A073869 -id:A073869

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page 1

A090909

Terms a(k) of A073869 for which a(k-1), a(k) and a(k+1) are distinct.

+20
23

0, 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, 159, 162

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Apart from the initial 0, is this the same as A001950? - Alec Mihailovs (alec(AT)mihailovs.com), Jul 23 2007

Identical to n + A066096(n)? - Ed Russell (times145(AT)hotmail.com), May 09 2009

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..10000

FORMULA

a(n) = floor(phi^2*n), where phi = (1+sqrt(5))/2. - Gary Detlefs, Mar 10 2011

MATHEMATICA

(* First program *)

A002251= Fold[Append[#1, #2 Ceiling[#2/GoldenRatio] -Total[#1]] &, {1}, Range[2, 500]] - 1; (* Birkas Gyorgy's code of A019444, modified *)

A090909= Join[{0}, Select[Partition[A002251, 2, 1], #[[2]] > #[[1]] &][[All, 2]]] (* G. C. Greubel, Sep 12 2023 *)

(* Second program *)

Floor[GoldenRatio^2*Range[0, 80]] (* G. C. Greubel, Sep 12 2023 *)

PROG

(Magma) [Floor((3+Sqrt(5))*n/2): n in [0..80]]; // G. C. Greubel, Sep 12 2023

(SageMath) [floor(golden_ratio^2*n) for n in range(81)] # G. C. Greubel, Sep 12 2023

CROSSREFS

Cf. A001622, A001950, A002251, A005206, A073869, A090908.

Cf. A004919, A004920, A004921, A004922, A004923, A004924, A004925.

Cf. A004926, A004927, A004928, A004929, A004930, A004931, A004932.

Cf. A004933, A004934, A004935, A004976, A066096.

KEYWORD

nonn

AUTHOR

Amarnath Murthy, Dec 14 2003

EXTENSIONS

More terms from R. J. Mathar, Sep 29 2017

STATUS

approved

A090908

Terms a(k) of A073869 for which a(k)=a(k+1).

+20
3

1, 3, 4, 6, 8, 9, 11, 12, 14, 16, 17, 19, 21, 22, 24, 25, 27, 29, 30, 32, 33, 35, 37, 38, 40, 42, 43, 45, 46, 48, 50, 51, 53, 55, 56, 58, 59, 61, 63, 64, 66, 67, 69, 71, 72, 74, 76, 77, 79, 80, 82, 84, 85, 87, 88, 90, 92, 93, 95, 97, 98, 100, 101, 103, 105, 106, 108, 110, 111

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Is this the same as A000201?

LINKS

Table of n, a(n) for n=1..69.

CROSSREFS

Cf. A002251, A005206, A073869, A090909, A000201.

KEYWORD

nonn

AUTHOR

Amarnath Murthy, Dec 14 2003

EXTENSIONS

More terms from R. J. Mathar, Sep 29 2017

STATUS

approved

A090638

Duplicate of A073869.

+20
0

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

LINKS

Table of n, a(n) for n=1..29.

KEYWORD

dead

STATUS

approved

A005206

Hofstadter G-sequence: a(0) = 0; a(n) = n - a(a(n-1)) for n > 0.
(Formerly M0436)

+10
82

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

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

\ /

To construct the tree: node n is connected with the node a(n) below

a(n)

For example, since a(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

\ /

can be repeatedly added on top of its own ends, to construct the tree from its root: e.g.,

/ \ /

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

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

N. J. A. Sloane and T. D. Noe, Table of n, a(n) for n = 0..20000 (the first 1000 terms were found by T. D. Noe)

L. Carlitz, Fibonacci Representations, Fibonacci Quarterly, volume 6, number 4, October 1968, pages 193-220. a(n) = e(n) at equation 1.10 or 2.11 and see equation 3.8 recurrence.

M. Celaya and F. Ruskey, Morphic Words and Nested Recurrence Relations, arXiv preprint arXiv:1307.0153 [math.CO], 2013.

M. Celaya and F. Ruskey, Another Property of Only the Golden Ratio, American Mathematical Monthly, Problem 11651, solutions volume 121, number 6, June-July 2014, pages 549-556.

