A048723 -id:A048723 - cp's OEIS frontend

A106445 Exponent-recursed cross-domain bijection from GF(2)[X] to N. Variant of A091203 and A106443.

0, 1, 2, 3, 4, 9, 6, 5, 8, 15, 18, 7, 12, 11, 10, 27, 16, 81, 30, 13, 36, 25, 14, 33, 24, 17, 22, 45, 20, 21, 54, 19, 512, 57, 162, 55, 60, 23, 26, 63, 72, 29, 50, 51, 28, 135, 66, 31, 48, 35, 34, 19683, 44, 39, 90, 37, 40, 99, 42, 41, 108, 43, 38, 75, 64, 225, 114, 47

Offset: 0

Antti Karttunen, May 09 2005

nonn

This map from the multiplicative domain of GF(2)[X] to that of N preserves 'superfactorized' structures, e.g. A106493(n) = A106490(a(n)), A106494(n) = A106491(a(n)), A106495(n) = A064372(a(n)). Shares with A091203 and A106443 the property that maps A014580(n) to A000040(n). Differs from the plain variant A091203 for the first time at n=32, where A091203(32)=32, while a(32)=512. Differs from the variant A106443 for the first time at n=48, where A106443(48)=768, while a(48)=48. Differs from a yet deeper variant A106447 for the first time at n=13, where A106447(13)=23, while a(13)=11.

			a(5) = 9, as 5 encodes the GF(2)[X] polynomial x^2+1, which is the square of the second irreducible GF(2)[X] polynomial x+1 (encoded as 3) and the square of the second prime is 3^2=9. a(32) = a(A048723(2,5)) = 2^a(5) = 2^9 = 512. a(48) = a(3 X A048723(2,4)) = 3 * 2^a(4) = 3 * 2^4 = 3 * 16 = 48.

Inverse: A106444.

a(0)=0, a(1)=1. For irreducible GF(2)[X] polynomials ir_i with index i (i.e. A014580(i)), a(ir_i) = A000040(i) and for composite polynomials n = A048723(ir_i, e_i) X A048723(ir_j, e_j) X A048723(ir_k, e_k) X ..., a(n) = a(ir_i)^a(e_i) * a(ir_j)^a(e_j) * a(ir_k)^a(e_k) * ... = A000040(i)^a(e_i) * A000040(j)^a(e_j) * A000040(k)^a(e_k), where X stands for carryless multiplication of GF(2)[X] polynomials (A048720) and A048723(n, y) raises the n-th GF(2)[X] polynomial to the y:th power, while * is the ordinary multiplication and ^ is the ordinary exponentiation.

A048757 Sum_{i=0..2n} (C(2n,i) mod 2)Fibonacci(i+2) = Sum_{i=0..n} (C(n,i) mod 2)Fibonacci(2i+2).

Original entry on oeis.org

1, 4, 9, 33, 56, 203, 441, 1596, 2585, 9353, 20304, 73461, 124033, 448756, 974169, 3524577, 5702888, 20633243, 44791065, 162055596, 273617239, 989956471, 2149017696, 7775219067, 12591974497, 45558191716, 98898651657

Offset: 0

Antti Karttunen, Jul 13 1999

The history of 1-D CA Rule 90 starting from the seed pattern 1 interpreted as Zeckendorffian expansion.

Also, product of distinct terms of A001566 and appropriate Fibonacci or Lucas numbers: a(n) = FL(n+2)Product(L(2^i)^bit(n,i),i=0..) Here L(2^i) = A001566 and FL(n) = n-th Fibonacci number if n even, n-th Lucas number if n odd. bit(n,i) is the i-th digit (0 or 1) in the binary expansion of n, with the least significant digit being bit(n,0).

			1 = Fib(2) = 1;
101 = Fib(4) + Fib(2) = 3 + 1 = 4;
10001 = Fib(6) + Fib(2) = 8 + 1 = 9;
1010101 = Fib(8) + Fib(6) + Fib(4) + Fib(2) = 21 + 8 + 3 + 1 = 33; etc.

Antti Karttunen, On Pascal's Triangle Modulo 2 in Fibonacci Representation, Fibonacci Quarterly, 42 (2004), 38-46.

a(n) = A022290(A038183(n)) = A022290(A048723(5, n)) = A003622(A051656(n)) = A075148(n, 2)*A050613(n). Third row of A050609, third column of A050610.

Cf. A054433.

Table[Sum[Mod[Binomial[2n, i], 2] Fibonacci[i + 2], {i, 0, 2n}], {n, 0, 19}] (* