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.

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A002487 Stern's diatomic series (or Stern-Brocot sequence): a(0) = 0, a(1) = 1; for n > 0: a(2*n) = a(n), a(2*n+1) = a(n) + a(n+1).

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 4, 5, 1, 6, 5, 9, 4, 11, 7, 10, 3, 11, 8, 13, 5, 12, 7, 9, 2, 9, 7, 12, 5, 13, 8, 11, 3, 10, 7, 11, 4, 9, 5, 6, 1, 7, 6, 11, 5, 14, 9, 13, 4, 15, 11, 18, 7, 17, 10, 13, 3, 14, 11, 19, 8, 21, 13, 18, 5, 17, 12, 19
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

Also called fusc(n) [Dijkstra].
a(n)/a(n+1) runs through all the reduced nonnegative rationals exactly once [Stern; Calkin and Wilf].
If the terms are written as an array:
column 0 1 2 3 4 5 6 7 8 9 ...
row 0: 0
row 1: 1
row 2: 1,2
row 3: 1,3,2,3
row 4: 1,4,3,5,2,5,3,4
row 5: 1,5,4,7,3,8,5,7,2,7,5,8,3,7,4,5
row 6: 1,6,5,9,4,11,7,10,3,11,8,13,5,12,7,9,2,9,7,12,5,13,8,11,3,10,...
...
then (ignoring row 0) the sum of the k-th row is 3^(k-1), each column is an arithmetic progression and the steps are nothing but the original sequence. - Takashi Tokita (butaneko(AT)fa2.so-net.ne.jp), Mar 08 2003
From N. J. A. Sloane, Oct 15 2017: (Start)
The above observation can be made more precise. Let A(n,k), n >= 0, 0 <= k <= 2^(n-1)-1 for k > 0, denote the entry in row n and column k of the left-justified array above.
The equations for columns 0,1,2,3,4,... are successively (ignoring row 0):
1 (n >= 1),
n (n >= 2),
n-1 (n >= 3),
2n-3 (n >= 3),
n-2 (n >= 4),
3n-7 (n >= 4),
...
and in general column k > 0 is given by
A(n,k) = a(k)*n - A156140(k) for n >= ceiling(log_2(k+1))+1, and 0 otherwise.
(End)
a(n) is the number of odd Stirling numbers S_2(n+1, 2r+1) [Carlitz].
Moshe Newman proved that the fraction a(n+1)/a(n+2) can be generated from the previous fraction a(n)/a(n+1) = x by 1/(2*floor(x) + 1 - x). The successor function f(x) = 1/(floor(x) + 1 - frac(x)) can also be used.
a(n+1) = number of alternating bit sets in n [Finch].
If f(x) = 1/(1 + floor(x) - frac(x)) then f(a(n-1)/a(n)) = a(n)/a(n+1) for n >= 1. If T(x) = -1/x and f(x) = y, then f(T(y)) = T(x) for x > 0. - Michael Somos, Sep 03 2006
a(n+1) is the number of ways of writing n as a sum of powers of 2, each power being used at most twice (the number of hyperbinary representations of n) [Carlitz; Lind].
a(n+1) is the number of partitions of the n-th integer expressible as the sum of distinct even-subscripted Fibonacci numbers (= A054204(n)), into sums of distinct Fibonacci numbers [Bicknell-Johnson, theorem 2.1].
a(n+1) is the number of odd binomial(n-k, k), 0 <= 2*k <= n. [Carlitz], corrected by Alessandro De Luca, Jun 11 2014
a(2^k) = 1. a(3*2^k) = a(2^(k+1) + 2^k) = 2. Sequences of terms between a(2^k) = 1 and a(2^(k+1)) = 1 are palindromes of length 2^k-1 with a(2^k + 2^(k-1)) = 2 in the middle. a(2^(k-1) + 1) = a(2^k - 1) = k+1 for k > 1. - Alexander Adamchuk, Oct 10 2006
The coefficients of the inverse of the g.f. of this sequence form A073469 and are related to binary partitions A000123. - Philippe Flajolet, Sep 06 2008
It appears that the terms of this sequence are the number of odd entries in the diagonals of Pascal's triangle at 45 degrees slope. - Javier Torres (adaycalledzero(AT)hotmail.com), Aug 06 2009
Let M be an infinite lower triangular matrix with (1, 1, 1, 0, 0, 0, ...) in every column shifted down twice:
1;
1, 0;
1, 1, 0;
0, 1, 0, 0;
0, 1, 1, 0, 0;
0, 0, 1, 0, 0, 0;
0, 0, 1, 1, 0, 0, 0;
...
Then this sequence A002487 (without initial 0) is the first column of lim_{n->oo} M^n. (Cf. A026741.) - Gary W. Adamson, Dec 11 2009 [Edited by M. F. Hasler, Feb 12 2017]
Member of the infinite family of sequences of the form a(n) = a(2*n); a(2*n+1) = r*a(n) + a(n+1), r = 1 for A002487 = row 1 in the array of A178239. - Gary W. Adamson, May 23 2010
Equals row 1 in an infinite array shown in A178568, sequences of the form
a(2*n) = r*a(n), a(2*n+1) = a(n) + a(n+1); r = 1. - Gary W. Adamson, May 29 2010
Row sums of A125184, the Stern polynomials. Equivalently, B(n,1), the n-th Stern polynomial evaluated at x = 1. - T. D. Noe, Feb 28 2011
The Kn1y and Kn2y triangle sums, see A180662 for their definitions, of A047999 lead to the sequence given above, e.g., Kn11(n) = A002487(n+1) - A000004(n), Kn12(n) = A002487(n+3) - A000012(n), Kn13(n) = A002487(n+5) - A000034(n+1) and Kn14(n) = A002487(n+7) - A157810(n+1). For the general case of the knight triangle sums see the Stern-Sierpiński triangle A191372. This triangle not only leads to Stern's diatomic series but also to snippets of this sequence and, quite surprisingly, their reverse. - Johannes W. Meijer, Jun 05 2011
Maximum of terms between a(2^k) = 1 and a(2^(k+1)) = 1 is the Fibonacci number F(k+2). - Leonid Bedratyuk, Jul 04 2012
Probably the number of different entries per antidiagonal of A223541. That would mean there are exactly a(n+1) numbers that can be expressed as a nim-product 2^x*2^y with x + y = n. - Tilman Piesk, Mar 27 2013
Let f(m,n) be the frequency of the integer n in the interval [a(2^(m-1)), a(2^m-1)]. Let phi(n) be Euler's totient function (A000010). Conjecture: for all integers m,n n<=m f(m,n) = phi(n). - Yosu Yurramendi, Sep 08 2014
Back in May 1995, it was proved that A000360 is the modulo 3 mapping, (+1,-1,+0)/2, of this sequence A002487 (without initial 0). - M. Jeremie Lafitte (Levitas), Apr 24 2017
Define a sequence chf(n) of Christoffel words over an alphabet {-,+}: chf(1) = '-'; chf(2*n+0) = negate(chf(n)); chf(2*n+1) = negate(concatenate(chf(n),chf(n+1))). Then the length of the chf(n) word is fusc(n) = a(n); the number of '-'-signs in the chf(n) word is c-fusc(n) = A287729(n); the number of '+'-signs in the chf(n) word is s-fusc(n) = A287730(n). See examples below. - I. V. Serov, Jun 01 2017
The sequence can be extended so that a(n) = a(-n), a(2*n) = a(n), a(2*n+1) = a(n) + a(n+1) for all n in Z. - Michael Somos, Jun 25 2019
Named after the German mathematician Moritz Abraham Stern (1807-1894), and sometimes also after the French clockmaker and amateur mathematician Achille Brocot (1817-1878). - Amiram Eldar, Jun 06 2021
It appears that a(n) is equal to the multiplicative inverse of A007305(n+1) mod A007306(n+1). For example, a(12) is 2, the multiplicative inverse of A007305(13) mod A007306(13), where A007305(13) is 4 and A007306(13) is 7. - Gary W. Adamson, Dec 18 2023

