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 1-10 of 25 results. Next

A134409 Second differences of A006336.

Original entry on oeis.org

0, 1, 1, 0, 2, 0, 3, 3, 0, 5, 5, 0, 8, 0, 11, 11, 0, 16, 0, 21, 21, 0, 29, 29, 0, 40, 0, 51, 51, 0, 67, 67, 0, 88, 0, 109, 109, 0, 138, 0, 167, 167, 0, 207, 207, 0, 258, 0, 309, 309, 0, 376, 0, 443, 443, 0, 531, 531, 0, 640, 0, 749, 749, 0, 887, 887, 0, 1054, 0, 1221, 1221, 0, 1428
Offset: 1

Views

Author

Reinhard Zumkeller, Oct 24 2007

Keywords

Comments

a(n) = A006336(n+2)-2*A006336(n+1)-A006336(n) = A134408(n+1)-A134408(n);
apart from 0 same range as A006336;
A001950(n) = Min(m: a(m) = A006336(n)).

Crossrefs

Cf. A001950, A006336, A134408.

A134408 First differences of A006336.

Original entry on oeis.org

1, 1, 2, 3, 3, 5, 5, 8, 11, 11, 16, 21, 21, 29, 29, 40, 51, 51, 67, 67, 88, 109, 109, 138, 167, 167, 207, 207, 258, 309, 309, 376, 443, 443, 531, 531, 640, 749, 749, 887, 887, 1054, 1221, 1221, 1428, 1635, 1635, 1893, 1893, 2202, 2511, 2511, 2887, 2887, 3330, 3773
Offset: 1

Views

Author

Reinhard Zumkeller, Oct 24 2007

Keywords

Comments

a(n) = A006336(n+1) - A006336(n);
a(n+1) - a(n) = A134409(n).

A001950 Upper Wythoff sequence (a Beatty sequence): a(n) = floor(n*phi^2), where phi = (1+sqrt(5))/2.

Original entry on oeis.org

2, 5, 7, 10, 13, 15, 18, 20, 23, 26, 28, 31, 34, 36, 39, 41, 44, 47, 49, 52, 54, 57, 60, 62, 65, 68, 70, 73, 75, 78, 81, 83, 86, 89, 91, 94, 96, 99, 102, 104, 107, 109, 112, 115, 117, 120, 123, 125, 128, 130, 133, 136, 138, 141, 143, 146, 149, 151, 154, 157
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Indices at which blocks (1;0) occur in infinite Fibonacci word; i.e., n such that A005614(n-2) = 0 and A005614(n-1) = 1. - Benoit Cloitre, Nov 15 2003
A000201 and this sequence may be defined as follows: Consider the maps a -> ab, b -> a, starting from a(1) = a; then A000201 gives the indices of a, A001950 gives the indices of b. The sequence of letters in the infinite word begins a, b, a, a, b, a, b, a, a, b, a, ... Setting a = 0, b = 1 gives A003849 (offset 0); setting a = 1, b = 0 gives A005614 (offset 0). - Philippe Deléham, Feb 20 2004
a(n) = n-th integer which is not equal to the floor of any multiple of phi, where phi = (1+sqrt(5))/2 = golden number. - Philippe LALLOUET (philip.lallouet(AT)wanadoo.fr), May 09 2007
Write A for A000201 and B for the present sequence (the upper Wythoff sequence, complement of A). Then the composite sequences AA, AB, BA, BB, AAA, AAB, ..., BBB, ... appear in many complementary equations having solution A000201 (or equivalently, the present sequence). Typical complementary equations: AB=A+B (=A003623), BB=A+2B (=A101864), BBB=3A+5B (=A134864). - Clark Kimberling, Nov 14 2007
Apart from the initial 0 in A090909, is this the same as that sequence? - Alec Mihailovs (alec(AT)mihailovs.com), Jul 23 2007
If we define a base-phi integer as a positive number whose representation in the golden ratio base consists only of nonnegative powers of phi, and if these base-phi integers are ordered in increasing order (beginning 1, phi, ...), then it appears that the difference between the n-th and (n-1)-th base-phi integer is phi-1 if and only if n belongs to this sequence, and the difference is 1 otherwise. Further, if each base-phi integer is written in linear form as a + b*phi (for example, phi^2 is written as 1 + phi), then it appears that there are exactly two base-phi integers with b=n if and only if n belongs to this sequence, and exactly three base-phi integers with b=n otherwise. - Geoffrey Caveney, Apr 17 2014
Numbers with an odd number of trailing zeros in their Zeckendorf representation (A014417). - Amiram Eldar, Feb 26 2021
Numbers missing from A066096. - Philippe Deléham, Jan 19 2023

Examples

			From _Paul Weisenhorn_, Aug 18 2012 and Aug 21 2012: (Start)
a(14) = floor(14*phi^2) = 36; a'(14) = floor(14*phi)=22;
with r=9 and j=1: a(13+1) = 34 + 2 = 36;
with r=8 and j=1: a'(13+1) = 21 + 1 = 22.
k=6 and a(5)=13 < n <= a(6)=15
a(14) = 3*14 - 6 = 36; a'(14) = 2*14 - 6 = 22;
a(15) = 3*15 - 6 = 39; a'(15) = 2*15 - 6 = 24. (End)

References

  • Claude Berge, Graphs and Hypergraphs, North-Holland, 1973; p. 324, Problem 2.
  • Eric Friedman, Scott M. Garrabrant, Ilona K. Phipps-Morgan, A. S. Landsberg and Urban Larsson, Geometric analysis of a generalized Wythoff game, in Games of no Chance 5, MSRI publ. Cambridge University Press, 2019.
  • Martin Gardner, Penrose Tiles to Trapdoor Ciphers, W. H. Freeman, 1989; see p. 107.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • I. M. Yaglom, Two games with matchsticks, pp. 1-7 of Qvant Selecta: Combinatorics I, Amer Math. Soc., 2001.

