|
%I #49 May 30 2022 23:35:31
%S 1,3,5,7,11,13,21,27,43,53,85,107,171,213,341,427,683,853,1365,1707,
%T 2731,3413,5461,6827,10923,13653,21845,27307,43691,54613,87381,109227,
%U 174763,218453,349525,436907,699051,873813,1398101,1747627,2796203,3495253
%N Locations of row maxima in "crushed" version of Stern's diatomic array.
%C From _Michel Marcus_, Jan 22 2015: (Start)
%C The Stern's diatomic array begins (see A049456).
%C 1...............................1
%C 1...............2...............1
%C 1.......3.......2.......3.......1
%C 1...4...3...5...2...5...3...4...1
%C 1.5.4.7.3.8.5.7.2.7.5.8.3.7.4.5.1
%C ...
%C The "crushed" version is obtained by removing the right column, and then squeezing everything to the left.
%C 1;
%C 1, 2;
%C 1, 3, 2, 3;
%C 1, 4, 3, 5, 2, 5, 3, 4;
%C 1, 5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 4, 5;
%C ...
%C This gives sequence 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, ... (cf. A002487).
%C The "crushed" array row maxima are: 1, 2, 3, 5, 8, ... (cf. A000045).
%C The indices of these values in A002487 are 1, 3, 5, 7, 11, ... : this sequence.
%C Note, for instance, that for 3rd row, the maximum which is 3, appears twice, at indices 5 and 7, giving 2 terms for this sequence.
%C (End)
%H S. Northshield, <a href="http://dx.doi.org/10.4169/000298910X496714">Stern's diatomic sequence 0, 1, 1, 2, 1, 3, 2, 3, 1, 4, ...</a>, Amer. Math. Monthly, 117 (2010), 581-598.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,0,2).
%F a(2n+1) + a(2n+2) = 3*2^(n+1), n>0 . - _Yosu Yurramendi_, Jun 29 2016
%F a(2n+3) = 3*2^(n+1) - a(n); a(2n+4) = 3*2^(n+1) + a(n), n>=0, a(0)=0 (new term), a(1)=1, a(2)=3 . - _Yosu Yurramendi_, Jun 29 2016
%F G.f.: x*(1 + 3*x + 4*x^2 + 4*x^3 + 4*x^4)/((1 + x^2)*(1 - 2*x^2)). - _Ilya Gutkovskiy_, Jun 29 2016
%F For n>1, a(n) = (2^(n/2 - 1)*(5 + 4*sqrt(2) + (-1)^n*(5 - 4*sqrt(2))) + cos(Pi*n/2) + sin(Pi*n/2))/3. - _Vaclav Kotesovec_, Jun 30 2016
%F a(2n) = a(2n-7) + 3*2^(n-1); a(2n-1) = a(2n-7) - 3*2^(n-1), n>=5 . - _Yosu Yurramendi_, Jul 06 2016
%F a(2n-1) = A168642(n), n>0; a(2n) = A048573(n), n>0; a(2n-1) = A026644(n) + 1, n>1; a(2n) = A084170(n) + 1, n>0 . - _Yosu Yurramendi_, Dec 11 2016
%e G.f. = x + 3*x^2 + 5*x^3 + 7*x^4 + 11*x^5 + 13*x^6 + 21*x^7 + 27*x^8 + 43*x^9 + ...
%t a[n_] := a[n] = If[n <= 5, {1, 3, 5, 7, 11}[[n]], a[n-2] + 2a[n-4]]; Array[a, 42] (* _Jean-François Alcover_, Dec 11 2016 *)
%o (PARI) fusc(n)=local(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); b; \\ from A002487
%o lista(nn) = {nb = 2^(nn+1)-1; vall = vector(nb, n, fusc(n)); for (n=1, nn, vmax = 0; for (j=2^(n-1), 2^n-1, if (vall[j] > vmax, vmax = vall[j]);); for (j=2^(n-1), 2^n-1, if (vall[j] == vmax, print1(j, ", "));););} \\ _Michel Marcus_, Jan 22 2015
%K nonn,easy
%O 1,2
%A _N. J. A. Sloane_, Aug 08 2010
%E More terms from _Michel Marcus_, Jan 22 2015
|
