cp's OEIS Frontend

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.

Showing 1-3 of 3 results.

A105753 Lexicographically earliest sequence of positive integers with the property that a(a(n)) = a(1)+a(2)+...+a(n).

Original entry on oeis.org

1, 3, 4, 8, 6, 22, 9, 16, 53, 11, 133, 13, 279, 15, 573, 69, 18, 1233, 20, 2486, 23, 44, 4995, 25, 10059, 27, 20145, 29, 40319, 31, 80669, 33, 161371, 35, 322777, 37, 645591, 39, 1291221, 41, 2582483, 43, 5165009, 5039, 46, 10335103, 48
Offset: 1

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Author

Eric Angelini, Aug 13 2006

Keywords

Comments

The Fibonacci 9-step numbers referenced in the Noe-Post paper are in A104144. - T. D. Noe, Oct 27 2008

Examples

			Sequence reads from the beginning:
- at position a(1)=1 we see the sum of all previously written terms [indeed, nil + 1=1]
- at position a(2)=3 we see the sum of all previously written terms [indeed, 1+ 3=4]
- at position a(3)=4 we see the sum of all previously written terms [indeed, 1+3+4=8]
- at position a(4)=8 we see the sum of all previously written terms [indeed, 1+3+4+8=16]
- at position a(5)=6 we see the sum of all previously written terms [indeed, 1+3+4+8+6=22]
- at position a(6)=22 we see the sum of all previously written terms [indeed, 1+3+4+8+6+22=44 and 44 is the 22nd term of S]
etc.
		

Crossrefs

Extensions

More terms from Max Alekseyev, Aug 14 2006
Edited by Max Alekseyev, Mar 08 2015

A121173 Sequence S with property that for n in S, a(n) = a(1) + a(2) +...+ a(n-1) and for n not in S, a(n) = n+1.

Original entry on oeis.org

2, 2, 4, 8, 6, 22, 8, 52, 10, 114, 12, 240, 14, 494, 16, 1004, 18, 2026, 20, 4072, 22, 8166, 24, 16356, 26, 32738, 28, 65504, 30, 131038, 32, 262108, 34, 524250, 36, 1048536, 38, 2097110, 40, 4194260, 42, 8388562, 44, 16777168, 46, 33554382
Offset: 1

Views

Author

Max Alekseyev, Aug 15 2006

Keywords

Comments

a(1)=1 cannot happen, so the sequence S starts with a(1)=2.
Note that a(n)=a(1)+a(2)+...+a(n-1) can hold even if n is not in S. The smallest example is n=3.
All terms are even. - Reinhard Zumkeller, Nov 06 2013

Crossrefs

Programs

  • Haskell
    a121173 n = a121173_list !! (n-1)
    a121173_list = f 1 [] where
       f x ys = y : f (x + 1) (y : ys) where
         y = if x `elem` ys then sum ys else x + 1
    -- Reinhard Zumkeller, Nov 06 2013
    
  • Mathematica
    s={2};Do[If[MemberQ[s,n],m=Total[s],m=n+1];AppendTo[s,m],{n,2,46}];s (* James C. McMahon, Oct 13 2024 *)
  • Python
    from itertools import count, islice
    def agen(): # generator of terms
        S, s, an = {2}, 2, 2
        for n in count(2):
            yield an
            an = s if n in S else n+1
            s += an
            S.add(an)
    print(list(islice(agen(), 50))) # Michael S. Branicky, Oct 13 2024

Formula

a(2*n) = A145654(n+1). - Reinhard Zumkeller, Nov 06 2013
a(2*n+1) = 2*n+2.
From Colin Barker, Jan 30 2016: (Start)
a(n) = 2*(2^(n/2+1)-2)-n for n even.
a(n) = n+1 for n odd.
a(n) = -a(n-1)+3*a(n-2)+3*a(n-3)-2*a(n-4)-2*a(n-5) for n>5.
G.f.: 2*x*(1+2*x) / ((1-x)*(1+x)^2*(1-2*x^2)). (End)
E.g.f.: (x - 4)*cosh(x) + 4*cosh(sqrt(2)*x) + (1 - x)*sinh(x). - Stefano Spezia, Oct 14 2024

A121175 Sequence S with the following properties: (i) a(1)=2; (ii) for n is S, a(n)=a(1)+a(2)+...+a(n-1); (iii) for n not in S, a(n)=the smallest number different from a(1), ..., a(n-1) not breaking property (ii).

Original entry on oeis.org

2, 2, 4, 8, 3, 7, 26, 52, 10, 114, 12, 240, 14, 494, 16, 1004, 18, 2026, 20, 4072, 22, 8166, 24, 16356, 27, 32739, 65478, 29, 130985, 31, 262001, 33, 524035, 35, 1048105, 37, 2096247, 39, 4192533, 41, 8385107, 43, 16770257, 45, 33540559
Offset: 1

Views

Author

Max Alekseyev, Aug 15 2006

Keywords

Comments

Taking a(1)=2 makes all terms distinct except for a(1)=a(2)=2

Crossrefs

Showing 1-3 of 3 results.