A244780 -id:A244780 - cp's OEIS frontend

A240808 a(0)=2, a(1)=1, a(2)=0; thereafter a(n) = a(n-1-a(n-1))+a(n-2-a(n-2)) unless a(n-1) <= n-1 or a(n-2) <= n-2 in which case the sequence terminates.

2, 1, 0, 2, 1, 3, 2, 1, 3, 5, 4, 3, 5, 4, 6, 8, 4, 6, 8, 7, 9, 8, 7, 12, 11, 7, 12, 14, 10, 12, 14, 10, 12, 17, 13, 12, 20, 16, 12, 20, 19, 15, 20, 19, 18, 23, 19, 21, 26, 19, 21, 26, 19, 24, 29, 19, 27, 32, 19, 27, 32, 22, 30, 32, 22, 30, 32, 25, 33, 32, 28, 36, 32, 31, 39, 32, 31, 42, 35, 31, 45, 38, 31, 45, 38, 31, 48, 41, 31, 51, 44, 31, 51, 47, 34

Offset: 0

N. J. A. Sloane, Apr 15 2014

a(A241218(n)) = n and a(m) <> n for m < A241218(n). - Reinhard Zumkeller, Apr 17 2014

Higham, Jeff and Tanny, Stephen, A tamely chaotic meta-Fibonacci sequence. Twenty-third Manitoba Conference on Numerical Mathematics and Computing (Winnipeg, MB, 1993). Congr. Numer. 99 (1994), 67-94. [Contains a detailed analysis of this sequence]

Reinhard Zumkeller, Table of n, a(n) for n = 0..20000
Rémy Sigrist, Colored scatterplot of a(n) for n = 0..20000 (where the color is function of n mod 3)
Index entries for Hofstadter-type sequences

A006949 and A240807 have the same recurrence but different initial conditions.

Trisections: A244780..A244782.

a240808 n = a240808_list !! n
a240808_list = 2 : 1 : 0 : zipWith (+) xs (tail xs)
   where xs = map a240808 $ zipWith (-) [1..] $ tail a240808_list
-- Reinhard Zumkeller, Apr 17 2014

a:=proc(n) option remember; global k;
if n = 0 then 2
elif n = 1 then 1
elif n = 2 then 0
else
if (a(n-1) <= n-1) and (a(n-2) <= n-2) then
a(n-1-a(n-1))+a(n-2-a(n-2));
else lprint("died with n =",n); return (-1);
fi;
fi; end;
[seq(a(n),n=0..100)];

a[n_] := a[n] = Switch[n, 0, 2, 1, 1, 2, 0, _,
   If[a[n - 1] <= n - 1 && a[n - 2] <= n - 2,
   a[n - 1 - a[n - 1]] + a[n - 2 - a[n - 2]],
   Print["died with n =", n]; Return[-1]]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Oct 02 2024 *)

A244819 Positive numbers primitively represented by the binary quadratic form (1, 0, 3).

Original entry on oeis.org

1, 3, 4, 7, 12, 13, 19, 21, 28, 31, 37, 39, 43, 49, 52, 57, 61, 67, 73, 76, 79, 84, 91, 93, 97, 103, 109, 111, 124, 127, 129, 133, 139, 147, 148, 151, 156, 157, 163, 169, 172, 181, 183, 193, 196, 199, 201, 211, 217, 219, 223, 228, 229, 237, 241, 244, 247, 259

Offset: 1

Peter Luschny, Jul 06 2014

nonn

Discriminant = -12.

x^2 + 3*y^2 represents positive k properly (gcd(x, y) = 1), with nonnegative x, and the following multiplicities m(k): m(1) = 1, m(3) = 1, m(4) = 2, and if k = 3^a*4^b*Product_{j=1..P1} p1(j)^e1(j), with p1(j) primes 1 (mod 6) (A002476), e1(j) nonnegative integer numbers, and a and b from {0, 1}, then m(k) = 2^(P1+b). Shown by the lifting theorem (e.g., Apostol) for prime powers. Note that for prime 2 there is one solution of j^2 + 3 == 0 (mod 2) this corresponds the imprimitive reduced form (2, 2, 2), not to the one reduced primitive form (1, 0, 3) for discriminant -12 (A000003(3) = 1). - Wolfdieter Lang, Mar 02 2021

			Proper solution to x^2 + 3*y^2 = a(n), with x nonnegative: a(12 = 3*4) with (x, y) = (3, pm 1), pm = +1 or -1, multiplicity m(12) = 2, (a, b, P1) = (1, 1, 0); a(21 = 3*7) with (3, pm 2), m(21) = 2, (a, b, P1) = (1, 0, 1); a(49 = 7^2) with (1, pm 4), m(49) = 2 (a, b, P1) = (0, 0, 1)). - _Wolfdieter Lang_, Mar 02 2021

Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976 (1986), Theorem 5.30, pp. 121-122.

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A002476, A092574, A244779, A244780.