F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.

F. M. Dekking, On Hofstadter's G-sequence, Journal of Integer Sequences 26 (2023), Article 23.9.2, 1-11.

Larry Ericksen and Peter G. Anderson, Patterns in differences between rows in k-Zeckendorf arrays, The Fibonacci Quarterly, Vol. 50, No. 1 (February 2012), pp. 11-18.

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. Also annotated scanned copy.

Martin Griffiths, A formula for an infinite family of Fibonacci-word sequences, Fib. Q., 56 (2018), 75-80.

H. W. Gould, J. B. Kim and V. E. Hoggatt, Jr., Sequences associated with t-ary coding of Fibonacci's rabbits, Fib. Quart., 15 (1977), 311-318.

Vincent Granville and Jean-Paul Rasson, A strange recursive relation, J. Number Theory 30 (1988), no. 2, 238--241. MR0961919(89j:11014).

J. Grytczuk, Another variation on Conway's recursive sequence, Discr. Math. 282 (2004), 149-161.

Nick Hobson, Python program for this sequence

D. R. Hofstadter, Eta-Lore [Cached copy, with permission]

D. R. Hofstadter, Pi-Mu Sequences [Cached copy, with permission]

D. R. Hofstadter and N. J. A. Sloane, Correspondence, 1977 and 1991

The IMO Compendium, Problem 1, 45th Czech and Slovak Mathematical Olympiad 1996.

Clark Kimberling and K. B. Stolarsky, Slow Beatty sequences, devious convergence, and partitional divergence, Amer. Math. Monthly, 123 (No. 2, 2016), 267-273.

P. Letouzey, Hofstadter's problem for curious readers, Technical Report, 2015.

Mustazee Rahman, A Combinatorial interpretation of Hofstadter's G-sequence, arXiv:1105.1718 [math.CO], 2011-2013.

B. Schoenmakers, A tight lower bound for top-down skew heaps, Information Processing Letters, 61(5): 279-284, 14 March 1997.

Torsten Sillke, Floor Recurrences

Th. Stoll, On Hofstadter's married functions, Fib. Q., 46/47 (2008/2009), 62-67. - N. J. A. Sloane, May 30 2009

Eric Weisstein's World of Mathematics, Hofstadter G-Sequence

Wikipedia, Hofstadter sequence

Index entries for Hofstadter-type sequences

Index entries for sequences from "Goedel, Escher, Bach"

Index to sequences related to Olympiads.

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

a(n + a(n)) = n. - Pierre Letouzey, Sep 09 2015

a(n) + a(a(n+1) - 1) = n. - Pierre Letouzey, Sep 09 2015

a(0) = 0, a(n+1) = a(n) + d(n) and d(0) = 1, d(n+1)=1-d(n)*d(a(n)). - Pierre Letouzey, Sep 09 2015

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)

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 *)

PROG

(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

(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

(Magma) [Floor((n+1)*(1+Sqrt(5))/2)-n-1: n in [0..80]]; // Vincenzo Librandi, Nov 19 2016

(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

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.

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

STATUS

approved

A002251

Start with the nonnegative integers; then swap L(k) and U(k) for all k >= 1, where L = A000201, U = A001950 (lower and upper Wythoff sequences).

+10
23

0, 2, 1, 5, 7, 3, 10, 4, 13, 15, 6, 18, 20, 8, 23, 9, 26, 28, 11, 31, 12, 34, 36, 14, 39, 41, 16, 44, 17, 47, 49, 19, 52, 54, 21, 57, 22, 60, 62, 24, 65, 25, 68, 70, 27, 73, 75, 29, 78, 30, 81, 83, 32, 86, 33, 89, 91, 35, 94, 96, 37, 99, 38, 102, 104, 40, 107, 109

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

(n,a(n)) are Wythoff pairs: (0,0), (1,2), (3,5), (4,7), ..., where each difference occurs once.