Examples

			Stern's diatomic array begins:
  1,1,
  1,2,1,
  1,3,2,3,1,
  1,4,3,5,2,5,3,4,1,
  1,5,4,7,3,8,5,7,2,7,5,8,3,7,4,5,1,
  1,6,5,9,4,11,7,10,3,11,8,13,5,12,7,9,2,9,7,12,5,13,8,11,3,10,7,11,4,9,...
  ...
a(91) = 19, because 91_10 = 1011011_2; b_6=b_4=b_3=b_1=b_0=1, b_5=b_2=0;  L=5; m_1=0, m_2=1, m_3=3, m_4=4, m_5=6; c_1=2, c_2=3, c_3=2, c_4=3; f(1)=1, f(2)=2, f(3)=5, f(4)=8, f(5)=19. - _Yosu Yurramendi_, Jul 13 2016
From _I. V. Serov_, Jun 01 2017: (Start)
a(n) is the length of the Christoffel word chf(n):
n  chf(n) A070939(n)   a(n)
1   '-'       1          1
2   '+'       2          1
3   '+-'      2          2
4   '-'       3          1
5   '--+'     3          3
6   '-+'      3          2
... (End)
G.f. = x + x^2 + 2*x^3 + x^4 + 3*x^5 + 2*x^6 + 3*x^7 + x^8 + ... - _Michael Somos_, Jun 25 2019

References

  • M. Aigner and G. M. Ziegler, Proofs from THE BOOK, 3rd ed., Berlin, Heidelberg, New York: Springer-Verlag, 2004, p. 97.
  • Elwyn R. Berlekamp, John H. Conway and Richard K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 114.
  • Krishna Dasaratha, Laure Flapan, Chansoo Lee, Cornelia Mihaila, Nicholas Neumann-Chun, Sarah Peluse and Matthew Stroegeny, A family of multi-dimensional continued fraction Stern sequences, Abtracts Amer. Math. Soc., Vol. 33 (#1, 2012), #1077-05-2543.
  • Edsger W. Dijkstra, Selected Writings on Computing, Springer, 1982, p. 232 (sequence is called fusc).
  • F. G. M. Eisenstein, Eine neue Gattung zahlentheoretischer Funktionen, welche von zwei Elementen abhaengen und durch gewisse lineare Funktional-Gleichungen definirt werden, Verhandlungen der Koenigl. Preuss. Akademie der Wiss. Berlin (1850), pp. 36-42, Feb 18, 1850. Werke, II, pp. 705-711.
  • Graham Everest, Alf van der Poorten, Igor Shparlinski and Thomas Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
  • Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 2.16.3; pp. 148-149.
  • Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 117.
  • Thomas Koshy, Fibonacci and Lucas numbers with applications, Wiley, 2001, p. 98.
  • 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

Cf. A000120, A000123, A000360, A001045, A002083, A011655, A020950, A026741, A037227, A046815, A070871, A070872, A071883, A073459, A084091, A101624, A126606, A168081, A174980, A174981, A178239, A178568, A212288, A213369, A260443, A277020, A277325, A287729, A287730, A293160.
Record values are in A212289.
If the 1's are replaced by pairs of 1's we obtain A049456.
Inverse: A020946.
Cf. a(A001045(n)) = A000045(n). a(A062092(n)) = A000032(n+1).
Cf. A064881-A064886 (Stern-Brocot subtrees).
A column of A072170.
Cf. A049455 for the 0,1 version of Stern's diatomic array.
Cf. A000119, A262097 for analogous sequences in other bases and A277189, A277315, A277328 for related sequences with similar graphs.
Cf. A086592 and references therein to other sequences related to Kepler's tree of fractions.