# Function PriRepBQF in A244779.
A244819 list := n -> PriRepBQF(1, 0, 3, n); A244819_list(259);

Reap[For[n = 1, n < 1000, n++, r = Reduce[x^2 + 3 y^2 == n, {x, y}, Integers]; If[r =!= False, If[AnyTrue[{x, y} /. {ToRules[r /. C[1] -> 0]}, CoprimeQ @@ # &], Sow[n]]]]][[2, 1]] (* Jean-François Alcover, Oct 31 2016 *)

A244779 Positive numbers primitively represented by the binary quadratic form (1, 1, 2).

Original entry on oeis.org

1, 2, 4, 7, 8, 11, 14, 16, 22, 23, 28, 29, 32, 37, 43, 44, 46, 53, 56, 58, 64, 67, 71, 74, 77, 79, 86, 88, 92, 106, 107, 109, 112, 113, 116, 121, 127, 128, 134, 137, 142, 148, 149, 151, 154, 158, 161, 163, 172, 176, 179, 184, 191, 193, 197, 203, 211, 212, 214

Offset: 1

Peter Luschny, Jul 06 2014

nonn

Discriminant = -7.

Robert Israel, Table of n, a(n) for n = 1..10000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A244780, A244819.

PriRepBQF := proc(a, b, c, n) local L,q,R,r,k;
q := a*x^2 + b*x*y + c*y^2; L := NULL;
for k from 1 to n do
   R := [isolve(q = k)];
   if R = [] then next fi;
   for r in R do
      igcd(op(2,r[1]), op(2,r[2]));
      if 1 = % then L := L,k; break fi od
od; L end:
A244779_list := n -> PriRepBQF(1, 1, 2, n); A244779_list(214);
# Alternate program
A244779_set:= proc(N) local A, B, y,x;
   A:= {};
   for y from 0 to floor(sqrt(4*N/7)) do
     for x from ceil(-y/2) to floor(-y/2 + sqrt(N - 7/4*y^2)) do
       if igcd(x,y) = 1 then
         A:= A union {x^2 + x*y + 2*y^2}
       fi
     od
    od;
A
end proc:
A244779_set(1000); # Robert Israel, Jul 06 2014

Reap[For[n = 1, n < 1000, n++, r = Reduce[x^2 + x y + 2 y^2 == n, {x, y}, Integers]; If[r =!= False, If[AnyTrue[{x, y} /. {ToRules[r /. C[1] -> 0]}, CoprimeQ @@ # &], Sow[n]]]]][[2, 1]] (* Jean-François Alcover, Oct 31 2016 *)

A244481 First trisection of A240808.

Original entry on oeis.org

1, 1, 1, 4, 4, 4, 7, 7, 7, 10, 10, 13, 16, 19, 19, 19, 19, 19, 19, 19, 22, 22, 25, 28, 31, 31, 31, 31, 31, 31, 31, 34, 37, 40, 40, 43, 46, 46, 49, 49, 52, 55, 58, 61, 64, 64, 67, 70, 73, 73, 76, 79, 82, 82, 82, 82, 82, 82, 82, 82, 82, 82, 82, 82, 82, 82, 82, 82, 82, 82, 82, 82, 82, 82, 82, 82, 85, 88, 91

Offset: 0

N. J. A. Sloane, Jul 03 2014

nonn

Higham, Jeff and Tanny, Stephen, A tamely chaotic meta-Fibonacci sequence. Twenty-third Manitoba Conference on Numerical Mathematics and Computing (Winnipeg, MB, 1993). Congr. Numer. 99 (1994), 67-94.

Index entries for Hofstadter-type sequences

Cf. A240808, A244780, A244782.

A244482 Second trisection of A240808.

Original entry on oeis.org

0, 3, 3, 3, 6, 6, 9, 12, 12, 12, 12, 12, 12, 15, 18, 21, 21, 24, 27, 27, 30, 30, 33, 36, 39, 42, 45, 45, 48, 51, 51, 51, 51, 51, 51, 51, 51, 51, 51, 51, 51, 51, 51, 51, 51, 51, 51, 51, 54, 54, 57, 60, 63, 66, 69, 69, 72, 75, 78, 81, 84, 84, 87, 90, 90, 93, 93, 96, 99, 102, 105, 108, 108, 111, 114, 114

Offset: 0

N. J. A. Sloane, Jul 03 2014

nonn

Higham, Jeff and Tanny, Stephen, A tamely chaotic meta-Fibonacci sequence. Twenty-third Manitoba Conference on Numerical Mathematics and Computing (Winnipeg, MB, 1993). Congr. Numer. 99 (1994), 67-94.

Index entries for Hofstadter-type sequences

Cf. A240808, A244780, A244781.

cp's OEIS Frontend

A240808 a(0)=2, a(1)=1, a(2)=0; thereafter a(n) = a(n-1-a(n-1))+a(n-2-a(n-2)) unless a(n-1) <= n-1 or a(n-2) <= n-2 in which case the sequence terminates.

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A244819 Positive numbers primitively represented by the binary quadratic form (1, 0, 3).

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A244779 Positive numbers primitively represented by the binary quadratic form (1, 1, 2).

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A244481 First trisection of A240808.

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A244482 Second trisection of A240808.

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