Links

Crossrefs

a(n) = greatest k such that s(k) = n, where s = A026242.
Complement of A000201 or A066096.
A002251 maps between A000201 and A001950, in that A002251(A000201(n)) = A001950(n), A002251(A001950(n)) = A000201(n).
Cf. A001622, A026352, A004976, A004919.
Let A = A000201, B = A001950. Then AA = A003622, AB = A003623, BA = A035336, BB = A101864.
First differences give (essentially) A076662.
Bisections: A001962, A001966.
Cf. A014417, A329825.
The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A000201 as the parent: A000201, A001030, A001468, A001950, A003622, A003842, A003849, A004641, A005614, A014675, A022342, A088462, A096270, A114986, A124841. - N. J. A. Sloane, Mar 11 2021

Programs

  • Haskell
    a001950 n = a000201 n + n  -- Reinhard Zumkeller, Mar 10 2013
  • Magma
    [Floor(n*((1+Sqrt(5))/2)^2): n in [1..80]]; // Vincenzo Librandi, Nov 19 2016
  • Maple
    A001950 := proc(n)
    floor(n*(3+sqrt(5))/2) ;
end proc:
seq(A001950(n),n=0..40) ; # R. J. Mathar, Jul 16 2024
  • Mathematica
    Table[Floor[N[n*(1+Sqrt[5])^2/4]], {n, 1, 75}]
Array[ Floor[ #*GoldenRatio^2] &, 60] (* Robert G. Wilson v, Apr 17 2010 *)
  • PARI
    a(n)=floor(n*(sqrt(5)+3)/2)
  • PARI
    A001950(n)=(sqrtint(n^2*5)+n*3)\2 \\ M. F. Hasler, Sep 17 2014
  • Python
    from math import isqrt
def A001950(n): return (n+isqrt(5*n**2)>>1)+n # Chai Wah Wu, Aug 10 2022

Formula

a(n) = n + floor(n*phi). In general, floor(n*phi^m) = Fibonacci(m-1)*n + floor(Fibonacci(m)*n*phi). - Benoit Cloitre, Mar 18 2003
a(n) = n + floor(n*phi) = n + A000201(n). - Paul Weisenhorn and Philippe Deléham
Append a 0 to the Zeckendorf expansion (cf. A035517) of n-th term of A000201.
a(n) = A003622(n) + 1. - Philippe Deléham, Apr 30 2004
a(n) = Min(m: A134409(m) = A006336(n)). - Reinhard Zumkeller, Oct 24 2007
If a'=A000201 is the ordered complement (in N) of {a(n)}, then a(Fib(r-2) + j) = Fib(r) + a(j) for 0 < j <= Fib(r-2), 3 < r; and a'(Fib(r-1) + j) = Fib(r) + a'(j) for 0 < j <= Fib(r-2), 2 < r. - Paul Weisenhorn, Aug 18 2012
With a(1)=2, a(2)=5, a'(1)=1, a'(2)=3 and 1 < k and a(k-1) < n <= a(k) one gets a(n)=3*n-k, a'(n)=2*n-k. - Paul Weisenhorn, Aug 21 2012

Extensions

Corrected by Michael Somos, Jun 07 2000

A005185 Hofstadter Q-sequence: a(1) = a(2) = 1; a(n) = a(n-a(n-1)) + a(n-a(n-2)) for n > 2.

Original entry on oeis.org

1, 1, 2, 3, 3, 4, 5, 5, 6, 6, 6, 8, 8, 8, 10, 9, 10, 11, 11, 12, 12, 12, 12, 16, 14, 14, 16, 16, 16, 16, 20, 17, 17, 20, 21, 19, 20, 22, 21, 22, 23, 23, 24, 24, 24, 24, 24, 32, 24, 25, 30, 28, 26, 30, 30, 28, 32, 30, 32, 32, 32, 32, 40, 33, 31, 38, 35, 33, 39, 40, 37, 38, 40, 39
Offset: 1

Views

Author

Simon Plouffe and N. J. A. Sloane, May 20 1991

Keywords

Comments

Rate of growth is not known. In fact it is not even known if this sequence is defined for all positive n.
Roman Pearce, Aug 29 2014, has computed that a(n) exists for n <= 10^10. - N. J. A. Sloane
a(n) exists for n <= 3*10^10. - M. Eric Carr, Jul 02 2023

Examples

			a(18) = 11 because a(17) is 10 and a(16) is 9, so we take a(18 - 10) + a(18 - 9) = a(8) + a(9) = 5 + 6 = 11.

References

  • B. W. Conolly, "Meta-Fibonacci sequences," in S. Vajda, editor, Fibonacci and Lucas Numbers and the Golden Section. Halstead Press, NY, 1989, pp. 127-138.
  • R. K. Guy, Unsolved Problems in Number Theory, Sect. E31.
  • D. R. Hofstadter, Goedel, Escher, Bach: an Eternal Golden Braid, Random House, 1980, p. 138.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • S. Vajda, Fibonacci and Lucas Numbers and the Golden Section, Wiley, 1989, see p. 129.
  • S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 129.

Links

Crossrefs

Cf. A004001, A005206, A005374, A005375, A005378.
Cf. A005379, A070867, A226222, A239913, A284019.
Cf. A081827 (first differences).
Cf. A226244, A226245 (record values and where they occur).
See A244477 for a different start.