Programs

  • Haskell
    a002487 n = a002487_list !! n
a002487_list = 0 : 1 : stern [1] where
   stern fuscs = fuscs' ++ stern fuscs' where
     fuscs' = interleave fuscs $ zipWith (+) fuscs $ (tail fuscs) ++ [1]
   interleave []     ys = ys
   interleave (x:xs) ys = x : interleave ys xs
-- Reinhard Zumkeller, Aug 23 2011
  • Julia
    using Nemo
function A002487List(len)
    a, A = QQ(0), [0,1]
    for n in 1:len
        a = next_calkin_wilf(a)
        push!(A, denominator(a))
    end
A end
A002487List(91) |> println # Peter Luschny, Mar 13 2018
  • Magma
    [&+[(Binomial(k, n-k-1) mod 2): k in [0..n]]: n in [0..100]]; // Vincenzo Librandi, Jun 18 2019
  • Maple
    A002487 := proc(n) option remember; if n <= 1 then n elif n mod 2 = 0 then procname(n/2); else procname((n-1)/2)+procname((n+1)/2); fi; end: seq(A002487(n),n=0..91);
A002487 := proc(m) local a,b,n; a := 1; b := 0; n := m; while n>0 do if type(n,odd) then b := a+b else a := a+b end if; n := floor(n/2); end do; b; end proc: seq(A002487(n),n=0..91); # Program adapted from E. Dijkstra, Selected Writings on Computing, Springer, 1982, p. 232. - Igor Urbiha (urbiha(AT)math.hr), Oct 28 2002. Since A007306(n) = a(2*n+1), this program can be adapted for A007306 by replacing b := 0 by b := 1.
A002487 := proc(n::integer) local k; option remember; if n = 0 then 0 elif n=1 then 1 else add(K(k,n-1-k)*procname(n - k), k = 1 .. n) end if end proc:
K := proc(n::integer, k::integer) local KC; if 0 <= k and k <= n and n-k <= 2 then KC:=1; else KC:= 0; end if; end proc: seq(A002487(n),n=0..91); # Thomas Wieder, Jan 13 2008
# next Maple program:
a:= proc(n) option remember; `if`(n<2, n,
      (q-> a(q)+(n-2*q)*a(n-q))(iquo(n, 2)))
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Feb 11 2021
fusc := proc(n) local a, b, c; a := 1; b := 0;
    for c in convert(n, base, 2) do
        if c = 0 then a := a + b else b := a + b fi od;
    b end:
seq(fusc(n), n = 0..91); # Peter Luschny, Nov 09 2022
Stern := proc(n, u) local k, j, b;
    b := j -> nops({seq(Bits:-Xor(k, j-k), k = 0..j)}):
    ifelse(n=0, 1-u, seq(b(j), j = 2^(n-1)-1..2^n-1-u)) end:
seq(print([n], Stern(n, 1)), n = 0..5); # As shown in the comments.
seq(print([n], Stern(n, 0)), n = 0..5); # As shown in the examples. # Peter Luschny, Sep 29 2024
  • Mathematica
    a[0] = 0; a[1] = 1; a[n_] := If[ EvenQ[n], a[n/2], a[(n-1)/2] + a[(n+1)/2]]; Table[ a[n], {n, 0, 100}] (* end of program *)
Onemore[l_] := Transpose[{l, l + RotateLeft[l]}] // Flatten;
NestList[Onemore, {1}, 5] // Flatten  (*gives [a(1), ...]*) (* Takashi Tokita, Mar 09 2003 *)
ToBi[l_] := Table[2^(n - 1), {n, Length[l]}].Reverse[l]; Map[Length,
Split[Sort[Map[ToBi, Table[IntegerDigits[n - 1, 3], {n, 500}]]]]]  (*give [a(1), ...]*) (* Takashi Tokita, Mar 10 2003 *)
A002487[m_] := Module[{a = 1, b = 0, n = m}, While[n > 0, If[OddQ[n], b = a+b, a = a+b]; n = Floor[n/2]]; b]; Table[A002487[n], {n, 0, 100}] (* Jean-François Alcover, Sep 06 2013, translated from 2nd Maple program *)
a[0] = 0; a[1] = 1;
Flatten[Table[{a[2*n] = a[n], a[2*n + 1] = a[n] + a[n + 1]}, {n, 0, 50}]] (* Horst H. Manninger, Jun 09 2021 *)
nmax = 100; CoefficientList[Series[x*Product[(1 + x^(2^k) + x^(2^(k+1))), {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 08 2022 *)
  • PARI
    {a(n) = n=abs(n); if( n<2, n>0, a(n\2) + if( n%2, a(n\2 + 1)))};
  • PARI
    fusc(n)=local(a=1,b=0);while(n>0,if(bitand(n,1),b+=a,a+=b);n>>=1);b \\ Charles R Greathouse IV, Oct 05 2008
  • PARI
    A002487(n,a=1,b=0)=for(i=0,logint(n,2),if(bittest(n,i),b+=a,a+=b));b \\ M. F. Hasler, Feb 12 2017, updated Feb 14 2019
  • Python
    from functools import lru_cache
@lru_cache(maxsize=None)
def a(n): return n if n<2 else a(n//2) if n%2==0 else a((n - 1)//2) + a((n + 1)//2) # Indranil Ghosh, Jun 08 2017; corrected by Reza K Ghazi, Dec 27 2021
  • Python
    def a(n):
    a, b = 1, 0
    while n > 0:
        if n & 1:
            b += a
        else:
            a += b
        n >>= 1
    return b
# Reza K Ghazi, Dec 29 2021
  • Python
    def A002487(n): return sum(int(not (n-k-1) & ~k) for k in range(n)) # Chai Wah Wu, Jun 19 2022
  • Python
    # (fast way for big vectors)
from math import log, ceil
import numpy
how_many_terms = 2**20  # (Powers of 2 recommended but other integers are also possible.)
A002487, A002487[1]  = numpy.zeros(2**(ce:=ceil(log(how_many_terms,2))), dtype=object), 1
for exponent in range(1,ce):
    L, L2 = 2**exponent, 2**(exponent+1)
    A002487[L2 - 1] = exponent + 1
    A002487[L:L2][::2] = A002487[L >> 1: L]
    A002487[L + 1:L2 - 2][::2] = A002487[L:L2 - 3][::2]  +  A002487[L + 2:L2 - 1][::2]
print(list(A002487[0:100])) # Karl-Heinz Hofmann, Jul 22 2025
  • R
    N <- 50 # arbitrary
a <- 1
for (n in 1:N)
{
  a[2*n    ] = a[n]
  a[2*n + 1] = a[n] + a[n+1]
  a
}
a
# Yosu Yurramendi, Oct 04 2014
  • R
    # Given n, compute a(n) by taking into account the binary representation of n
a <- function(n){
  b <- as.numeric(intToBits(n))
  l <- sum(b)
  m <- which(b == 1)-1
  d <- 1
  if(l > 1) for(j in 1:(l-1)) d[j] <- m[j+1]-m[j]+1
  f <- c(0,1)
  if(l > 1) for(j in 3:(l+1)) f[j] <- d[j-2]*f[j-1]-f[j-2]
  return(f[l+1])
} # Yosu Yurramendi, Dec 13 2016
  • R
    # computes the sequence as a vector A, rather than function a() as above.
A <- c(1,1)
maxlevel <- 5 # by choice
for(m in 1:maxlevel) {
  A[2^(m+1)] <- 1
  for(k in 1:(2^m-1)) {
    r <- m - floor(log2(k)) - 1
    A[2^r*(2*k+1)] <- A[2^r*(2*k)] + A[2^r*(2*k+2)]
}}
A # Yosu Yurramendi, May 08 2018
  • Sage
    def A002487(n):
    M = [1, 0]
    for b in n.bits():
        M[b] = M[0] + M[1]
    return M[1]
print([A002487(n) for n in (0..91)])
# For a dual see A174980. Peter Luschny, Nov 28 2017
  • Scheme
    ;; An implementation of memoization-macro definec can be found for example in: http://oeis.org/wiki/Memoization
(definec (A002487 n) (cond ((<= n 1) n) ((even? n) (A002487 (/ n 2))) (else (+ (A002487 (/ (- n 1) 2)) (A002487 (/ (+ n 1) 2))))))
;; Antti Karttunen, Nov 05 2016

