Search: id:A002487 Results 1-1 of 1 results found. %I A002487 M0141 N0056 %S A002487 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, %T A002487 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, %U A002487 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 %N A002487 Stern's diatomic series: a(0) = 0, a(1) = 1; for n >= 0, a(2n) = a(n), a(2n+1) = a(n) + a(n+1). %C A002487 Also called fusc(n). %C A002487 a(n)/a(n+1) runs through all the reduced nonnegative rationals exactly once [Stern; Calkin and Wilf] %C A002487 If the terms are written as an array: %C A002487 1 %C A002487 1,2 %C A002487 1,3,2,3 %C A002487 1,4,3,5,2,5,3,4 %C A002487 1,5,4,7,3,8,5,7,2,7,5,8,3,7,4,5 %C A002487 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 %C A002487 then the sum of the k-th row is 3^(k-1), each columns 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 %C A002487 Number of odd Stirling numbers S_2(n+1,2r+1) [Carlitz] %C A002487 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. %C A002487 a(n+1) = number of alternating bit sets in n (see Finch). %C A002487 If f(x):=1/(1+floor(x)-frac(x)) then f(a(n-1)/a(n))=a(n)/a(n+1) except for n=0 or n=-1. If T(x):=-1/x and f(x)=y, then f(T(y))=T(x) except for x=0. - Michael Somos Sep 03 2006 %C A002487 a(n+1) = 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] %C A002487 a(n+1) = partitions of the n-th integer expressible as the sum of distinct even-subscripted Fibonacci numbers (=A054204(n)), into sums of distinct Fibonacci numbers numbers. [Bicknell-Johnson] %C A002487 a(n+1) = number of odd binomial(n-k,k),0<=2k1. - Alexander Adamchuk (alex(AT)kolmogorov.com), Oct 10 2006 %C A002487 The coefficients of the inverse of the GF of this sequence form A073469 and are related to binary partitions A000123. - Philippe Flajolet (Philippe.Flajolet(AT)inria.fr), Sep 06 2008 %C A002487 It appears that are also the odd entries in diagonals in Pascal's triangle at 45 degrees slope [From Javier Torres (adaycalledzero(AT)hotmail.com), Aug 06 2009] %D A002487 M. Aigner and G. M. Ziegler, Proofs from THE BOOK, 3rd ed., Berlin, Heidelberg, New York: Springer-Verlag, 2004, p. 97. %D A002487 J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197. %D A002487 J.-P. Allouche and J. Shallit, The ring of k-regular sequences, II, Theoret. Computer Sci., 307 (2003), 3-29. %D A002487 E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 114. %D A002487 M. Bicknell-Johnson, Stern's Diatomic Array Applied to Fibonacci Representations, " Fibonacci Quarterly 41 (2003), pp. 169-180. %D A002487 N. Calkin and H. S. Wilf, Recounting the rationals, Amer. Math. Monthly, 107 (No. 4, 2000), pp. 360-363. %D A002487 L. Carlitz, A problem in partitions related to the Stirling numbers, Bull. Amer. Math. Soc., 70(2) (1964), pp. 275-278. %D A002487 L. Carlitz, A problem in partitions related to the Stirling numbers, Riv. Mat. Univ. Parma, (2) 5 (1964), 61-75. %D A002487 Courtright, K. M. and Sellers, J. A., Arithmetic Properties for Hyper m-ary Partitions, INTEGERS 4 (2004), Article A6 %D A002487 G. De Rham, Un peu de mathematiques a propos d'une courbe plane, Elemente der Math. 2 (1947), pp. 73-76 and 89-97. %D A002487 E. Dijkstra, Selected Writings on Computing, Springer, 1982, p. 232 (sequence is called fusc). %D A002487 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) 36-42, Feb 18, 1850. Werke, II, pp. 705-711. %D A002487 S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 2.16.3; pp. 148-149. %D A002487 G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255. %D A002487 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 117. %D A002487 Hinz, A. M.; Klavzar, S; Milutinovic, U.; Parisse, D.; and Petr, C., Metric properties of the Tower of Hanoi graphs and Stern's diatomic sequence. European J. Combin. 