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 A270270 The number of n-digit numbers in A270048. 2
 4, 6, 17, 45, 131, 381, 1123, 3334, 9973, 29991, 90601, 274746, 835844, 2549874, 7797469, 23894630, 73358721, 225589420, 694745922, 2142444490, 6614766985, 20445300258, 63256499281, 195890524486, 607136782567, 1883199766658, 5845450449249, 18156369461770, 56429925440218 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Conjecture: lim_{n -> infinity} a(n)/a(n-1) = sqrt(10). (Similar to A265108, where we count the n-digit numbers of A264847, pluritriangular numbers.) It is not possible to count some hundred-digit numbers without a "climbing algorithm" (see also Program and Links). LINKS Francesco Di Matteo, Table of n, a(n) for n = 1..300 Francesco Di Matteo, Calculating the A270270 terms EXAMPLE a(1) = 4 because in A270048 there are 4 numbers with 1 digit (0, 1, 3, 6). a(2) = 6 because in A270048 there are 8 numbers with 2 digits (10, 20, 32, 46, 62, 80). PROG (Python) # init values seq = [4] # the output list somme = [4] # the n-value list after the adding of the last seq term last = [10] # last a(n) term, or the first k-digit number (10 with k=2) # CLIMBING loop, put a bigger value if you want for n in range (1, 30): k = (len(somme)+1) # the digits number limit = 10**k # the newest value to achieve base = (somme[-1]+1)*k # this is the (n+1)*k value hypo = seq[-1]*3 # to obtain rapidly the limite value rid = 10**(len(str(hypo))-1) # the reduction factor # Adjustment LOOP # s, p, m = 0, 0, 0 while s < 1: diff_1 = (base + (base + (k*(hypo-1))))*float(hypo)/2 tot = last[-1] + diff_1 if tot < limit: p = 1 if m == 1 and rid > 1: m = 0; rid = rid/10 hypo = hypo + rid else: diff_2 = (base + (base + (k*(hypo-2))))*float(hypo-1)/2 tot = last[-1] + diff_2 if tot > limit: m = 1 if p == 1 and rid > 1: p = 0; rid = rid/10 hypo = hypo - rid else: s = 1 # escape value # lists updating seq.append(hypo) somme.append(somme[-1]+ hypo) last.append(last[-1]+ diff_1) # if you want to prove the conjecture values, uncomment next line #print(seq[-1], float(seq[-1])/seq[-2]) print(seq) CROSSREFS Cf. A264847, A265108, A270048. Sequence in context: A034492 A125691 A302878 * A211949 A079803 A061361 Adjacent sequences: A270267 A270268 A270269 * A270271 A270272 A270273 KEYWORD nonn,base AUTHOR Francesco Di Matteo, Mar 14 2016 STATUS approved

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Last modified April 16 05:35 EDT 2024. Contains 371697 sequences. (Running on oeis4.)