Programs

  • C
    #include 
#define LIM 20
int Qa[LIM];
int Q(int n){if (n==1 || n==2){return 1;} else{return Qa[n-Qa[n-1]]+Qa[n-Qa[n-2]];}}
int main(){int i;printf("n\tQ\n");for(i=1; iGonzalo Ciruelos, Aug 01 2013
  • Haskell
    a005185 n = a005185_list !! (n-1)
a005185_list = 1 : 1 : zipWith (+)
   (map a005185 $ zipWith (-) [3..] a005185_list)
   (map a005185 $ zipWith (-) [3..] $ tail a005185_list)
-- Reinhard Zumkeller, Jun 02 2013, Sep 15 2011
  • Magma
    I:=[1,1]; [n le 2 select I[n] else Self(n-Self(n-1))+Self(n-Self(n-2)): n in [1..90]]; // Vincenzo Librandi, Aug 08 2014
  • Maple
    A005185 := proc(n) option remember;
    if n<=2 then 1
    elif n > procname(n-1) and n > procname(n-2) then
        RETURN(procname(n-procname(n-1))+procname(n-procname(n-2)));
    else
        ERROR(" died at n= ", n);
    fi; end proc;
# More generally, the following defines the Hofstadter-Huber sequence Q(r,s) - N. J. A. Sloane, Apr 15 2014
r:=1; s:=2;
a:=proc(n) option remember; global r,s;
if n <= s then  1
else
    if (a(n-r) <= n) and (a(n-s) <= n) then
    a(n-a(n-r))+a(n-a(n-s));
    else lprint("died with n =",n); return (-1);
    fi; fi; end;
[seq(a(n), n=1..100)];
  • Mathematica
    a[1]=a[2]=1; a[n_]:= a[n]= a[n -a[n-1]] + a[n -a[n-2]]; Table[ a[n], {n,70}]
  • MuPAD
    q:=proc(n) option remember; begin if n<=2 then 1 else q(n-q(n-1))+q(n-q(n-2)) end_if; end_proc: q(i)$i=1..100; // Zerinvary Lajos, Apr 03 2007
  • PARI
    {a(n)= local(A); if(n<1, 0, A=vector(n,k,1); for(k=3, n, A[k]= A[k-A[k-1]]+ A[k-A[k-2]]); A[n])} /* Michael Somos, Jul 16 2007 */
  • Python
    from functools import lru_cache
@lru_cache(maxsize=None)
def a(n):
    if n < 3: return 1
    return a(n - a(n-1)) + a(n - a(n-2))
print([a(n) for n in range(1, 75)]) # Michael S. Branicky, Jul 26 2021
  • Sage
    @CachedFunction
def a(n):
    if (n<3): return 1
    else: return a(n -a(n-1)) + a(n -a(n-2))
[a(n) for n in (1..70)] # G. C. Greubel, Feb 13 2020
  • Scheme
    (define q (lambda (n) (cond ( (eqv? n 0) 1) ( (eqv? n 1) 1) ( #t (+ (q (- n (q (- n 1)))) (q (- n (q (- n 2)))))))))
  • Scheme
    ;; An implementation of memoization-macro definec can be found for example in: http://oeis.org/wiki/Memoization
(definec (A005185 n) (if (<= n 2) 1 (+ (A005185 (- n (A005185 (- n 1)))) (A005185 (- n (A005185 (- n 2)))))))
;; Antti Karttunen, Mar 22 2017

A005206 Hofstadter G-sequence: a(0) = 0; a(n) = n - a(a(n-1)) for n > 0.

