9},{11, 30},{12, 15},{13, 4},{14, 9},{15, 30},{16, 97},{17, 84},
    {18, 84},{19, 26},{20, 15},{21, 127},{22, 308},{23, 30},{24, 15},
    {25, 127},{26, 898},{27, 24},{28, 913}, {29, 97},{30, 24},{31, 913},
    {32, 308},{33, 69},{34, 2},{35, 4},{36, 9},{37, 30},{38, 2996},
    {39, 4217},{40, 308},{41, 560},{42, 97},{43, 69},{44, 1040},{45, 11},
    {46, 69},{47, 868},{48, 9},{49, 30},{50, 2996},{51, 7327},{52, 14566},
    {53, 13618},{54, 39}, {55, 11544},{56, 26},{57, 1938},{58, 913},
    {59, 2875}, {60,64695},{61, 4292},{62, 97},{63, 4288},{64, 1},{65, 2},
    {66, 4},{67, 9},{68, 30},{69, 2996},{70, 64721}, {71, 11852},{72,13618}
    {73, 11852},{74, 84},{75, 30},{76, 2996},{77, 211582},{78, 13698},
    {79, 3909},{80, 146861},{81, 26},{82, 1938},{83, 2972}, {84, 4217},
    {85, 39494},{86, 433},{87, 676711},{88, 481203},{89, 146861},
    {90, 223434},{91, 2972},{92, 2875},{93, 676711}, {94, 11},{95, 69},
    {96, 146861},{97, 8},{98, 26},{99, 1938},{100, 7327}


If the sequence has closed form, finding of it,is difficult.


Regards,

Farideh Firoozbakht
UniverSity of Isfahan,Iran 
f.firoozbakht at sci.ui.ac.ir


Quoting Leroy Quet <qqquet at mindspring.com>:
> I just posted the below to sci.math.
> (This particular sequence is, in my opinion, far from the most 
> interesting/fundamental sequence I have yet to post to the EIS. But my
> 
> computer's copy of Mathematica is not working properly after a crash;
> and 
> I am too lazy even to suggest the other sequences on this email list. 
> ...but maybe someday..)
> 
> ---- 
> 
> sci.math post:"a[m] = sum{j=0 to a[m-1](mod m)} a[j]"
> 
> If I did not make a mistake calculating the first few terms by hand,
> here is a recursively-defined sequence which is not in the EIS yet.
> 
> a[0] = 1;
> 
> and for m >= 1, 
> 
> a[m] = sum{j=0 to a[m-1](mod m)} a[j]
> 
> Ascii-art:
>        a[m-1](mod m)
>         ---
>         \
> a[m] =  /      a[k]
>         ---
>         k=0
> 
> 
> And,  0 <= a[m-1](mod m) <= m-1.
> 
> 
> The sequence begins (maybe):
> 
> 1, 1, 2, 4, 1, 2, 4, 9, 2, 4, 9, 30, 15,...
> 
> What can be said about this sequence? Does it have a closed-form (ie
> nonrecursive) representation?
> 
> Also, other sequences can be based on the same idea: a partial sum
> somehow involving earlier terms, BOTH in the general term and in the
> limit of the indexes used in the sum.
> 
> Thanks,
> Leroy Quet
>  
> 


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