Formula

a(n+1) = (2*k+1)*a(n) - a(n-1) where k = floor(a(n-1)/a(n)). - David S. Newman, Mar 04 2001
Let e(n) = A007814(n) = exponent of highest power of 2 dividing n. Then a(n+1) = (2k+1)*a(n)-a(n-1), n > 0, where k = e(n). Moreover, floor(a(n-1)/a(n)) = e(n), in agreement with D. Newman's formula. - Dragutin Svrtan (dsvrtan(AT)math.hr) and Igor Urbiha (urbiha(AT)math.hr), Jan 10 2002
Calkin and Wilf showed 0.9588 <= limsup a(n)/n^(log(phi)/log(2)) <= 1.1709 where phi is the golden mean. Does this supremum limit = 1? - Benoit Cloitre, Jan 18 2004. Coons and Tyler show the limit is A246765 = 0.9588... - Kevin Ryde, Jan 09 2021
a(n) = Sum_{k=0..floor((n-1)/2)} (binomial(n-k-1, k) mod 2). - Paul Barry, Sep 13 2004
a(n) = Sum_{k=0..n-1} (binomial(k, n-k-1) mod 2). - Paul Barry, Mar 26 2005
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v^3 + 2*u*v*w - u^2*w. - Michael Somos, May 02 2005
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1^3*u6 - 3*u1^2*u2*u6 + 3*u2^3*u6 - u2^3*u3. - Michael Somos, May 02 2005
G.f.: x * Product_{k>=0} (1 + x^(2^k) + x^(2^(k+1))) [Carlitz].
a(n) = a(n-2) + a(n-1) - 2*(a(n-2) mod a(n-1)). - Mike Stay, Nov 06 2006
A079978(n) = (1 + e^(i*Pi*A002487(n)))/2, i=sqrt(-1). - Paul Barry, Jan 14 2005
a(n) = Sum_{k=1..n} K(k, n-k)*a(n - k), where K(n,k) = 1 if 0 <= k AND k <= n AND n-k <= 2 and K(n,k) = 0 else. (When using such a K-coefficient, several different arguments to K or several different definitions of K may lead to the same integer sequence. For example, if we drop the condition k <= n in the above definition, then we arrive at A002083 = Narayana-Zidek-Capell numbers.) - Thomas Wieder, Jan 13 2008
a(k+1)*a(2^n - k) - a(k)*a(2^n - (k+1)) = 1; a(2^n - k) + a(k) = a(2^(n+1) + k). Both formulas hold for 0 <= k <= 2^n - 1. G.f.: G(z) = a(1) + a(2)*z + a(3)*z^2 + ... + a(k+1)*z^k + ... Define f(z) = (1 + z + z^2), then G(z) = lim f(z)*f(z^2)*f(z^4)* ... *f(z^(2^n))*... = (1 + z + z^2)*G(z^2). - Arie Werksma (werksma(AT)tiscali.nl), Apr 11 2008
a(k+1)*a(2^n - k) - a(k)*a(2^n - (k+1)) = 1 (0 <= k <= 2^n - 1). - Arie Werksma (werksma(AT)tiscali.nl), Apr 18 2008
a(2^n + k) = a(2^n - k) + a(k) (0 <= k <= 2^n). - Arie Werksma (werksma(AT)tiscali.nl), Apr 18 2008
Let g(z) = a(1) + a(2)*z + a(3)*z^2 + ... + a(k+1)*z^k + ..., f(z) = 1 + z + z^2. Then g(z) = lim_{n->infinity} f(z)*f(z^2)*f(z^4)*...*f(z^(2^n)), g(z) = f(z)*g(z^2). - Arie Werksma (werksma(AT)tiscali.nl), Apr 18 2008
For 0 <= k <= 2^n - 1, write k = b(0) + 2*b(1) + 4*b(2) + ... + 2^(n-1)*b(n-1) where b(0), b(1), etc. are 0 or 1. Define a 2 X 2 matrix X(m) with entries x(1,1) = x(2,2) = 1, x(1,2) = 1 - b(m), x(2,1) = b(m). Let P(n)= X(0)*X(1)* ... *X(n-1). The entries of the matrix P are members of the sequence: p(1,1) = a(k+1), p(1,2) = a(2^n - (k+1)), p(2,1) = a(k), p(2,2) = a(2^n - k). - Arie Werksma (werksma(AT)tiscali.nl), Apr 20 2008
Let f(x) = A030101(x); if 2^n + 1 <= x <= 2^(n + 1) and y = 2^(n + 1) - f(x - 1) then a(x) = a(y). - Arie Werksma (Werksma(AT)Tiscali.nl), Jul 11 2008
a(n) = A126606(n + 1) / 2. - Reikku Kulon, Oct 05 2008