Original entry on oeis.org

0, 1, 1, 2, 3, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 9, 10, 11, 11, 12, 12, 13, 14, 14, 15, 16, 16, 17, 17, 18, 19, 19, 20, 21, 21, 22, 22, 23, 24, 24, 25, 25, 26, 27, 27, 28, 29, 29, 30, 30, 31, 32, 32, 33, 33, 34, 35, 35, 36, 37, 37, 38, 38, 39, 40, 40, 41, 42, 42, 43, 43, 44, 45, 45, 46, 46, 47
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Rule for finding n-th term: a(n) = An, where An denotes the Fibonacci antecedent to (or right shift of) n, which is found by replacing each F(i) in the Zeckendorf expansion (obtained by repeatedly subtracting the largest Fibonacci number you can until nothing remains) by F(i-1) (A1=1). For example: 58 = 55 + 3, so a(58) = 34 + 2 = 36. - Diego Torres (torresvillarroel(AT)hotmail.com), Nov 24 2002
From Albert Neumueller (albert.neu(AT)gmail.com), Sep 28 2006: (Start)
A recursively built tree structure can be obtained from the sequence (see Hofstadter, p. 137):
14 15 16 17 18 19 20 21
\ / / \ / \ / /
9 10 11 12 13
\ / / \ /
6 7 8
\ / /
\ / /
\ / /
4 5
\ /
\ /
\ /
\ /
\ /
3
/
2
\ /
1
To construct the tree: node n is connected with the node a(n) below
n
/
a(n)
For example, since a(7) = 4:
7
/
4
If the nodes of the tree are read from bottom to top, left to right, one obtains the positive integers: 1, 2, 3, 4, 5, 6, ... The tree has a recursive structure, since the construct
/
x
\ /
x
can be repeatedly added on top of its own ends, to construct the tree from its root: e.g.,
/
x
/ \ /
x x
\ / /
x x
\ /
\ /
x
When moving from a node to a lower connected node, one is moving to the parent. Parent node of n: floor((n+1)/tau). Left child of n: floor(tau*n). Right child of n: floor(tau*(n+1))-1 where tau=(1+sqrt(5))/2. (See the Sillke link.)
(End)
The number n appears A001468(n) times; A001468(n) = floor((n+1)*Phi) - floor(n*Phi) with Phi = (1 + sqrt 5)/2. - Philippe Deléham, Sep 22 2005
Number of positive Wythoff A-numbers A000201 not exceeding n. - N. J. A. Sloane, Oct 09 2006
Number of positive Wythoff B-numbers < A000201(n+1). - N. J. A. Sloane, Oct 09 2006
From Bernard Schott, Apr 23 2022: (Start)
Properties coming from the 1st problem proposed during the 45th Czech and Slovak Mathematical Olympiad in 1996 (see IMO Compendium link):
-> a(n) >= a(n-1) for any positive integer n,
-> a(n) - a(n-1) belongs to {0,1},
-> No integer n exists such that a(n-1) = a(n) = a(n+1). (End)
For n >= 1, find n in the Wythoff array (A035513). a(n) is the number that precedes n in its row, using the preceding column of the extended Wythoff array (A287870) if n is at the start of the (unextended) row. - Peter Munn, Sep 17 2022
See my 2023 publication on Hofstadter's G-sequence for a proof of the equality of (a(n)) with the sequence A073869. - Michel Dekking, Apr 28 2024
From Michel Dekking, Dec 16 2024: (Start)
Focus on the pairs of duplicate values, i.e., the pairs (a(n-1),a(n)) with a(n-1) = a(n). Directly from Theorem 1 in Kimberling and Stolarsky (2016) one derives that the m-th pair of duplicate values (a(n-1),a(n)) occurs at n = U(m), where U = 2,5,7,10,... is the upper Wythoff sequence. For example, (3,3) is the second pair, and occurs at U(2) = 5.
This property can be used to give a simple construction for (a(n)) -- ignoring the superfluous a(0) = 0.
Let 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,... be the sequence of positive natural numbers. Double all the elements of the lower Wythoff sequence (L(n)) = 1,3,4,6,8,9,11,....:
(x(n)) := 1,1, 2, 3,3, 4,4, 5, 6,6, 7, 8,8, 9,9, 10,....
Claim: the result is (a(n)). This follows since because of the doubling, the m-th pair of duplicate values (a(n-1),a(n)) occurs in x at n = L(m) + m = U(m). The second equality by a well-known formula.
It follows from this by Theorem 1 of K&S, that a(n-1) = x(n-1), and a(n) = x(n) if n = U(m), for all m. But since L and U are complementary sequences, a(n) = x(n) will also hold if n = L(k), for all k. For example, L(4) = 6, and a(6) = x(6) = 4.
Corollary: for n >= 2 replace every pair of duplicate values (a(n-1),a(n)) by 0, and all the remaining elements of (a(n)) by 1. Then the result is the Fibonacci word 0,1,0,0,1,0,1,0,0... This is implied by the fact that L gives the positions of the 0s, and U the position of the 1's in the Fibonacci word. (End)
For all n >= 0, a(n) <= A005374(n), as proved in Letouzey-Li-Steiner link. Last equality occurs at n = 12, while a(n) < A005374(n) afterwards. - Pierre Letouzey, Feb 20 2025

References

  • D. R. Hofstadter, Goedel, Escher, Bach: an Eternal Golden Braid, Random House, 1980, p. 137.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Apart from initial terms, same as A060143. Cf. A123070.
a(n):=Sum{k=1..n} h(k), n >= 1, with h(k):= A005614(k-1) and a(0):=0.
Cf. A060144, A019446, A072649, A213676, A000201.
Cf. A035513, A287870.

Programs

  • Haskell
    a005206 n = a005206_list !! n
a005206_list = 0 : zipWith (-) [1..] (map a005206 a005206_list)
-- Reinhard Zumkeller, Feb 02 2012, Aug 07 2011
  • Haskell
    a005206 = sum . zipWith (*) a000045_list . a213676_row . a000201 . (+ 1)
-- Reinhard Zumkeller, Mar 10 2013
  • Magma
    [Floor((n+1)*(1+Sqrt(5))/2)-n-1: n in [0..80]]; // Vincenzo Librandi, Nov 19 2016
  • Maple
    H:=proc(n) option remember; if n=0 then 0 elif n=1 then 1 else n-H(H(n-1)); fi; end proc: seq(H(n),n=0..76);
  • Mathematica
    a[0] = 0; a[n_] := a[n] = n - a[a[n - 1]]; Array[a, 77, 0]
(* Second program: *)
Fold[Append[#1, #2 - #1[[#1[[#2]] + 1 ]] ] &, {0}, Range@ 76] (* Michael De Vlieger, Nov 13 2017 *)
  • PARI
    first(n)=my(v=vector(n)); v[1]=1; for(k=2,n, v[k]=k-v[v[k-1]]); concat(0,v) \\ Charles R Greathouse IV, Sep 02 2015
  • Python
    from math import isqrt
def A005206(n): return (n+1+isqrt(5*(n+1)**2)>>1)-n-1 # Chai Wah Wu, Aug 09 2022

Formula

a(n) = floor((n+1)*tau) - n - 1 = A000201(n+1)-n-1, where tau = (1+sqrt(5))/2; or a(n) = floor(sigma*(n+1)) where sigma = (sqrt(5)-1)/2.
a(0)=0, a(1)=1, a(n) = n - a(floor(n/tau)). - Benoit Cloitre, Nov 27 2002
a(n) = A019446(n) - 1. - Reinhard Zumkeller, Feb 02 2012
a(n) = n - A060144(n+1). - Reinhard Zumkeller, Apr 07 2012
a(n) = Sum_{k=1..A072649(m)} A000045(m)*A213676(m,k): m=A000201(n+1). - Reinhard Zumkeller, Mar 10 2013
From Pierre Letouzey, Sep 09 2015: (Start)
a(n + a(n)) = n.
a(n) + a(a(n+1) - 1) = n.
a(0) = 0, a(n+1) = a(n) + d(n) and d(0) = 1, d(n+1)=1-d(n)*d(a(n)). (End)
a(n) = A293688(n)/(n+1) for n >= 0 (conjectured). - Enrique Navarrete, Oct 15 2017
A generalization of Diego Torres's 2002 comment as a formula: if n = Sum_{i in S} A000045(i+1), where S is a set of positive integers, then a(n) = Sum_{i in S} A000045(i). - Peter Munn, Sep 28 2022
Conjectures from Chunqing Liu, Aug 01 2023: (Start)
a(A000201(n)-1) = n-1.
a(A001950(n)-1) = a(A001950(n)) = A000201(n). (End)

Extensions

a(0) = 0 added in the Name by Bernard Schott, Apr 23 2022

A005228 Sequence and first differences (A030124) together list all positive numbers exactly once.

