Semiprimonacci numbers, analogue of A078465

[Jonathan Post]

Re your point c)
it had explicitly occurred to me that this is a new transform.

Not every substitution of "semiprime" for prime is well-defined, let
alone interesting. But some are.  I ask, to compensate for my biases.

I am quite interested in your conjecture:  "that for any increasing
seq. d of positive integers (i.e.  any set),  my nonLinGenFib() gives
the Number of compositions of n into sums of such numbers."

If so, this is an important property of the new transform.

re: PS2, though I flunked Russian in college, I do read a lot in
translation, and my father's father was a Pasternak.

Best,

Jonathan Vos Post

On 6/22/08, Maximilian Hasler <maximilian.hasler at gmail.com> wrote:
> On Sun, Jun 22, 2008 at 14:40, Jonathan Post <jvospost3 at gmail.com> wrote:
>  > Is this interesting to anyone but me?
>  > 1, 1, 1, 1, 1, 1, 3, 5, 5, 6, 7, 7, 13, 17, 18, 20, 30, 37, 47
>  > Semiprimonacci numbers:
>  > a(n)=a(n-4)+a(n-6)+a(n-9)+a(n-10)+a(n-14)+...+a(n-sp(k))+... until n >
>  > sp(k), where sp(k) is the k-th semprime = A001358(k);
>  > a(1)=a(2)=a(3)=1.
>
>
> I'd say : maybe not during the "Summer Rules" period...? ;-)
>
>  Also, I think
>  a)
>  A078465 Primonacci numbers 1,1,1,2,2,4,...
>  is neither very natural concerning the name (the seq. has nothing in
>  common with Fibonacci numbers except for the values a(1) and a(2)),
>  nor natural for the start: why not a(0)=0, a(1)=1 (and then use the
>  general rule), like for Fibonacci?
>  Then it gives A023360: see below
>
>  b)
>  if you define a1=a2=a3 = 1 then a4 = 0 (we have to stop the sum before
>  n=4 since a0 is undefined (or =0 as in Fibonacci)) and your seq. would
>  go :
>  1, 1, 1, 0, 1, 1, 2, 1, 2, 2, 5, 4, 5, 4, 10, 11, 15, 12, 20, 24, 37,...
>  Fixing one more term=1 I get something closer to your seq., but still
>  not the same values:
>  1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 5, 5, 6, 7, 10, 13, 17, 18, 23, 30, 40, 47...
>  But wouldn't it be more natural to take a0=0, a1=1 and then use the rules ?
>  with this I get:
>  0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 3, 0, 2, 2, 6, 3, 6, 3, 11, 10, 16,
>  10, 23, 23, 40...
>
>  c)
>  I understand that your motivation is to transpose every existing seq.
>  definition using "primes" to the analogue using "semiprimes", but
>  (especially in this case) isn't this a bit arbitrary ? shouldnt it
>  then logically be done for any increasing sequence? (e.g., squares,
>  even/odd numbers (maybe the same), cubes, factorials, ...)
>
>  i.e. (highly unoptimized):
>  nonLinGenFib( d, a=[0,1] /*initial vals*/) = {
>   for( i = #a, d[ #d ], a=concat( a,0);
>   for( j=1, #d, if( d[j] + 1 >= #a, break);
>   a[#a] += a[ #a - d[j] ])); a }
>
>  nonLinGenFib(vector(10,i,prime(i)),[1,1])
>  /* this is "primonacci" */
>  [1, 1, 1, 2, 2, 4, 5, 8, 12, 16, 26, 36, 55, 81, 118, 177, ...]
>
>  nonLinGenFib(vector(10,i,prime(i)),[0,1])
>  /* this would be the natural version of "primo..." */
>  [0, 1, 0, 1, 1, 1, 3, 2, 6, 6, 10, 16, 20, 35, 46, 72, 105,...]
>  =A023360: Number of compositions of n into sums of primes.
>
>  nonLinGenFib([4,6,9,10,14,15,21],[0,1])
>  /* this would be the natural version of semiprimo...*/
>  [0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 3, 0, 2, 2, 6, 3, 6, 3, 11, 10,...]
>  I conjecture (i.e. don't want to write out the proof...) that this is
>  NEW SEQ: Number compositions of n into sums of semiprimes.
>
>  nonLinGenFib([4,6,9,10,14,15,21,22,25],[1,1])
>  /*the 1,1 version*/
>  [1, 1, 0, 0, 1, 1, 1, 1, 1, 2, 4, 3, 2, 4, 8, 9, 9, 9, 14, 21, 26, 26,
>  33, 46, 63, 74]
>
>  nonLinGenFib([4,6,9,10,14,15,21,22,25],[1,1])
>  /*the 1,1,1 version*/
>  [1, 1, 1, 0, 1, 1, 2, 1, 2, 2, 5, 4, 5, 4, 10, 11, 15, 12, 20, 24, 37,
>  36, 49...]
>
>  nonLinGenFib([4,6,9,10,14,15,21,22,25,26,33],[1,1,1,1])
>  /*the 1,1,1,1 version : this should be yours */
>  [1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 5, 5, 6, 7, 10, 13, 17, 18, 23, 30, ...]
>
>  So I get other values , which I beleive correct. In order to get a 3
>  as first element >1 you have to impose 9 initial 1's, but then you get
>  a series of 4's after that 3.
>
>  nonLinGenFib(vector(5,i,i^2)) \\ "squaronacci"??
>  [0, 1, 1, 1, 1, 2, 3, 4, 5, 7, 11, 16, 22, 30, 43, 62, 88,...]
>  =A006456: Number of compositions of n into sums of squares.
>
>  nonLinGenFib(vector(5,i,i^3)) \\ "cubonacci"??
>  [0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 14, 18, 23...]
>  = A023358 : Number of compositions into sums of cubes.
>
>  I conjecture that for any increasing seq. d of positive integers (i.e.
>  any set),
>  my nonLinGenFib() gives the Number of compositions of n into sums of
>  such numbers.
>
>  Maximilian
>
>  PS1: the following version allows to produce linear sequences by
>  appending a number of zeros to the vector of differences:
>
>  nonLinGenFib( d, a=[0,1] /*initial vals*/) = {
>   for( i = #a, max( vecmax(d), #d ), a=concat( a,0);
>   for( j=1, #d, if( d[j] + 1 >= #a || !d[j], break);
>   a[#a] += a[ #a - d[j] ])); a }
>
>  nonLinGenFib([1,2,0,0,0,0,0,0,0])
>  => [0, 1, 1, 2, 3, 5, 8, 13, 21, 34]
>
>  PS2 [unrelated]: yes, of course, you found the correct author for the
>  Russian poem !
>