This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
2 + 20 = 22 which is the concatenation of 2 and 2. 2 + 20 + 198 = 220 which is the concatenation of 2, 2 and 0. 2 + 20 + 198 + 1981 = 2201 which is the concatenation of 2, 2, 0 and 1.
a(n,show=1,a=2,c=a,d=[c])={for(n=2,n,show&&print1(a",");a=-c+c=c*10+d[1];d=concat(d[^1],if(n>2,digits(a))));a} \\ M. F. Hasler, Feb 22 2018
9 + 90 = 99 which is the concatenation of 9 and 9. 9 + 90 + 891 = 990 which is the concatenation of 9, 9 and 0. 9 + 90 + 891 + 8918 = 9908 which is the concatenation of 9, 9, 0 and 8. From n = 3 on, a(n) can be computed directly as c(n) - c(n-1), cf. formula: a(3) = 990 - 99 = 891, a(4) = 9908 - 990 = 8918, etc. - _M. F. Hasler_, Feb 22 2018
a(n,show=1,a=9,c=a,d=[a])={for(n=2,n,show&&print1(a",");a=-c+c=c*10+d[1];d=concat(d[^1],if(n>2,digits(a))));a} \\ M. F. Hasler, Feb 22 2018
For n=5, 1 + 2 + 3 + 7 + 14 = 1_2 + 10_2 + 11_2 + 111_2 + 1110_2 = 11011_2, the first five bits of the sequence.
a[1]=1;a[2]=2;a[n_]:=a[n]=FromDigits[Flatten[IntegerDigits[#,2]&/@Table[a[k],{k,n-1}]][[;;n]],2]-Total@Table[a[m],{m,n-1}]
Table[a[l],{l,40}] (* Giorgos Kalogeropoulos, Mar 30 2021 *)
def aupton(terms):
alst, bstr = [1, 2], "110"
for n in range(3, terms+1):
an = int(bstr[:n], 2) - int(bstr[:n-1], 2)
alst, bstr = alst + [an], bstr + bin(an)[2:]
return alst
print(aupton(34)) # Michael S. Branicky, Mar 30 2021
def first_bits(n, seq); seq.map { |i| i.to_s(2) }.join[0...n].to_i(2) end
def next_term(n, seq); first_bits(n,seq) - first_bits(n-1,seq) end
def a308092_list(n)
(3..n).reduce([1,2]) { |accum, i| accum << next_term(i, accum) }
end
a(2) = 10 as a(1) + 10 = 1 + 10 = 11 which is a substring of "1" + "10" = "110". a(3) = 98 as a(1) + a(2) + 98 = 1 + 10 + 98 = 109 which is a substring of "1" + "10" + "98" = "11098". a(4) = 767 as a(1) + a(2) + a(3) + 767 = 1 + 10 + 98 + 767 = 876 which is a substring of "1" + "10" + "98" + "767" = "11098767".
from itertools import islice
def agen(): # generator of terms
s, mink, aset, concat = 1, 2, {1}, "1"
yield from [1]
while True:
an = mink
while an in aset or not str(s+an) in concat+str(an): an += 1
aset.add(an); s += an; concat += str(an); yield an
while mink in aset: mink += 1
print(list(islice(agen(), 60))) # Michael S. Branicky, Feb 08 2024
3 + 30 = 33 which is the concatenation of 3 and 3. 3 + 30 + 297 = 330 which is the concatenation of 3, 3 and 0. 3 + 30 + 297 + 2972 = 3302 which is the concatenation of 3, 3, 0 and 2. From n = 3 on, a(n) can be computed directly as c(n) - c(n-1), cf. formula: a(3) = 330 - 33 = 297, a(4) = 3302 - 330 = 2972, etc. - _M. F. Hasler_, Feb 22 2018
a(n,show=1,a=3,c=a,d=[a])={for(n=2,n,show&&print1(a",");a=-c+c=c*10+d[1];d=concat(d[^1],if(n>2,digits(a))));a} \\ M. F. Hasler, Feb 22 2018
8 + 80 = 88 which is the concatenation of 8 and 8. 8 + 80 + 792 = 880 which is the concatenation of 8, 8 and 0. 8 + 80 + 792 + 7927 = 8807 which is the concatenation of 8, 8, 0 and 7. From n = 3 on, a(n) can be computed directly as c(n) - c(n-1), cf. formula: a(3) = 880 - 88 = 792, a(4) = 8807 - 880 = 7927, etc. - _M. F. Hasler_, Feb 22 2018
a(n,show=1,a=8,c=a,d=[a])={for(n=2,n,show&&print1(a",");a=-c+c=c*10+d[1];d=concat(d[^1],if(n>2,digits(a))));a} \\ M. F. Hasler, Feb 22 2018
a(1) + a(2) = 1 + 10 = 11 and “11” is visible in [1,10] a(2) + a(3) = 10 + 99 = 109 and “109” is visible in [10,99] a(3) + a(4) = 99 + 11 = 110 and “110” is visible in [1,10] a(4) + a(5) = 11 + 80 = 91 and “91” is visible in [99,11] a(5) + a(6) = 80 + 19 = 99 and “99” is visible in [99] a(6) + a(7) = 19 + 61 = 80 and “80” is visible in [80] ...
Nest[Function[a, Append[a, Block[{k = 1, d}, While[Nand[FreeQ[a, k], SequenceCount[Flatten@ IntegerDigits[Append[a, k]], IntegerDigits[a[[-1]] + k]] > 0], k++]; k]]], {1}, 75] (* Michael De Vlieger, Feb 22 2018 *)
a(n,show=1,a=1,s=a,u=[a],t,m)={for(n=2,n, show&&print1(a","); for(k=u[1]+1,oo, setsearch(u,k)&&next;m=Mod(a+k,10^#Str(a+k));t=s*10^#Str(k)+k; until(k>=t\=10,t==m&&(a=k)&&break(2)));s=s*10^#Str(a)+a;u=setunion(u,[a]); u[2]==u[1]+1&&u=u[^1]);a} \\ M. F. Hasler, Feb 22 2018
4 + 40 = 44 which is the concatenation of 4 and 4. 4 + 40 + 396 = 440 which is the concatenation of 4, 4, and 0. 4 + 40 + 396 + 3963 = 4403 which is the concatenation of 4, 4, 0 and 3. From n = 3 on, a(n) can be computed directly as c(n) - c(n-1), cf. formula: a(3) = 440 - 4 = 396, a(4) = 4403 - 440 = 3963, etc. - _M. F. Hasler_, Feb 22 2018
a(n,show=1,a=4,c=a,d=[a])={for(n=2,n,show&&print1(a",");a=-c+c=c*10+d[1];d=concat(d[^1],if(n>2,digits(a))));a} \\ M. F. Hasler, Feb 22 2018
5 + 50 = 55 which is the concatenation of 5 and 5. 5 + 50 + 495 = 550 which is the concatenation of 5, 5 and 0. 5 + 50 + 495 + 4954 = 5504 which is the concatenation of 5, 5, 0 and 4. From n = 3 on, a(n) can be computed directly as c(n) - c(n-1), cf. formula: a(3) = 550 - 55 = 495, a(4) = 5504 - 550 = 4954, etc. - _M. F. Hasler_, Feb 22 2018
a(n,show=1,a=5,c=a,d=[a])={for(n=2,n,show&&print1(a",");a=-c+c=c*10+d[1];d=concat(d[^1],if(n>2,digits(a))));a} \\ M. F. Hasler, Feb 22 2018
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