A272569 A variation on Stern's diatomic sequence.

0, 0, 1, 0, 2, 1, 2, 0, 3, 2, 6, 1, 6, 2, 3, 0, 4, 3, 10, 2, 15, 6, 12, 1, 12, 6, 15, 2, 10, 3, 4, 0, 5, 4, 14, 3, 24, 10, 21, 2, 28, 15, 40, 6, 35, 12, 20, 1, 20, 12, 35, 6, 40, 15, 28, 2, 21, 10, 24, 3, 14, 4, 5, 0, 6, 5, 18, 4, 33, 14, 30, 3, 44, 24, 65, 10

Offset

1,5

Comments

This sequence has an analogous relationship to A001654 as A002487 has to A000045; maxima between a(2^n) and a(2^n+1) = A001654(n).

For 2^k<=n<=2^k+1: a(n) = A002487(2^(k+1)-n)*A002487(n-2^k).

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Sam Northshield, Three analogues of Stern's diatomic sequence, arXiv preprint arXiv:1503.03433 [math.CO], 2015; also, Proceedings of the 16th International Conference on Fibonacci Numbers and Their Applications, Rochester Institute of Technology, Rochester, New York, July 20-27, 2014.

Sam Northshield, Some generalizations of a formula of Reznick, SUNY Plattsburgh (2022).

Formula

a(2n) = a(n), a(2n+1) = a(n) + a(n+1) + (4a(n)*a(n+1)+1)^(1/2).

Mathematica

nn = 100;
a[_] = 0; a[1] = 0; Do[a[n] = If[EvenQ[n], a[n/2], m = (n-1)/2; a[m] + a[m + 1] + Sqrt[1 + 4 a[m] a[m+1]] // Floor], {n, 2, nn}];
Array[a, nn] (* Jean-François Alcover, Sep 25 2018, from PARI *)

Prog

(PARI) lista(nn) = {va = vector(nn); va[1] = 0; for (n=2, nn, if (n % 2 == 0, va[n] = va[n/2], m = (n-1)/2; va[n] = va[m] + va[m+1] + sqrtint(1 + 4*va[m]*va[m+1])); ); va; } \\ Michel Marcus, May 03 2016

Crossrefs

Cf. A002487, A001654.

Sequence in context: A365385 A366788 A290537 * A344788 A372687 A068076

Adjacent sequences: A272566 A272567 A272568 * A272570 A272571 A272572

Keyword

nonn

Author

Max Barrentine, May 02 2016

Status

approved