A169969 Locations of row maxima in "crushed" version of Stern's diatomic array.

1, 3, 5, 7, 11, 13, 21, 27, 43, 53, 85, 107, 171, 213, 341, 427, 683, 853, 1365, 1707, 2731, 3413, 5461, 6827, 10923, 13653, 21845, 27307, 43691, 54613, 87381, 109227, 174763, 218453, 349525, 436907, 699051, 873813, 1398101, 1747627, 2796203, 3495253, 5592405

Offset: 1

Author

N. J. A. Sloane, Aug 08 2010

Keywords

• nonn
• easy

Comments

From Michel Marcus, Jan 22 2015: (Start)

The Stern's diatomic array begins (see A049456).

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

...

The "crushed" version is obtained by removing the right column, and then squeezing everything to the left.

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;

...

This gives sequence 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, ... (cf. A002487).

The "crushed" array row maxima are: 1, 2, 3, 5, 8, ... (cf. A000045).

The indices of these values in A002487 are 1, 3, 5, 7, 11, ... : this sequence.

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.

(End)

Examples

			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 + ...
		

Links

• S. Northshield, Stern's diatomic sequence 0, 1, 1, 2, 1, 3, 2, 3, 1, 4, ..., Amer. Math. Monthly, 117 (2010), 581-598.
• Index entries for linear recurrences with constant coefficients, signature (0,1,0,2).

Crossrefs

Cf. A000079, A026644, A048573, A049456, A084170, A168642.

Programs

Mathematica

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 *)

PARI

fusc(n)=local(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); b; \\ from A002487
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

Formula

a(2n+1) + a(2n+2) = 3*2^(n+1), n>0 . - Yosu Yurramendi, Jun 29 2016

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

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

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

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

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

Extensions

More terms from Michel Marcus, Jan 22 2015