A080118 -id:A080118 - cp's OEIS frontend

A080119 Positions of A080118 in A014486.

1, 2, 7, 33, 81, 74395, 8369196, 215802898, 414859094165, 520973680640109, 4064761999842441067, 517978450857911919447, 4255027826896017770661, 5222501054779098990032001033, 718000720375918750838217734094612383

Offset: 1

Antti Karttunen, Feb 11 2003

nonn

A080119 := n -> CatalanRankGlobal(A080118(n));

def A080119(n) : return A215406(A080118(n))
[A080119(n) for n in (1..15)] # Peter Luschny, Aug 10 2012

a(n) = A215406(A080118(n)). - Peter Luschny, Aug 10 2012

A080117 Binary encoding of quadratic residue set formed for n-th prime, coerced to "complementarily symmetric binary sequence" with A080261 if the prime is of the form 4k+1.

Original entry on oeis.org

2, 10, 52, 738, 2866, 53620, 162438, 4023888, 166243974, 921787428, 48034443442, 935251508324, 2558696229078, 68055676507664, 2655011787909270, 210067141980993186, 831463106366605026, 42882922858578320598

Offset: 2

Antti Karttunen, Feb 11 2003

nonn

Cf. A080118, A080146, A080148.

with(numtheory,ithprime); A080117 := proc(n) local c,p; p := ithprime(n); c := A055094(p); if(3 = (p mod 4)) then RETURN(c); else RETURN(A080261(c)); fi; end;

A055094[n_] := With[{rr = Table[Mod[k^2, n], {k, 1, n-1}] // Union}, Boole[ MemberQ[rr, #]] & /@ Range[n-1]] // FromDigits[#, 2]&;
A080261[n_] := Module[{bb = IntegerDigits[n, 2]}, lg = Length[bb]; Do[ bb[[i]] = 1 - bb[[i]], {i, lg, lg - Floor[lg/2] + 1, -1}]; FromDigits[ bb, 2]];
a[n_] := Module[{c, p = Prime[n]}, c = A055094[p]; If[Mod[p, 4] == 3, c, A080261[c]]]; Table[a[n], {n, 2, 20}] (* Jean-François Alcover, Mar 05 2016, adapted from Maple *)

# uses[A080261]
def A080117(n) :
    p = nth_prime(n)
    c = A055094(p)
    return c if 3 == p%4 else A080261(c)
[A080117(n) for n in (2..19)] # Peter Luschny, Aug 09 2012

a(A080148(n)) = A080146(A080148(n))

A080120 Dyck path encodings of Legendre's candelabras formed for primes in A080114. (I.e., symmetric rooted plane trees constructed from their quadratic residue sets.)

Original entry on oeis.org

10, 1010, 110100, 1011100010, 101100110010, 1111010110011001010000, 110110111100010101110000100100, 101100101111000100110111000010110010, 1111011110010101110010011011000101011000010000

Offset: 1

Antti Karttunen, Feb 11 2003

nonn

For the 2nd, 5th and 8th term of the sequence, the quadratic residue set of the corresponding prime (5,13,37, of the form 4k+1) has been converted from symmetric to complementarily symmetric as 1001->1010, 101100001101->101100110010, 101100101111000100001000111101001101->101100101111000100110111000010110010, for the others (of the form 4k+3), it is the quadratic residue set encoded as in A055094 (with +1 mapped to 1 and -1 to 0).

Antti Karttunen, Illustration of initial terms

Same sequence in decimal: A080118. Cf. A080114.

A080120 := n -> convert(A080118(n),binary);

a(n) = A063171(A080119(n)).

cp's OEIS Frontend

A080119 Positions of A080118 in A014486.

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A080117 Binary encoding of quadratic residue set formed for n-th prime, coerced to "complementarily symmetric binary sequence" with A080261 if the prime is of the form 4k+1.

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A080120 Dyck path encodings of Legendre's candelabras formed for primes in A080114. (I.e., symmetric rooted plane trees constructed from their quadratic residue sets.)

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