Self-inverse when considered as a permutation or function, i.e., a(a(n)) = n. - Howard A. Landman, Sep 25 2001

If the offset is 1, the sequence can also be obtained by rearranging the natural numbers so that sum of n terms is a multiple of n, or equivalently so that the arithmetic mean of the first n terms is an integer. - Amarnath Murthy, Aug 16 2002

For n = 1, 2, 3, ..., let p(n)=least natural number not already an a(k), q(n) = n + p(n); then a(p(n)) = q(n), a(q(n)) = p(n). - Clark Kimberling

Also, indices of powers of 2 in A086482. - Amarnath Murthy, Jul 26 2003

There is a 7-state Fibonacci automaton (see a002251_1.pdf) that accepts, in parallel, the Zeckendorf representations of n and a(n). - Jeffrey Shallit, Jul 14 2023

REFERENCES

E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 76.

LINKS

T. D. Noe, Table of n, a(n) for n = 0..10000

F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, Queens in exile: non-attacking queens on infinite chess boards, Electronic J. Combin., 27:1 (2020), #P1.52.

Eric Duchene, Aviezri S. Fraenkel, Vladimir Gurvich, Nhan Bao Ho, Clark Kimberling, and Urban Larsson, Wythoff Wisdom, 43 pages, no date, unpublished.

Eric Duchene, Aviezri S. Fraenkel, Vladimir Gurvich, Nhan Bao Ho, Clark Kimberling, and Urban Larsson, Wythoff Wisdom, unpublished, no date [Cached copy, with permission]

Alex Meadows and B. Putman, A New Twist on Wythoff's Game, arXiv preprint arXiv:1606.06819 [math.CO], 2016.

Gabriel Nivasch, More on the Sprague-Grundy function for Wythoff’s game, pages 377-410 in "Games of No Chance 3, MSRI Publications Volume 56, 2009.

Jeffrey Shallit, Automaton for A002251

Jeffrey Shallit, Proving properties of some greedily-defined integer recurrences via automata theory, arXiv:2308.06544 [cs.DM], 2023.

R. Silber, Wythoff's Nim and Fibonacci Representations, Fibonacci Quarterly #14 (1977), pp. 85-88.

N. J. A. Sloane, Scatterplot of first 100 terms [The points are symmetrically placed about the diagonal, although that is hard to see here because the scales on the axes are different]

Index entries for sequences that are permutations of the natural numbers

FORMULA

a(n) = A019444(n+1) - 1.

MATHEMATICA

With[{n = 42}, {0}~Join~Take[Values@ #, LengthWhile[#, # == 1 &] &@ Differences@ Keys@ #] &@ Sort@ Flatten@ Map[{#1 -> #2, #2 -> #1} & @@ # &, Transpose@ {Array[Floor[# GoldenRatio] &, n], Array[Floor[# GoldenRatio^2] &, n]}]] (* Michael De Vlieger, Nov 14 2017 *)

PROG

(PARI) A002251_upto(N, c=0, A=Vec(0, N))={for(n=1, N, A[n]||(#A<A[n]=n+c++)|| A[n+c]=n); A} \\ The resulting vector starts with A002251[1]=2, a(0)=0 is not included. - M. F. Hasler, Nov 27 2019, replacing earlier code from Sep 17 2014

CROSSREFS

The sequence maps between A000201 and A001950, in that a(A000201(n)) = A001950(n), a(A001950(n)) = A000201(n).

Row 0 of A018219.

Cf. A073869, A342297.

KEYWORD

nonn,easy,nice

AUTHOR

Michael Kleber

EXTENSIONS

Edited by Christian G. Bower, Oct 29 2002

STATUS

approved

A367068

a(n) = ((Sum_{i=1..n} A340510(i))-1)/(n+1).

+10
2

0, 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 7, 8, 9, 9, 10, 11, 11, 12, 12, 13, 14, 14, 15, 16, 16, 17, 17, 18, 19, 19, 20, 20, 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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

Define the Avdispahić-Zejnulahi means sequence AZM(k) by a(n) = ((Sum_{i=1..n} b(i))-k)/(n+k). This is the AZM(1) sequence.