26 (2005), no. 5, 693-708. %D A002487 D. E. Knuth, Problem 10906, American Mathematical Monthly; solution by Moshe Newman, American Mathematical Monthly, 110 (No. 7, 2003), 642-643. %D A002487 T. Koshy, Fibonacci and Lucas numbers with applications, Wiley, 2001, p. 98. %D A002487 D. H. Lehmer, On Stern's Diatomic Series, Amer. Math. Monthly 36(1) 1929, pp. 59-67. %D A002487 B. Reznick, Some binary partition functions, in "Analytic number theory" (Conf. in honor P. T. Bateman, Allerton Park, IL, 1989), 451-477, Progr. Math., 85, Birkhaeuser Boston, Boston, MA, 1990. %D A002487 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). %D A002487 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). %D A002487 M. A. Stern, Ueber eine zahlentheoretische Funktion, J. Reine Angew. Math., 55 (1858), 193-220. %H A002487 T. D. Noe, Table of n, a(n) for n = 0..10000 %H A002487 J.-P. Allouche, M. Mendes France, A. Lubiw, A.J. van der Poorten and J. Shallit, Convergents of folded continued fractions %H A002487 J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197. %H A002487 J.-P. Allouche and J. Shallit, The Ring of k-regular Sequences, II %H A002487 L. Carlitz, A problem in partitions related to the Stirling numbers (abstract) %H A002487 E. W. Dijkstra, An exercise for Dr. R. M. Burstall %H A002487 E. W. Dijkstra, More about the function ``fusc'' %H A002487 S. R. Finch, P. Sebah and Z.-Q. Bai, Odd Entries in Pascal's Trinomial Triangle (arXiv:0802.2654) %H A002487 Michael Gilleland, Some Self-Similar Integer Sequences %H A002487 B. Hayes, On the Teeth of Wheels %H A002487 N. J. A. Sloane, Stern-Brocot or Farey Tree %H A002487 R. Stephan, Divide-and-conquer generating functions. I. Elementary sequences %H A002487 Eric Weisstein's World of Mathematics, Stern's Diatomic Series %H A002487 Eric Weisstein's World of Mathematics, Calkin-Wilf Tree %H A002487 Thomas Wieder, HomePage. %H A002487 H. S. Wilf, Recounting the Rationals (with N. Calkin) %H A002487 Index entries for sequences related to Stern's sequences %H A002487 Index entries for "core" sequences %H A002487 Index entries for sequences related to trees %F A002487 a(0) = 0; a(1) = 1; a(2k) = a(k); a(2k+1) = a(k) + a(k+1) generates the same sequence with an initial 0 - David W. Wilson %F A002487 a(n+1) = (2k+1)a(n) - a(n-1) where k = [a(n-1) / a(n) ] is the largest integer smaller than or equal to a(n-1)/a(n) - David Newman (dznewman(AT)netvision.net.il), Mar 04, 2001 %F A002487 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. %F A002487 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 (benoit7848c(AT)orange.fr), Jan 18 2004 %F A002487 a(n)=sum{k=0..floor((n-1)/2), mod(binomial(n-k-1, k), 2)} - Paul Barry (pbarry(AT)wit.ie), Sep 13 2004 %F A002487 a(n)=sum{k=0..n-1, binomial(k, n-k-1) mod 2}; - Paul Barry (pbarry(AT)wit.ie), Mar 26 2005 %F A002487 G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^4)) where f(u, v, w)=v^3+2uvw-u^2w. - Michael Somos May 02 2005 %F A002487 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 %F A002487 For all integer n, a(-n)=-(-1)^n*a(n), a(n)=a(2n)sign(n)^n, a(n-1)+a(n)sign(n)=a(2n-1)sign(n)^n. - Michael Somos Sep 03 2006 %F A002487 G.f.: x Product_{k>=0} (1+x^2^k+x^2^{k+1}). [Carlitz] %F A002487 a(n)=a(n-2)+a(n-1)-2*(a(n-2) mod a(n-1)) where x mod y gives the remainder after dividing x by y - Mike Stay (mike(AT)math.ucr.edu), Nov 