Equals infinite convolution product of [1,1,1,0,0,0,0,0,0] aerated A000079 - 1 times, i.e., [1,1,1,0,0,0,0,0,0] * [1,0,1,0,1,0,0,0,0] * [1,0,0,0,1,0,0,0,1]. - Mats Granvik and Gary W. Adamson, Oct 02 2009; corrected by Mats Granvik, Oct 10 2009
a(2^(p+2)*n+2^(p+1)-1) - a(2^(p+1)*n+2^p-1) = A007306(n+1), p >= 0 and n >= 0. - Johannes W. Meijer, Feb 07 2013
a(2*n-1) = A007306(n), n > 0. - Yosu Yurramendi, Jun 23 2014
a(n*2^m) = a(n), m>0, n > 0. - Yosu Yurramendi, Jul 03 2014
a(k+1)*a(2^m+k) - a(k)*a(2^m+(k+1)) = 1 for m >= 0, 0 <= k < 2^m. - Yosu Yurramendi, Nov 07 2014
a(2^(m+1)+(k+1))*a(2^m+k) - a(2^(m+1)+k)*a(2^m+(k+1)) = 1 for m >= 0, 0 <= k < 2^m. - Yosu Yurramendi, Nov 07 2014
a(5*2^k) = 3. a(7*2^k) = 3. a(9*2^k) = 4. a(11*2^k) = 5. a(13*2^k) = 5. a(15*2^k) = 4. In general: a((2j-1)*2^k) = A007306(j), j > 0, k >= 0 (see Adamchuk's comment). - Yosu Yurramendi, Mar 05 2016
a(2^m+2^m'+k') = a(2^m'+k')*(m-m'+1) - a(k'), m >= 0, m' <= m-1, 0 <= k' < 2^m'. - Yosu Yurramendi, Jul 13 2016
From Yosu Yurramendi, Jul 13 2016: (Start)
Let n be a natural number and [b_m b_(m-1) ... b_1 b_0] its binary expansion with b_m=1.
Let L = Sum_{i=0..m} b_i be the number of binary digits equal to 1 (L >= 1).
Let {m_j: j=1..L} be the set of subindices such that b_m_j = 1, j=1..L, and 0 <= m_1 <= m_2 <= ... <= m_L = m.
If L = 1 then c_1 = 1, otherwise let {c_j: j=1..(L-1)} be the set of coefficients such that c_(j) = m_(j+1) - m_j + 1, 1 <= j <= L-1.
Let f be a function defined on {1..L+1} such that f(1) = 0, f(2) = 1, f(j) = c_(j-2)*f(j-1) - f(j-2), 3 <= j <= L+1.
Then a(n) = f(L+1) (see example). (End)
a(n) = A001222(A260443(n)) = A000120(A277020(n)). Also a(n) = A000120(A101624(n-1)) for n >= 1. - Antti Karttunen, Nov 05 2016
(a(n-1) + a(n+1))/a(n) = A037227(n) for n >= 1. - Peter Bala, Feb 07 2017
a(0) = 0; a(3n) = 2*A000360(3n-1); a(3n+1) = 2*A000360(3n) - 1; a(3n+2) = 2*A000360(3n+1) + 1. - M. Jeremie Lafitte (Levitas), Apr 24 2017
From I. V. Serov, Jun 14 2017: (Start)
a(n) = A287896(n-1) - 1*A288002(n-1) for n > 1;
a(n) = A007306(n-1) - 2*A288002(n-1) for n > 1. (End)
From Yosu Yurramendi, Feb 14 2018: (Start)
a(2^(m+2) + 2^(m+1) + k) - a(2^(m+1) + 2^m + k) = 2*a(k), m >= 0, 0 <= k < 2^m.
a(2^(m+2) + 2^(m+1) + k) - a(2^(m+1) + k) = a(2^m + k), m >= 0, 0 <= k < 2^m.
a(2^m + k) = a(k)*(m - floor(log_2(k)) - 1) + a(2^(floor(log_2(k))+1) + k), m >= 0, 0 < k < 2^m, a(2^m) = 1, a(0) = 0. (End)
From Yosu Yurramendi, May 08 2018: (Start)
a(2^m) = 1, m >= 0.
a(2^r*(2*k+1)) = a(2^r*(2*k)) + a(2^r*(2*k+2)), r < - m - floor(log_2(k)) - 1, m > 0, 1 <= k < 2^m. (End)
Trow(n) = [card({k XOR (j-k): k=0..j}) for j = 2^(n-1)-1..2^n-2] when regarded as an irregular table (n >= 1). - Peter Luschny, Sep 29 2024
a(n) = A000120(A168081(n)). - Karl-Heinz Hofmann, Jun 16 2025

Extensions

Additional references and comments from Len Smiley, Joshua Zucker, Rick L. Shepherd and Herbert S. Wilf
Typo in definition corrected by Reinhard Zumkeller, Aug 23 2011
Incorrect formula deleted and text edited by Johannes W. Meijer, Feb 07 2013

A101625 A bisection of the Stern-Jacobsthal numbers.

