cp's OEIS Frontend

A191372 The Sierpinski-Stern triangle.

%I A191372 #10 Nov 25 2019 01:00:19
%S A191372 0,1,2,1,2,1,3,2,3,2,3,1,4,2,3,2,3,1,4,3,4,2,3,1,4,3,5,2,3,1,4,3,5,2,
%T A191372 5,3,4,3,5,2,5,3,4,1,6,3,6,4,5,2,4,3,5,2,5,3,4,1,5,4,7,4,6,2,5,3,5,2,
%U A191372 5,3,4,1,5,4,7,3
%N A191372 The Sierpinski-Stern triangle.
%C A191372 The knight sums of the first and second kind Kn1y(n) = Kn2y(n), y >= 1, see A180662 for their definitions, of Sierpinski's triangle A047999 lead to the formula Kn1y(n) = A002487(n+(2*y-1)) - AS2S2S2(n,d) where the AS2S2S2(n,d) is the infinite concatenation of a S2(T, d = y-1) sequence; see for the first ten S2(T, d) and the first four Kn1y(n) the examples.
%C A191372 The A191372 sequence is the concatenation of all S2(T, d) sequences, d >= 0. The lengths of the S2(T, d) sequences are 2^ceiling(log(d)/log(2)) for d >= 1 while the length of S2(T, d=0) is 1.
%C A191372 Both the concatenation of the S2(T, d = 2^p) sequences, p >= 0, and the concatenation of the S2(T, d = 2^p-1) sequences, p >= 0, lead to Stern’s diatomic series A002487(n), n >= 2.
%C A191372 The differences of the sequences (AS2S2S2(T, 2^p-delta) - AS2S2S2(T, 2^(p-1)-delta)), T from 0 to (2^(p-1) -1) and 1 <= delta <= (2^(p-1)-1) (take care that p <= pmax), lead to sequences that are snippets of A002487 and, surprisingly, their reverse; see the examples.
%C A191372 The row sums of the Sierpinski-Stern triangle are given by the terms of A191487.
%H A191372 Sam Northshield, <a href="http://faculty.plattsburgh.edu/sam.northshield/08-0412.pdf">Stern's Diatomic Sequence 0,1,1,2,1,3,2,3,1,4,...</a>, Amer. Math. Month., Vol. 117 (7), pp. 581-598, 2010.
%H A191372 Igor Urbiha, <a href="http://hrcak.srce.hr/file/1591">Some properties of a function studied by De Rham, Carlitz and Dijkstra and its relation to the (Eisenstein -) Stern's diatomic sequence</a>, Math. Commun. 6, pp. 181-198, 2002.
%F A191372 The first few S2(T, d) rows of the Sierpinski-Stern triangle are:
%F A191372 d=0: [0]
%F A191372 d=1: [1]
%F A191372 d=2: [2, 1]
%F A191372 d=3: [2, 1, 3, 2]
%F A191372 d=4: [3, 2, 3, 1]
%F A191372 d=5: [4, 2, 3, 2, 3, 1, 4, 3]
%F A191372 d=6: [4, 2, 3, 1, 4, 3, 5, 2]
%F A191372 d=7: [3, 1, 4, 3, 5, 2, 5, 3]
%F A191372 d=8: [4, 3, 5, 2, 5, 3, 4, 1]
%F A191372 d=9: [6, 3, 6, 4, 5, 2, 4, 3, 5, 2, 5, 3, 4, 1, 5, 4]
%F A191372 The first four Kn1y(n), y = d+1, sequences:
%F A191372 Kn11(n) = A002487(n+1) - A000004(n)
%F A191372 Kn12(n) = A002487(n+3) - A000012(n)
%F A191372 Kn13(n) = A002487(n+5) - A000034(n+1)
%F A191372 Kn14(n) = A002487(n+7) - A157810(n+1)
%F A191372 Three (AS2S2S2(T, 2^p-delta) - AS2S2S2(T, 2^(p-1)-delta)) sequences for p=6:
%F A191372 delta =  1: [1, 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]
%F A191372 delta =  8: [4, 3, 5, 2, 5, 3, 4, 1, 3, 2, 3, 1, 2, 1, 1, 0, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1]
%F A191372 delta = 16: [5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 4, 5, 1, 4, 3, 5, 2, 5, 3, 4, 1, 3, 2, 3, 1, 2, 1, 1, 0]
%p A191372 nmax:=2^5; pmax:=log(nmax)/log(2)-1; A047999:=proc(n,k) option remember; A047999(n,k) :=binomial(n,k) mod 2 end: 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: d:=0: for n from 0 to nmax-d-1 do Kn1(n,d):= add(A047999(n-k+d, k+d),k=0..floor(n/2)): AS2S2S2(n,d):= A002487(n+1+2*d)-Kn1(n,d): od: for p from 1 to pmax do for d from 2^(p-1) to 2^p do for n from 0 to nmax-d-1 do Kn1(n,d):=add(A047999(n-k+d, k+d),k=0..floor(n/2)): AS2S2S2(n,d):= A002487(n+1+2*d)-Kn1(n,d) od: od: od: S2(0,0):=AS2S2S2(0,0): a(0):=S2(0,0): for d from 1 to 2^pmax do for Tx from 0 to 2^ceil(log(d)/log(2))-1 do S2(Tx,d):=AS2S2S2(Tx,d) od: od: Ty:=0: for d from 1 to 2^pmax do for Tx from 0 to 2^ceil(log(d)/log(2))-1 do Ty:=Ty+1: a(Ty):=S2(Tx,d) od: od: S2(0,0); for d from 1 to 2^pmax do seq(S2(Tx,d), Tx=0..2^ceil(log(d)/ log(2))-1) od; seq(a(n),n=0..Ty);
%Y A191372 Cf. A047999 (Sierpinski), A002487 (Stern).
%K A191372 nonn,tabf
%O A191372 0,3
%A A191372 _Johannes W. Meijer_, Jun 05 2011