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A003622 The Wythoff compound sequence AA: [n*phi^2] - 1, where phi = (1+sqrt(5))/2.
(Formerly M3278)
49
1, 4, 6, 9, 12, 14, 17, 19, 22, 25, 27, 30, 33, 35, 38, 40, 43, 46, 48, 51, 53, 56, 59, 61, 64, 67, 69, 72, 74, 77, 80, 82, 85, 88, 90, 93, 95, 98, 101, 103, 106, 108, 111, 114, 116, 119, 122, 124, 127, 129, 132, 135, 137, 140, 142, 145, 148, 150, 153, 156, 158, 161, 163, 166 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Also, integers with "odd" Zeckendorf expansions (end with ...+F1 = ...+1) (Fibonacci-odd numbers); first column of Wythoff array A035513; from a 3-way splitting of positive integers.

Also, numbers n such that A005206(n) = A005206(n+1). Also n such that A022342(A005206(n)) = n+1 (for all other n's this is n). - Michele Dondi (bik.mido(AT)tiscalenet.it), Dec 30 2001

Also, positions of 1's in A139764, the smallest term in Zeckendorf representation of n. - John W. Layman, Aug 25 2011

REFERENCES

A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 62.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 307-308 of 2nd edition.

C. Kimberling, "Stolarsky interspersions," Ars Combinatoria 39 (1995) 129-138.

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

D. R. Morrison, ``A Stolarsky array of Wythoff pairs,'' in A Collection of Manuscripts Related to the Fibonacci Sequence. Fibonacci Assoc., Santa Clara, CA, 1980, pp. 134-136.

J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 10.

N. J. A. Sloane and Simon Plouffe, Encyclopedia of Integer Sequences, Academic Press, 1995: this sequence appears twice, as both M3277 and M3278.

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

Larry Ericksen and Peter G. Anderson, Patterns in differences between rows in k-Zeckendorf arrays, The Fibonacci Quarterly, Vol. 50, February 2012. - N. J. A. Sloane, Jun 10 2012

C. Kimberling, Interspersions

C. Kimberling, Complementary equations and Wythoff Sequences, JIS 11 (2008) 08.3.3.

L. Lindroos, A. Sills and H. Wang, Odd fibbinary numbers and the golden ratio, Fib. Q., 52 (2014), 61-65.

M. Rigo, P. Salimov, and E. Vandomme, Some Properties of Abelian Return Words, Journal of Integer Sequences, Vol. 16 (2013), #13.2.5.

N. J. A. Sloane, Classic Sequences

FORMULA

a(n) = [(n+1)*phi] + n; a(n) = [[n*phi]*phi] = A000201(A000201(n)).

a(n) = 1 + A022342(1+A022342(n)).

G.f.: 1 - (1-x)*sum_{n=1..inf} x^a(n) = 1/1 + x/1 + x^2/1 + x^3/1 + x^5/1 + x^8/1 +...+ x^F(n)/1 +... (continued fraction where F(n)=n-th Fibonacci number). - Paul D. Hanna, Aug 16 2002

a(n) = A001950(n) - 1. - Philippe Deléham, Apr 30 2004

a(n) = A022342(n) + n. - Philippe Deléham, May 03 2004

MAPLE

A003622 := proc(n)

    n+floor(n*(1+sqrt(5))/2)-1 ;

end proc: # R. J. Mathar, Jan 25 2015

# Maple code for the Wythoff compound sequences, from N. J. A. Sloane, Mar 30 2016

# The Wythoff compound sequences: Let A = A000201, B = A001950. Then AA = A003622, AB = A003623, BA = A035336, BB = A101864. The eight triples AAA, AAB, ..., BBB are A134859, A134860, A035337, A134862, A134861, A134863, A035338, A134864, resp.

# Assume files out1, out2 contain lists of the terms in the base sequences A and B from their b-files

read out1; read out2; b[0]:=b1: b[1]:=b2:

w2:=(i, j, n)->b[i][b[j][n]];

w3:=(i, j, k, n)->b[i][b[j][b[k][n]]];

for i from 0 to 1 do

lprint("name=", i);

lprint([seq(b[i][n], n=1..100)]):

od:

for i from 0 to 1 do for j from 0 to 1 do

lprint("name=", i, j);

lprint([seq(w2(i, j, n), n=1..100)]);

od: od:

for i from 0 to 1 do for j from 0 to 1 do for k from 0 to 1 do

lprint("name=", i, j, k);

lprint([seq(w3(i, j, k, n), n=1..100)]);

od: od: od:

MATHEMATICA

With[{c=GoldenRatio^2}, Table[Floor[n c]-1, {n, 70}]] (* Harvey P. Dale, Jun 11 2011 *)

PROG

(PARI) a(n)=floor(n*(sqrt(5)+3)/2)-1

(Haskell)

a003622 n = a003622_list !! (n-1)

a003622_list = filter ((elem 1) . a035516_row) [1..]

-- Reinhard Zumkeller, Mar 10 2013

CROSSREFS

Positions of 1's in A003849.

Cf. A003623, A022342, A000201, A035336, A066094-A066097, A139764, A035516, A026273.

The Wythoff compound sequences: Let A = A000201, B = A001950. Then AA = A003622, AB = A003623, BA = A035336, BB = A101864. The eight triples AAA, AAB, ..., BBB are A134859, A134860, A035337, A134862, A134861, A134863, A035338, A134864, resp.

Sequence in context: A190304 A189366 A066095 * A007073 A189533 A047408

Adjacent sequences:  A003619 A003620 A003621 * A003623 A003624 A003625

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane, Mira Bernstein, Marc LeBrun

STATUS

approved

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Last modified May 31 00:01 EDT 2016. Contains 273533 sequences.