A272569 - OEIS

%I #25 Dec 28 2022 01:57:38

%S 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,

%T 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,

%U 10,24,3,14,4,5,0,6,5,18,4,33,14,30,3,44,24,65,10

%N A variation on Stern's diatomic sequence.

%C 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).

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

%H Seiichi Manyama, <a href="/A272569/b272569.txt">Table of n, a(n) for n = 1..10000</a>

%H Sam Northshield, <a href="http://arxiv.org/abs/1503.03433">Three analogues of Stern's diatomic sequence</a>, arXiv preprint arXiv:1503.03433 [math.CO], 2015; <a href="https://www.fq.math.ca/Papers1/52-5/Northshield.pdf">also</a>, Proceedings of the 16th International Conference on Fibonacci Numbers and Their Applications, Rochester Institute of Technology, Rochester, New York, July 20-27, 2014.

%H Sam Northshield, <a href="https://www.researchgate.net/profile/Sam-Northshield/publication/366321658_SOME_GENERALIZATIONS_OF_A_FORMULA_OF_REZNICK/">Some generalizations of a formula of Reznick</a>, SUNY Plattsburgh (2022).

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

%t nn = 100;

%t 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}];

%t Array[a, nn] (* _Jean-François Alcover_, Sep 25 2018, from PARI *)

%o (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

%Y Cf. A002487, A001654.

%K nonn

%O 1,5

%A _Max Barrentine_, May 02 2016