Original entry on oeis.org

0, 1, 1, 5, 1, 21, 17, 69, 1, 277, 273, 1349, 257, 5141, 4113, 16453, 1, 65813, 65809, 329029, 65793, 1381397, 1118225, 4538437, 65537, 18088213, 17826065, 88081733, 16777473, 335549461, 268439569, 1073758277, 1, 4295033109, 4295033105
Offset: 0

Views

Author

Paul Barry, Dec 10 2004

Keywords

Links

Crossrefs

Cf. A002450.

Programs

  • Python
    prpr = 0
prev = 1
for i in range(99):
    current = (prev)^(prpr*4)
    print(prpr, end=',')
    prpr = prev
    prev = current
# Alex Ratushnyak, May 06 2012

Formula

a(n) = Sum_{k=0..n} (binomial(2n-k, k-1) mod 2)2^(k-1);
a(n) = A101624(2n+1).
a(0)=0, a(1)=1, a(n) = a(n-1) XOR (a(n-2)*4), where XOR is the bitwise exclusive-OR operator. - Alex Ratushnyak, May 06 2012
a(n+1) = Sum_{k=0..n} A106344(n,k)*4^(n-k). - Philippe Deléham, May 27 2012

A006921 Diagonals of Pascal's triangle mod 2 interpreted as binary numbers.

Original entry on oeis.org

1, 1, 3, 2, 7, 5, 13, 8, 29, 21, 55, 34, 115, 81, 209, 128, 465, 337, 883, 546, 1847, 1301, 3357, 2056, 7437, 5381, 14087, 8706, 29443, 20737, 53505, 32768, 119041, 86273, 226051, 139778, 472839, 333061, 859405, 526344, 1903901, 1377557, 3606327
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A011973, A000079, A047999 (Sierpiński), A007318, A101624.
Cf. A168081, A260022.
Cf. A257971 (first differences).

Programs

  • Haskell
    a006921 = sum . zipWith (*)
                a000079_list . map (flip mod 2) . reverse . a011973_row
-- Reinhard Zumkeller, Jul 14 2015
  • Maple
    b2:=(n,k)->binomial(n,k) mod 2;
H:=n->add(b2(n-r,r)*2^( floor(n/2)-r ), r=0..floor(n/2));
[seq(H(n),n=0..30)]; # N. J. A. Sloane, Jul 14 2015
  • Python
    def A006921(n): return sum(int(not r & ~(n-r))*2**(n//2-r) for r in range(n//2+1)) # Chai Wah Wu, Jun 20 2022

Formula

a(2*n) = A260022(n); a(2*n+1) = A168081(n+1). - Reinhard Zumkeller, Jul 14 2015
a(n) = Sum_{r=0..n/2} binomial(n-r,r){mod 2} * 2^(floor(n/2)-r). - _N. J. A. Sloane, Jul 14 2015

A099902 Multiplies by 2 and shifts right under the XOR BINOMIAL transform (A099901).

Original entry on oeis.org

1, 3, 7, 11, 23, 59, 103, 139, 279, 827, 1895, 2955, 5655, 14395, 24679, 32907, 65815, 197435, 460647, 723851, 1512983, 3881019, 6774887, 9142411, 18219287, 54002491, 123733863, 192940939, 369104407, 939538491, 1610637415, 2147516555
Offset: 0

Views

Author

Paul D. Hanna, Oct 30 2004

Keywords

Comments

Equals the XOR BINOMIAL transform of A099901. Also, equals the main diagonal of the XOR difference triangle A099900, in which the central terms of the rows form the powers of 2.
Bisection of A101624. - Paul Barry, May 10 2005

Links

Crossrefs

Cf. A099884, A099900, A099901.

Programs

  • Maple
    a:= n -> add((binomial(n-k+floor(k/2),floor(k/2)) mod 2)*2^k, k=0..n):
map(a, [$0..100]); # Robert Israel, Jan 24 2016
  • PARI
    {a(n)=local(B);B=0;for(k=0,n,B=bitxor(B,binomial(n-k+k\2,k\2)%2*2^k));B}
  • PARI
    a(n)=sum(k=0,n,binomial(n-k+k\2,k\2)%2*2^k)
  • Python
    def A099902(n): return sum(int(not ~((n<<1)-k)&k)<Chai Wah Wu, Jul 30 2025

Formula

a(n) = SumXOR_{k=0..n} (binomial(n-k+floor(k/2), floor(k/2)) mod 2)*2^k for n >= 0.
a(n) = SumXOR_{i=0..n} (C(n, i) mod 2)*A099901(n-i), where SumXOR is the analog of summation under the binary XOR operation and C(i, j) mod 2 = A047999(i, j).
a(n) = Sum_{k=0..n} A047999(n-k+floor(k/2), floor(k/2)) * 2^k.
From Paul Barry, May 10 2005: (Start)
a(n) = Sum_{k=0..2n} (binomial(k, 2n-k) mod 2)*2^(2n-k);
a(n) = Sum_{k=0..n} (binomial(2n-k, k) mod 2)*2^k. (End)
a(n) = Sum_{k=0..2n} A106344(2n,k)*2^(2n-k). - Philippe Deléham, Dec 18 2008

A166555 Triangle read by rows, T(n, k) = 2^k * A047999(n, k).