Original entry on oeis.org

1, 3, 7, 12, 18, 26, 35, 45, 56, 69, 83, 98, 114, 131, 150, 170, 191, 213, 236, 260, 285, 312, 340, 369, 399, 430, 462, 495, 529, 565, 602, 640, 679, 719, 760, 802, 845, 889, 935, 982, 1030, 1079, 1129, 1180, 1232, 1285, 1339, 1394, 1451, 1509, 1568, 1628, 1689
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

This is the lexicographically earliest sequence that together with its first differences (A030124) contains every positive integer exactly once.
Hofstadter introduces this sequence in his discussion of Scott Kim's "FIGURE-FIGURE" drawing. - N. J. A. Sloane, May 25 2013
A225850(a(n)) = 2*n-1, cf. A167151. - Reinhard Zumkeller, May 17 2013
In view of the definition of A075326: start with a(0) = 0, and extend by rule that the next term is the sum of the predecessor and the most recent non-member of the sequence. - Reinhard Zumkeller, Oct 26 2014

Examples

			Sequence reads 1 3 7 12 18 26 35 45..., differences are 2 4 5, 6, 8, 9, 10 ... and the point is that every number not in the sequence itself appears among the differences. This property (together with the fact that both the sequence and the sequence of first differences are increasing) defines the sequence!

References

  • E. Angelini, "Jeux de suites", in Dossier Pour La Science, pp. 32-35, Volume 59 (Jeux math'), April/June 2008, Paris.
  • D. R. Hofstadter, Goedel, Escher, Bach: an Eternal Golden Braid, Random House, 1980, p. 73.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A030124 (complement), A037257, A056731, A056738, A140778, A225687.
The following are a group of related sequences: A005132, A006509, A037257, A037258, A037259, A081145, A093903, A099004, A100707, A129198, A129199, A140778, A225376, A225377, A225378, A225385, A225386, A225387.
Cf. A075326, A095115.
Cf. A225850, A232746, A232747 (inverse), A232739, A232740, A232750 and also permutation pair A232751/A232752 constructed from this sequence and its complement.
Cf. A001651 (analog with sums instead of differences), A121229 (analog with products).
The same recurrence a(n) = a(n-1) + c(n-1) with different starting conditions: A061577 (starting with 2), A022935 (3), A022936 (4), A022937 (5), A022938 (6).
Related recurrences:
a(n-1) + c(n+1) - A022953, A022954.
a(n-1) + c(n) - A022946 to A022952.
a(n-1) + c(n-2) - A022940, A022941.
a(n-2) + c(n-1) - A022942 to A022944.
a(n-2) + c(n-2) - A022939.
a(n-3) + c(n-3) - A022955.
a(n-4) + c(n-4) - A022956.
a(n-5) + c(n-5) - A022957.

Programs

  • Haskell
    a005228 = scanl (+) 1 a030124
a030124 = go 1 a005228 where go x ys | x < head ys = x     : go (x + 1) ys
                                     | otherwise   = x + 1 : go (x + 2) (tail ys)
-- Maks Verver, Jun 30 2025
  • Maple
    maxn := 5000; h := array(1..5000); h[1] := 1; a := [1]; i := 1; b := []; for n from 2 to 1000 do if h[n] <> 1 then b := [op(b), n]; j := a[i]+n; if j < maxn then a := [op(a),j]; h[j] := 1; i := i+1; fi; fi; od: a; b; # a is A005228, b is A030124.
A030124 := proc(n)
    option remember;
    local a,fnd,t ;
    if n <= 1 then
        op(n+1,[2,4]) ;
    else
        for a from procname(n-1)+1 do
            fnd := false;
            for t from 1 to n+1 do
                if A005228(t)  = a then
                    fnd := true;
                    break;
                end if;
            end do:
            if not fnd then
                return a;
            end if;
        end do:
    end if;
end proc:
A005228 := proc(n)
    option remember;
    if n <= 2 then
        op(n,[1,3]) ;
    else
        procname(n-1)+A030124(n-2) ;
    end if;
end proc: # R. J. Mathar, May 19 2013
  • Mathematica
    a = {1}; d = 2; k = 1; Do[ While[ Position[a, d] != {}, d++ ]; k = k + d; d++; a = Append[a, k], {n, 1, 55} ]; a
(* Second program: *)
(* Program from Larry Morris, Jan 19 2017: *)
d = 3; a = {1, 3, 7, 12, 18}; While[ Length[a = Join[a, a[[-1]] + Accumulate[Range[a[[d]] + 1, a[[++d]] - 1]]]] < 50]; a
(* Comment: This adds as many terms to the sequence as there are numbers in each set of sequential differences. Consequently, the list of numbers it produces may be longer than the limit provided. With the limit of 50 shown, the sequence produced has length 60. *)
  • PARI
    A005228(n,print_all=0,s=1,used=0)={while(n--,used += 1<M. F. Hasler, Feb 05 2013

