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 A137743 Number T(m,n) of different strings of length n obtained from "123...m" by iteratively duplicating any substring; formatted as upper right triangle. 12
 1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 8, 8, 4, 1, 1, 16, 21, 13, 5, 1, 1, 32, 54, 40, 19, 6, 1, 1, 64, 138, 119, 66, 26, 7, 1, 1, 128, 355, 348, 218, 100, 34, 8, 1, 1, 256, 924, 1014, 700, 360, 143, 43, 9, 1, 1, 512, 2432, 2966, 2218, 1246, 555, 196, 53, 10, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS The sequence T(m,m+3) = 1,8,21,40,66,100,143,196,260,... = A137742. LINKS FORMULA T(m,n)=0 for n < m, T(m,m)=T(1,n)=1, T(m,m+1)=m, T(m,m+2)=C(m+2,2)-2 = A034856(m); T(2,2+n)=2^n. For m > 3, T(m,m+2) = T(m-1,m+1) + T(m,m+1) + T(m+1,m+1). - Thomas Anton, Nov 05 2018 EXAMPLE The full matrix is: [ 1, 1, 1, 1, 1, 1, 1,_ 1,_ 1,__ 1,__ 1,...] (= A000012) [[], 1, 2, 4, 8,16,32, 64,128, 256, 512,...] (= A000079) [[],[], 1, 3, 8,21,54,138,355, 924,2432,...] (= A135473) [[],[],[], 1, 4,13,40,119,348,1014,2966,...] (= A137744) [[],[],[],[], 1, 5,19, 66,218, 700,2218,...] (= A137745) [[],[],[],[],[], 1, 6, 26,100, 360,1246,...] (= A137746) [[],[],[],[],[],[], 1,_ 7, 34, 143, 555,...] (= A137747) ... PROG (PARI) A135473(Nmax, d=3 /* # digits in the initial string = row of the triangular matrix */)={ local( t, tt, ee, lsb, L=vector(Nmax, i, []) /*store separately words of given length*/, w=log(d-.5)\log(2)+1/* bits required to code d distinct digits*/); L[d]=Set(sum(i=1, d-1, i<<(w*i))); for( i=d, Nmax-1, for( j=1, #t=eval(L[i]), forstep( ee=w, w*i, w, /*upper cutting point*/ forstep( len=w, min(ee, w*(Nmax-i)), w, /* length of substring */ lsb = bitand( tt=t[j], 1<

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Last modified June 14 13:39 EDT 2021. Contains 345025 sequences. (Running on oeis4.)