Original entry on oeis.org

1, 1, 2, 1, 0, 4, 1, 2, 4, 8, 1, 0, 0, 0, 16, 1, 2, 0, 0, 16, 32, 1, 0, 4, 0, 16, 0, 64, 1, 2, 4, 8, 16, 32, 64, 128, 1, 0, 0, 0, 0, 0, 0, 0, 256, 1, 2, 0, 0, 0, 0, 0, 0, 256, 512, 1, 0, 4, 0, 0, 0, 0, 0, 256, 0, 1024, 1, 2, 4, 8, 0, 0, 0, 0, 256, 512, 1024, 2048, 1, 0, 0, 0, 16, 0, 0, 0, 256, 0, 0, 0, 4096
Offset: 0

Views

Author

Gary W. Adamson, Oct 17 2009

Keywords

Comments

Number of positive terms in n-th row (n>=0) equals to A000120(n). - Vladimir Shevelev, Oct 25 2010
Former name: Triangle read by rows, Sierpinski's gasket, A047999 * (1,2,4,8,...) diagonalized. - G. C. Greubel, Dec 02 2024

Examples

			First few rows of the triangle are:
  1;
  1, 2;
  1, 0, 4;
  1, 2, 4, 8;
  1, 0, 0, 0, 16;
  1, 2, 0, 0, 16,.32;
  1, 0, 4, 0, 16,..0,..64;
  1, 2, 4, 8, 16,.32,..64,..128;
  1, 0, 0, 0,..0,..0,...0,....0,..256;
  1, 2, 0, 0,..0,..0,...0,....0,..256,...512;
  1, 0, 4, 0,..0,..0,...0,....0,..256,.....0,...1024;
  1, 2, 4, 8,..0,..0,...0,....0,..256,...512,...1024,...2048;
  1, 0, 0, 0, 16,..0,...0,....0,..256,.....0,......0,......0,..4096;
  ...

Links

Crossrefs

Cf. A000007, A000079, A000120, A038183, A147999.
Sums include: A001317 (row), A101624 (diagonal), A101625 (odd rows of signed diagonal).

Programs

  • Magma
    A166555:= func< n,k | 2^k*( Binomial(n,k) mod 2) >;
[A166555(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Dec 01 2024
  • Mathematica
    A166555[n_, k_]:= 2^k*Mod[Binomial[n, k], 2];
Table[A166555[n,k], {n,0,14}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 01 2024 *)
  • SageMath
    def A166555(n,k): return 2^k*int(not ~n & k) if kA166555(n,k) for k in range(n+1)] for n in range(15)])) # G. C. Greubel, Dec 01 2024

Formula

Triangle read by rows, A047999 * Q. A047999 = Sierpinski's gasket, Q = an infinite lower triangular matrix with (1,2,4,8,...) as the main diagonal and the rest zeros.
Sum_{k=0..n} T(n, k) = A001317(n).
From G. C. Greubel, Dec 02 2024: (Start)
T(n, k) = 2^k * (binomial(n,k) mod 2).
T(n, n) = A000079(n).
T(2*n, n) = A000007(n).
Sum_{k=0..n} (-1)^k*T(n, k) = (1/2)*( (1+(-1)^n)*A038183(n/2) - (1-(-1)^n) *A038183((n-1)/2) ).
Sum_{k=0..floor(n/2)} T(n-k, k) = A101624(n).
Sum_{k=0..floor((2*m+1)/2)} T(2*m-k+1, k) = A101625(m+1), m >= 0. (End)

Extensions

New name by G. C. Greubel, Dec 02 2024

A101623 Modular binomial transform of 10^n.

Original entry on oeis.org

1, 1, 11, 1, 111, 101, 1011, 1, 10111, 10101, 111011, 10001, 1100111, 1000101, 10001011, 1, 100010111, 100010101, 1100111011, 100010001, 11101100111, 10101000101, 101110001011, 100000001, 1011000010111, 1010000010101, 11100000111011, 1000000010001, 110000001100111
Offset: 0

Views

Author

Paul Barry, Dec 10 2004

Keywords

Crossrefs

A101624 in binary.

Programs

  • PARI
    a(n) = sum(k=0, n, lift(Mod(binomial(k, n-k), 2))*10^(n-k)); \\ Michel Marcus, Jul 31 2025
  • Python
    def A101623(n): return sum(int(not k & ~(n-k))*10**k for k in range((n>>1)+1)) # Chai Wah Wu, Jul 30 2025

Formula

a(n) = Sum_{k=0..floor(n/2)} mod(binomial(n-k, k), 2)*10^k.
a(n) = Sum_{k=0..n} mod(binomial(k, n-k), 2)*10^(n-k).

Extensions

More terms from Michel Marcus, Jul 31 2025

A204771 a(n) = a(n-1) XOR (a(n-2)*3).

Original entry on oeis.org

0, 1, 1, 2, 1, 7, 4, 17, 29, 46, 121, 243, 408, 833, 1929, 3658, 6353, 12815, 30844, 61009, 100133, 216534, 514233, 930107, 1686288, 3352737, 8264081, 15163506, 27077825, 53153175, 133991380, 243114769, 428343405, 854649182, 2120804377, 3870970883, 6937439304
Offset: 0

Views

Author

Alex Ratushnyak, May 07 2012

Keywords

Links

Crossrefs

Cf. A101624: a(n) = a(n-1) XOR (a(n-2)*2).
Cf. A101625: a(n) = a(n-1) XOR (a(n-2)*4).

Programs

  • Python
    prpr = 0
prev = 1
for i in range(99):
    current = (prev)^(prpr*3)
    print(prpr, end=',')
    prpr = prev
    prev = current

Formula

a(0)=0, a(1)=1, a(n) = a(n-1) XOR (a(n-2)*3), where XOR is the bitwise exclusive-OR operator.

A210680 a(n) = (3*a(n-1)) XOR a(n-2).

