A297928 -id:A297928 - cp's OEIS frontend

A375256 Number of pairs of antipodal vertices in the level n Hanoi graph.

3, 12, 39, 129, 453, 1677, 6429, 25149, 99453, 395517, 1577469, 6300669, 25184253, 100700157, 402726909, 1610760189, 6442745853, 25770393597, 103080394749, 412319219709, 1649272160253, 6597079203837, 26388297940989, 105553154015229, 422212540563453, 1688850011258877, 6755399743045629

Offset: 1

Allan Bickle, Aug 07 2024

nonn

A level 1 Hanoi graph is a triangle. Level n+1 is formed from three copies of level n by adding edges between pairs of corner vertices of each pair of triangles. This graph represents the allowable moves in the Towers of Hanoi problem with n disks.

Antipodal vertices are a pair of vertices at maximum distance in a graph. The diameter of the level n Hanoi graph is 2^n - 1.

			2 example graphs:
                           o
                          / \
                         o---o
                        /     \
             o         o       o
            / \       / \     / \
           o---o     o---o---o---o
Graph:      H_1           H_2
Since the level 1 Hanoi graph is a triangle, a(1) = 3.

Paolo Xausa, Table of n, a(n) for n = 1..1000
Allan Bickle, Properties of Sierpinski Triangle Graphs, Springer PROMS 448 (2021) 295-303.
A. Hinz, S. Klavzar, and S. Zemljic, A survey and classification of Sierpinski-type graphs, Discrete Applied Mathematics 217 3 (2017), 565-600.
Eric Weisstein's World of Mathematics, Hanoi Graph
Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Cf. A000225, A029858, A058809 (Hanoi graphs).
Cf. A370933 (antipodal pairs in Sierpiński triangle graphs).
Cf. A193233, A297928.

A375256[n_] := 3*(2^(2*n - 3) + 3*2^(n - 2) - 1);
Array[A375256, 30] (* or *)
LinearRecurrence[{7, -14, 8}, {3, 12, 39}, 30] (* Paolo Xausa, Sep 23 2024 *)

a(n) = 3*(2^(2*n-3)+3*2^(n-2)-1); \\ Michel Marcus, Aug 08 2024

a(n) = 3*(2^(2n-3)+3*2^(n-2)-1).
a(n) = A370933(n+1) - 3.
a(n) = 3*A297928(n-2) for n>=2. - Alois P. Heinz, Sep 23 2024

More terms from Michel Marcus, Aug 08 2024

A296807 Take a prime, convert it to base 2. Consider it as a string of digits and delete its leftmost and rightmost digit. Leading zeros are kept. Repeat the process. a(n) is the least prime that, in the first n steps of this process, generates a string that is a prime read in base 2.

Original entry on oeis.org

2, 13, 43, 151, 2143, 2143, 12479, 57727, 246527, 4267455487, 276009615632383, 4469780781584383, 576406542684520447

Offset: 0

Paolo P. Lava, Paolo Iachia, Dec 21 2017

a(n) >= 2*4^n + 3*2^n - 1 = A297928(n) >= 2*(4^n + 2^n) + 1 = A085601(n), n > 0. - Iain Fox, Dec 29 2017 (edited by Iain Fox, Jan 08 2018)

a(17) <= 2^163 + 361736822347711983585853439 (probably much smaller), building on a Cunningham chain of length 17 found by Jaroslaw Wroblewski. a(n) exists for n <= 17, and probably for all n. - Jens Kruse Andersen, Jan 21 2018

			a(1) = 13 because 13 in base 2 is 1101 and 10 is 2 and 13 is the least number with this property;
a(2) = 43 because 43 in base 2 is 101011 while 0101 is 5 and 10 is 2 and 43 is the least number with this property;
a(3) = 151 because 151 in base 2 is 10010111 while 001011 is 11, 0101 is 5 and 10 is 2 and 151 is the least number with this property.

Cf. A000030, A010879, A004676, A296806.

with(numtheory): P:=proc(q) local a,b,c,i,j,k,n,ok,x; x:=5; for k from 1 to q do for n from x to q do a:=convert(ithprime(n),base,2); ok:=1; for i from 1 to k do b:=nops(a)-i; while a[b]=0 do b:=b-1; od;
c:=0; for j from b by -1 to i+1 do c:=2*c+a[j]; od;if not isprime(c) then ok:=0; break; fi; od;if ok=1 then x:=n; print(ithprime(n)); break; fi; od; od; end: P(10^20);

Table[SelectFirst[Prime@ Range[#, # + 10^5] &@ PrimePi[2 (4^n + 2^n) + 1], AllTrue[Map[FromDigits[#, 2] &, Rest@ NestWhileList[Most@ Rest@ # &, IntegerDigits[#, 2], Length@ # > 2 &]], PrimeQ] &], {n, 8}] (* Michael De Vlieger, Dec 29 2017 *)

a(n) = if(!n, return(2)); forprime(p=2*4^n + 3*2^n - 1, , my(b=p); for(x=1, n, b = (b - (b>=4*2^(logint(p, 2) - 2*x))*4*2^(logint(p, 2) - 2*x) - 1)/2; if(!isprime(b) || (b==2 && x!=n), next(2))); return(p)) \\ Iain Fox, Dec 29 2017 (corrected by Iain Fox, Oct 26 2019)

Definition corrected, a(10)-a(12) by Jens Kruse Andersen, Jan 21 2018

cp's OEIS Frontend

A375256 Number of pairs of antipodal vertices in the level n Hanoi graph.

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