Formula

a(n) = a(n-1) + c(n-1) for n >= 2, where a(1)=1, a( ) increasing, c( ) = complement of a( ) (c is the sequence A030124).
Let a(n) = this sequence, b(n) = A030124 prefixed by 0. Then b(n) = mex{ a(i), b(i) : 0 <= i < n}, a(n) = a(n-1) + b(n) + 1. (Fraenkel)
a(1) = 1, a(2) = 3; a( ) increasing; for n >= 3, if a(q) = a(n-1)-a(n-2)+1 for some q < n then a(n) = a(n-1) + (a(n-1)-a(n-2)+2), otherwise a(n) = a(n-1) + (a(n-1)-a(n-2)+1). - Albert Neumueller (albert.neu(AT)gmail.com), Jul 29 2006
a(n) = n^2/2 + n^(3/2)/(3*sqrt(2)) + O(n^(5/4)) [proved in Jubin link]. - Benoit Jubin, May 13 2015
For all n >= 1, A232746(a(n)) = n and A232747(a(n)) = n. [Both sequences work as left inverses of this sequence.] - Antti Karttunen, May 14 2015

Extensions

Additional comments from Robert G. Wilson v, Oct 24 2001
Incorrect formula removed by Benoit Jubin, May 13 2015

A006337 An "eta-sequence": a(n) = floor( (n+1)*sqrt(2) ) - floor( n*sqrt(2) ).

Original entry on oeis.org

1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1
Offset: 1

Views

Author

D. R. Hofstadter, Jul 15 1977

Keywords

Comments

Defined by: (i) a(1) = 1; (ii) sequence consists of single 2's separated by strings of 1's; (iii) the sequence of lengths of runs of 1's in the sequence is equal to the sequence.
Equals its own "derivative", which is formed by counting the strings of 1's that lie between 2's.
First differences of A001951 (with a different offset). - Philippe Deléham, May 29 2006
Or number of perfect squares in interval (2*n^2, 2*(n+1)^2). In view of the uniform distribution mod 1 of sequence {sqrt(2)*n}, the density of 1's is 2-sqrt(2). - Vladimir Shevelev, Aug 05 2011
a(n) = number of repeating n's in A049472. - Reinhard Zumkeller, Jul 03 2015
Fixed point of the morphism 1 -> 12; 2 -> 121. - Jeffrey Shallit, Jan 19 2017
Also, let S be the increasing sequence of elements of the union N U N*sqrt(2), where N = {1, 2, 3, ...}. Then a(n) = { 1 if S(n) is integer, 2 if S(n) is irrational }. See A245222 for the analog with sqrt(3). - M. F. Hasler, Feb 06 2025

References

  • Douglas Hofstadter, "Fluid Concepts and Creative Analogies", Chapter 1: "To seek whence cometh a sequence".
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A006338. Exchanging 1's and 2's gives A080763. Essentially same as A004641 + 1.
Cf. A049472.
The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A003151 as the parent: A003151, A001951, A001952, A003152, A006337, A080763, A082844 (conjectured), A097509, A159684, A188037, A245219 (conjectured), A276862. - N. J. A. Sloane, Mar 09 2021
Cf. A245222 (an analog with sqrt(3) instead of sqrt(2)).

Programs

  • Haskell
    a006337 n = a006337_list !! (n-1)
a006337_list = f [1] where
   f xs = ys ++ f ys where
          ys = concatMap (\z -> if z == 1 then [1,2] else [1,1,2]) xs
-- Reinhard Zumkeller, May 06 2012
  • Maple
    Digits := 100; sq2 := sqrt(2.); A006337 := n->floor((n+1)*sq2)-floor(n*sq2);
  • Mathematica
    Flatten[ Table[ Nest[ Flatten[ # /. {1 -> {1, 2}, 2 -> {1, 1, 2}}] &, {1}, n], {n, 5}]] (* Robert G. Wilson v, May 06 2005 *)
Differences[ Table[ Floor[ n*Sqrt[2]], {n, 1, 106}]] (* Jean-François Alcover, Apr 06 2012 *)
  • PARI
    a(n)=sqrt(2)*(n+1)\1-sqrt(2)*n\1 \\ Charles R Greathouse IV, Apr 06 2012
  • PARI
    a(n)=sqrtint(2*n^2+4*n+2)-sqrtint(2*n^2) \\ Charles R Greathouse IV, Apr 06 2012
  • Python
    from math import isqrt
def A006337(n): return -isqrt(m:=n*n<<1)+isqrt(m+(n<<2)+2) # Chai Wah Wu, Aug 03 2022

Formula

Let S(0) = 1; obtain S(k) from S(k-1) by applying 1 -> 12, 2 -> 112; sequence is S(0), S(1), S(2), ... - Matthew Vandermast, Mar 25 2003
a(A003152(n)) = 1 and a(A003151(n)) = 2. - Philippe Deléham, May 29 2006
a(n) = A159684(n-1) + 1. - Filip Zaludek, Oct 28 2016

A001468 There are a(n) 2's between successive 1's.

Original entry on oeis.org

1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

The Fibonacci word on the alphabet {2,1}, with an extra 1 in front. - Michel Dekking, Nov 26 2018
Start with 1, apply 1->12, 2->122, take limit. - Philippe Deléham, Sep 23 2005
Also number of occurrences of n in Hofstadter G-sequence (A005206) and in A019446. - Reinhard Zumkeller, Feb 02 2012, Aug 07 2011
A block-fractal sequence: every block occurs infinitely many times. Also a reverse block-fractal sequence. See A280511. - Clark Kimberling, Jan 06 2017