Original entry on oeis.org

0, 1, 3, 8, 27, 89, 272, 873, 2859, 8936, 25491, 67665, 228192, 752241, 2167859, 6833896, 18464907, 52758345, 142005584, 440498361, 1186119547, 3461957320, 9357060899, 26968655777, 72945663424, 226371206881, 613739200867, 1752444795592, 4702791627067, 14623717009785
Offset: 0

Views

Author

Alex Ratushnyak, May 09 2012

Keywords

Links

Crossrefs

Cf. A168081: a(n) = (2*a(n-1)) XOR a(n-2), a(0)=0, a(1)=1.
Cf. A101624: a(n) = (2*a(n-2)) XOR a(n-1), a(0)=0, a(1)=1.
Cf. A204771: a(n) = (3*a(n-2)) XOR a(n-1), a(0)=0, a(1)=1.

Programs

  • Mathematica
    RecurrenceTable[{a[0]==0,a[1]==1,a[n]==BitXor[3a[n-1],a[n-2]]},a,{n,30}] (* Harvey P. Dale, Nov 27 2014 *)

Formula

a(n) = (3*a(n-1)) XOR a(n-2), a(0)=0, a(1)=1.

A260534 Square array read by ascending antidiagonals, T(n,k) = Sum_{j=0..k} n^j*(C(k-j,j) mod 2).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 4, 1, 3, 1, 1, 1, 5, 1, 7, 2, 1, 1, 1, 6, 1, 13, 5, 3, 1, 1, 1, 7, 1, 21, 10, 11, 1, 1, 1, 1, 8, 1, 31, 17, 31, 1, 4, 1, 1, 1, 9, 1, 43, 26, 69, 1, 23, 3, 1, 1, 1, 10, 1, 57, 37, 131, 1, 94, 21, 5, 1, 1, 1, 11
Offset: 0

Views

Author

Peter Luschny, Sep 20 2015

Keywords

Comments

A parametrization of Stern's diatomic series (which is here T(1,k)). (For other generalizations of Dijkstra's fusc function see the Luschny link.)

Examples

			Array starts:
n\k[0, 1,  2, 3,  4,  5,   6, 7,    8,    9,    10]
[0] 1, 1,  1, 1,  1,  1,   1, 1,    1,    1,     1, ...
[1] 1, 1,  2, 1,  3,  2,   3, 1,    4,    3,     5, ... [A002487]
[2] 1, 1,  3, 1,  7,  5,  11, 1,   23,   21,    59, ... [A101624]
[3] 1, 1,  4, 1, 13, 10,  31, 1,   94,   91,   355, ...
[4] 1, 1,  5, 1, 21, 17,  69, 1,  277,  273,  1349, ... [A101625]
[5] 1, 1,  6, 1, 31, 26, 131, 1,  656,  651,  3881, ...
[6] 1, 1,  7, 1, 43, 37, 223, 1, 1339, 1333,  9295, ...
[7] 1, 1,  8, 1, 57, 50, 351, 1, 2458, 2451, 19559, ...
[8] 1, 1,  9, 1, 73, 65, 521, 1, 4169, 4161, 37385, ...
-,-,-,-,A002061,A002522,A071568,-,-,A059826,-,A002523,

Links

Crossrefs

Cf. A002061, A002487, A002522, A002523, A059826, A071568, A101624, A101625.

Programs

  • Maple
    T := (n,k) -> add(modp(binomial(k-j,j),2)*n^j, j=0..k):
seq(lprint(seq(T(n,k),k=0..10)),n=0..5);
  • Mathematica
    Table[If[(n - k) == 0, 1, Sum[(n - k)^j Mod[Binomial[k - j, j], 2], {j, 0, k}]], {n, 0, 10}, {k, 0, n}] (* Michael De Vlieger, Sep 21 2015 *)
  • Python
    def A260534_T(n,k):
    return sum(0 if ~(k-j) & j else n**j for j in range(k+1)) # Chai Wah Wu, Feb 08 2016

A385114 a(n) is the least k with k > a(n-1) such that A000120(k) = A002487(n), with a(0) = 0.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 9, 11, 16, 23, 25, 31, 33, 47, 49, 51, 64, 79, 83, 127, 131, 255, 271, 319, 320, 351, 355, 383, 385, 415, 417, 419, 512, 543, 551, 767, 771, 2047, 2111, 2559, 2561, 3071, 3135, 8191, 8207, 10239, 10271, 10367, 12288, 12415, 12431, 13311, 13315, 14335
Offset: 0

Views

Author

Karl-Heinz Hofmann, Jun 18 2025

Keywords

Examples

			  n |  a(n)_2   a(n)   A002487(n)
----+-----------------------------
  0 |       0     0       0
  1 |       1     1       1
  2 |      10     2       1
  3 |      11     3       2
  4 |     100     4       1
  5 |     111     7       3
  6 |    1001     9       2
  7 |    1011    11       3
  8 |   10000    16       1
  9 |   10111    23       4
 10 |   11001    25       3
 11 |   11111    31       5
 12 |  100001    33       2
 13 |  101111    47       5
 14 |  110001    49       3

Links

Crossrefs

Cf. A000120, A002487, A101624, A168081, A277020.
Cf. A212288, A212289.

Programs

  • Mathematica
    k = s[0] = 0; s[1] = 1; s[n_] := s[n] = If[EvenQ[n], s[n/2], s[(n - 1)/2] + s[(n + 1)/2]]; Array[s, 2^10]; {k}~Join~Monitor[Table[Set[k, Max[k + 1, 2^s[n] - 1]]; While[DigitCount[k, 2, 1] != s[n], k++]; k, {n, 100}], n] (* Michael De Vlieger, Jul 14 2025 *)
  • PARI
    lista(nn) = my(list = List([0]), last = 0); for (n=1, nn, my(k=last+1, d=dia(n)); while (hammingweight(k) != d, k++); last = k; listput(list, k);); Vec(list); \\ Michel Marcus, Jul 12 2025
  • Python
    # see links

Formula

a(A212288(n)) >= 2^A212289(n+1) - 1. - Hugo Pfoertner, Jul 15 2025
Showing 1-10 of 10 results.

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