Is this a duplicate of A005379? For n<=1300 at least a(n)=A005379(n). - R. J. Mathar, Jan 30 2024

LINKS

Table of n, a(n) for n=1..68.

Muharem Avdispahić and Faruk Zejnulahi, An integer sequence with a divisibility property, Fibonacci Quarterly, Vol. 58:4 (2020), 321-333.

FORMULA

For n>2, a(n) = a(n-1) if a(n-1) <> A340510(k) (for k=1..n-1) and a(n) = a(n-1)+1=A340510(n)-n otherwise. (See Proposition 3.1. of Avdispahić and Zejnulahi in the link above).

MAPLE

A367068 := proc(n)

add(A340510(i), i=1..n)-1 ;

%/(n+1) ;

end proc:

seq(A367068(n), n=1..50) ; # R. J. Mathar, Jan 30 2024

MATHEMATICA

zlist = {-1, 1, 3};

mlist = {-1, 0, 1};

For[n = 3, n <= 101, n++,

If[MemberQ[zlist, mlist[[n]]], AppendTo[mlist, mlist[[n]] + 1];

AppendTo[zlist, mlist[[n + 1]] + n]; , AppendTo[mlist, mlist[[n]]];

AppendTo[zlist, mlist[[n + 1]]]; ]; ];

mlist = Drop[mlist, 1]; mlist

PROG

(Python)

z_list=[-1, 1, 3]

m_list=[-1, 0, 1]

n=2

for n in range(2, 100):

if m_list[n] in z_list:

m_list.append(m_list[n] + 1)

z_list.append(m_list[n+1] + n+1)

else:

m_list.append(m_list[n])

z_list.append(m_list[n+1])

print(m_list[1:])

CROSSREFS

Cf. A340510.

Cf. A073869 (AZM(0)), A367066 (AZM(2)).

KEYWORD

nonn

AUTHOR

Zenan Sabanac, Dec 17 2023

STATUS

approved

A367069

a(n) = ((Sum_{i=1..n} A367067(i))-3)/(n+3).

+10
0

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

For a positive integer k define the Avdispahić-Zejnulahi sequence AZ(k) by b(1)=k, and thereafter b(n) is the least positive integer not yet in the sequence such that Sum_{i=1..n} b(i) == k (mod n+k). Define the Avdispahić-Zejnulahi means sequence AZM(k) by a(n) = ((Sum_{i=1..n} b(i))-k)/(n+k). This is the AZM(3) sequence.

LINKS

Table of n, a(n) for n=1..69.

Muharem Avdispahić and Faruk Zejnulahi, An integer sequence with a divisibility property, Fibonacci Quarterly, Vol. 58:4 (2020), 321-333.

MATHEMATICA

zlist = {-1, 3, 5};

mlist = {-1, 0, 1};

For[n = 3, n <= 101, n++,

If[MemberQ[zlist, mlist[[n]]], AppendTo[mlist, mlist[[n]] + 1];

AppendTo[zlist, mlist[[n + 1]] + n + 2]; ,

AppendTo[mlist, mlist[[n]]]; AppendTo[zlist, mlist[[n + 1]]]; ]; ];

mlist = Drop[mlist, 1]; mlist

PROG

(Python)

z_list=[-1, 3, 5]

m_list=[-1, 0, 1]

n=2

for n in range(2, 100):

if m_list[n] in z_list:

m_list.append(m_list[n] + 1)

z_list.append(m_list[n+1] + n+3)

else:

m_list.append(m_list[n])

z_list.append(m_list[n+1])

print(m_list[1:])

CROSSREFS

Cf. A367067.

Cf. A073869 (AZM(0)), A367068 (AZM(1)), A367066 (AZM(2)).

KEYWORD

nonn

AUTHOR

Zenan Sabanac, Dec 17 2023

STATUS

approved

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