References

  • D. Gault and M. Clint, "Curiouser and curiouser" said Alice. Further reflections on an interesting recursive function, Internat. J. Computer Math., 26 (1988), 35-43. See Table 2.
  • D. R. Hofstadter, personal communication, Jul 15 1977.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Same as A014675 if initial 1 is deleted. Cf. A003849, A000201, A280511.
The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A000201 as the parent: A000201, A001030, A001468, A001950, A003622, A003842, A003849, A004641, A005614, A014675, A022342, A088462, A096270, A114986, A124841. - N. J. A. Sloane, Mar 11 2021

Programs

  • Haskell
    import Data.List (group)
a001468 n = a001468_list !! n
a001468_list = map length $ group a005206_list
-- Reinhard Zumkeller, Aug 07 2011
  • Maple
    Digits := 100: t := evalf( (1+sqrt(5))/2); A001468 := n-> floor((n+1)*t)-floor(n*t);
  • Mathematica
    Table[Floor[GoldenRatio*(n + 1)] - Floor[GoldenRatio*n], {n, 0, 80}] (* Joseph Biberstine (jrbibers(AT)indiana.edu), Aug 14 2006 *)
Nest[ Flatten[# /. {1 -> {1, 2}, 2 -> {1, 2, 2}}] &, {1}, 6] (* Robert G. Wilson v, May 20 2014 and corrected Apr 24 2017 following Clark Kimberling's email of Mar 22 2017 *)
SubstitutionSystem[{1->{1,2},2->{1,2,2}},{1},{6}][[1]] (* Harvey P. Dale, Jan 31 2022 *)
  • PARI
    a=[1];for(i=1,30,a=concat([a,vector(a[i],j,2),1]));a \\ Or compute as A001468(n)=A201(n+1)-A201(n) with A201(n)=(n+sqrtint(5*n^2))\2, working for n>=0 although A000201 is defined for n>=1. - M. F. Hasler, Oct 13 2017
  • Python
    def A001468(length):
    a = [1]
    for i in range(length):
        for _ in range(a[i]):
            a.append(2)
        a.append(1)
        if len(a)>=length:
            break
    return a[:length] # Nicholas Stefan Georgescu, Jun 02 2022
  • Python
    from math import isqrt
def A001468(n): return (n+1+isqrt(m:=5*(n+1)**2)>>1)-(n+isqrt(m-10*n-5)>>1) # Chai Wah Wu, Aug 25 2022

Formula

a(n) = [(n+1) tau] - [n tau], tau = (1 + sqrt 5)/2 = A001622, [] = floor function.
a(n) = A000201(n+1) - A000201(n) = A022342(n+1) - A022342(n), n >= 1; i.e., the first term discarded, this yields the first differences of A000201 and A022342. - M. F. Hasler, Oct 13 2017

Extensions

Rechecked by N. J. A. Sloane, Nov 07 2001

A005374 Hofstadter H-sequence: a(n) = n - a(a(a(n-1))).

Original entry on oeis.org

0, 1, 1, 2, 3, 4, 4, 5, 5, 6, 7, 7, 8, 9, 10, 10, 11, 12, 13, 13, 14, 14, 15, 16, 17, 17, 18, 18, 19, 20, 20, 21, 22, 23, 23, 24, 24, 25, 26, 26, 27, 28, 29, 29, 30, 31, 32, 32, 33, 33, 34, 35, 35, 36, 37, 38, 38, 39, 40, 41, 41, 42, 42, 43, 44, 45, 45, 46, 46, 47, 48, 48, 49, 50
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Rule for constructing the sequence: a(n) = An, where An denotes the Lamé antecessor to (or right shift of) n, which is found by replacing each Lm(i) in the Zeckendorffian expansion (obtained by repeatedly subtracting the largest Lamé number (A000930) you can until nothing remains) by Lm(i-1) (A1=1). For example: 58 = 41 + 13 + 4, so a(58)= 28 + 9 + 3 = 40.
From Albert Neumueller (albert.neu(AT)gmail.com), Sep 28 2006: (Start)
As is shown on page 137 of Goedel, Escher, Bach, a recursively built tree structure can be obtained from this sequence:
20.21..22..23.24.25.26.27.28
.\./.../.../...\./...\./../
..14.15..16....17....18..19
...\./.../..../.......\./
....10.11...12........13
.....\./.../........./
......7...8........9.
.......\./......./
........5......6
.........\.../
...........4
........../
.........3
......../
.......2
....\./
.....1
To construct the tree: node n is connected to the node a(n) below it:
...n
../
a(n)
For example:
...8
../
.5
since a(8) = 5. If the nodes of the tree are read from bottom-to-top, left-to-right, we obtain the natural numbers: 1, 2, 3, 4, 5, 6, ...
The tree has a recursive structure, since the following construct
....../
.....x
..../
...x
\./
.x
can be repeatedly added on top of its own ends, to construct the tree from its root: E.g.,
................../
.................x
................/
...............x
......../...\./
.......x.....x
....../...../
.....x.....x
..\./...../
...x.....x
....\.../
......x
(End)
From Pierre Letouzey, Feb 20 2025: (Start)
For all n >= 0, A005206(n) <= a(n) <= A005375(n), as proved in Letouzey-Li-Steiner link. Last equality A005206(n) = a(n) occurs at n = 12; last equality a(n) = A005375(n) occurs at n = 18.
For all n >= 0, |a(n)-c*n| < 0.996, where c is the real root of x^3 + x - 1 = 0, c = 0.682327803828019327369483739... Proved in Letouzey link. (End)
The bound for |a(n)-c*n| is improved to 0.862 in Shallit (2025). - Jeffrey Shallit, Mar 09 2025

References

  • D. R. Hofstadter, Goedel, Escher, Bach: an Eternal Golden Braid, Random House, 1980, p. 137.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A202340, A202341, A202342.