06 2006 %F A002487 A079978(n)=(1+e^(i*pi*A002487(n)))/2, i=sqrt(-1); - Paul Barry (pbarry(AT)wit.ie), Jan 14 2005 %F A002487 a(n)=sum(K(k, n-k)*a(n - k),k=1..n), 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 (thomas.wieder(AT)t-online.de), Jan 13 2008 %F A002487 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 %F A002487 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 %F A002487 a(2^n + k) = a(2^n - k) + a(k) (0 <= k <= 2^n). - Arie Werksma (werksma(AT)tiscali.nl), Apr 18 2008 %F A002487 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->00) 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 %F A002487 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 by 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 %F A002487 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 %F A002487 a(n) = A126606(n + 1) / 2. [From Reikku Kulon (reikku(AT)gmail.com), Oct 05 2008] %F A002487 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]. [From Mats Granvik, Gary W. Adamson (mats.granvik(AT)abo.fi), Oct 02 2009] %p A002487 A002487 := proc(n) option remember; if n <= 1 then n elif n mod 2 = 0 then A002487(n/2); else A002487((n-1)/2)+A002487((n+1)/2); fi; end; %p A002487 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: # 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(2n+1), this program can be adapted for A007306 by replacing b := 0 by b := 1. %p A002487 a := proc(n::integer) # A002487 Stern's diatomic series: a(0) = 0, a(1) = 1; for n >= 0, a(2n) = a(n), a(2n+1) = a(n) + a(n+1). local k; option remember; if n = 0 then 1 else add(K(k,n-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; - Thomas Wieder (thomas.wieder(AT)t-online.de), Jan 13 2008 %t A002487 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}] %t A002487 Onemore[l_] := Transpose[{l, l+RotateLeft[l]}]//Flatten NestList[Onemore, {1}, 5]//Flatten gives [a(1), ...] - Takashi Tokita Mar 09 2003 %t A002487 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 %o A002487 (PARI) a(n)=if(n<2,n>0,a(n\2)+if(n%2,a(n\2+1))) %o A002487 (PARI) {a(n)=if(n<0, -(-1)^n*a(-n), if(n<2, n>0, a(n\2)+if(n%2, a(n\2+1))))} /* Michael Somos Sep 03 2006 */ %o A002487 (PARI) fusc(n)=local(a=1,b=0);while(n>0,if(bitand(n,1),b+=a,a+=b);n>> =1);b [From Charles R Greathouse IV Oct 05 2008] %Y A002487 Cf. A084091. %Y A002487 If the 1's are replaced by pairs of 1's we obtain A049456. %Y A002487 Inverse: A020946. Cf. A020950, A046815, A070871, A070872, A071883. %Y A002487 A002487(A001045(n))=A000045(n). A002487(A062092(n))=A000032(n+1). %Y A002487 See also A064881-A064886 (Stern-Brocot subtrees). A column of A072170. %Y A002487 Cf. A001045, A002083. %Y A002487 Cf. A073459, A000123. %Y A002487 Cf. A126606, A049455 [From Reikku Kulon (reikku(AT)gmail.com), Oct 05 2008] %Y A002487 A011655. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Dec 27 2008] %Y A002487 Sequence in context: A070940 A020651 A160232 this_sequence A060162 A026730 A075256 %Y A002487 Adjacent sequences: A002484 A002485 A002486 this_sequence A002488 A002489 A002490 %K A002487 nonn,easy,nice,core %O A002487 0,4 %A A002487 N. J. A. Sloane (njas(AT)research.att.com). %E A002487 Additional references and comments from Leonard Smiley (smiley(AT)math.uaa.alaska.edu), Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), Rick L. Shepherd (rshepherd2(AT)hotmail.com) and Herb Wilf (wilf(AT)math.upenn.edu). %E A002487 More terms from Arie Werksma (Werksma(AT)Tiscali.nl), Jul 11 2008 %E A002487 Formula corrected by Mats Granvik (mats.granvik(AT)abo.fi), Oct 10 2009 Search completed in 0.003 seconds