Programs

  • Haskell
    a005374 n = a005374_list !! n
a005374_list = 0 : 1 : zipWith (-)
   [2..] (map (a005374 . a005374) $ tail a005374_list)
-- Reinhard Zumkeller, Dec 17 2011
  • Maple
    A005374 := proc(n) option remember: if n<1 then 0 else n-A005374(A005374(A005374(n-1))) fi end: # Francisco Salinas (franciscodesalinas(AT)hotmail.com), Jan 06 2002
H:=proc(n) option remember; if n=1 then 1 else n-H(H(H(n-1))); fi; end proc;
  • Mathematica
    a[n_]:= a[n]= n - a[a[a[n-1]]]; a[0] = 0; Table[a[n], {n, 0, 73}] (* Jean-François Alcover, Jul 28 2011 *)
  • PARI
    first(m)=my(v=vector(m));v[1]=1;for(i=2,m,v[i]=i-v[v[v[i-1]]]);concat([0],v) \\ Anders Hellström, Dec 07 2015
  • SageMath
    @CachedFunction # a = A005374
def a(n): return 0 if (n==0) else n - a(a(a(n-1)))
[a(n) for n in range(101)] # G. C. Greubel, Nov 14 2022

Formula

Conjecture: a(n) = floor(c*n) + 0 or 1, where c is the real root of x^3+x-1 = 0, c=0.682327803828019327369483739... - Benoit Cloitre, Nov 05 2002 [Proved by Letouzey, see Letouzey link. - Pierre Letouzey, Feb 20 2025], [Also proved in Shallit (2025). - Jeffrey Shallit, Mar 09 2025]
a(n) = A020942(n) - 2*A064105(n) + A064106(n) (e.g. for n = 30 we get 20 = 93 - 2*137 + 201), and a(n) = 2*A020942(n) - A064105(n) - A023443(n) (e.g. for n = 30 we get 20 = 2*93 - 137 - 29). [Corrected by N. J. A. Sloane, Apr 29 2024 at the suggestion of A.H.M. Smeets.]
Also: a(n) = a(n-1) + 1 if n-1 belongs to sequence A064105-A020942-A000012, a(n-1) otherwise.
Recurrence: a(n) = a(n-1) if n-1 belongs to sequence A020942, a(n-1) + 1 otherwise.
Recurrence for n>=3: a(n) = Lm(k-1) + a(n-Lm(k)), where Lm(n) denotes Lamé sequence A000930(n) (Lm(n) = Lm(n-1) + Lm(n-3)) and k is such that Lm(k)< n <= Lm(k+1). Special case: a(Lm(n)) = Lm(n-1) for n>=1.
For n > 0: a(A136495(n)) = n. - Reinhard Zumkeller, Dec 17 2011

Extensions

More terms from James Sellers, Jul 12 2000
Additional comments and formulas from Diego Torres (torresvillarroel(AT)hotmail.com), Nov 23 2002

A001030 Fixed under 1 -> 21, 2 -> 211.

Original entry on oeis.org

2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

If treated as the terms of a continued fraction, it converges to approximately
2.57737020881617828717350576260723346479894963737498275232531856357441\
7024804797827856956758619431996. - Peter Bertok (peter(AT)bertok.com), Nov 27 2001
There are a(n) 1's between successive 2's. - Eric Angelini, Aug 19 2008
Same sequence where 1's and 2's are exchanged: A001468. - Eric Angelini, Aug 19 2008

References

  • Midhat J. Gazale, Number: From Ahmes to Cantor, Section on 'Cleavages' in Chapter 6, Princeton University Press, Princeton, NJ 2000, pp. 203-211.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Length of the sequence after 'n' substitution steps is given by the terms of A000129.
Equals A004641(n) + 1.
The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A000201 as the parent: A000201, A001030, A001468, A001950, A003622, A003842, A003849, A004641, A005614, A014675, A022342, A088462, A096270, A114986, A124841. - N. J. A. Sloane, Mar 11 2021

Programs

  • Haskell
    Following Spage's PARI program.
a001030 n = a001030_list !! (n-1)
a001030_list = [2, 1, 1, 2] ++ f [2] [2, 1, 1, 2] where
   f us vs = ws ++ f vs (vs ++ ws) where
             ws = 1 : us ++ 1 : vs
-- Reinhard Zumkeller, Aug 04 2014
  • Mathematica
    ('n' is the number of substitution steps to perform.) Nest[Flatten[ # /. {1 -> {2, 1}, 2 -> {2, 1, 1}}] &, {1}, n]
SubstitutionSystem[{1->{2,1},2->{2,1,1}},{2},{6}][[1]] (* Harvey P. Dale, Feb 15 2022 *)
  • PARI
    /* Fast string concatenation method giving e.g. 5740 terms in 8 iterations */
a="2";b="2,1,1,2";print1(b);for(x=1,8,c=concat([",1,",a,",1,",b]);print1(c);a=b;b=concat(b,c)) \\ K. Spage, Oct 08 2009
  • Python
    from math import isqrt
def A001030(n): return [2, 1, 1, 2, 1, 2, 1, 2][n-1] if n < 9 else -isqrt(m:=(n-9)*(n-9)<<1)+isqrt(m+(n-9<<2)+2) # Chai Wah Wu, Aug 25 2022

Formula

a(n) = -1 + floor(n*(1+sqrt(2))+1/sqrt(2))-floor((n-1)*(1+sqrt(2))+1/sqrt(2)). - Benoit Cloitre, Jun 26 2004. [I don't know if this is a theorem or a conjecture. - N. J. A. Sloane, May 14 2008]
This is a theorem, following from Hofstadter's Generalized Fundamental Theorem of eta-sequences on page 10 of Eta-Lore. See also de Bruijn's paper from 1981 (hint from Benoit Cloitre). - Michel Dekking, Jan 22 2017

Extensions

More terms from Peter Bertok (peter(AT)bertok.com), Nov 27 2001
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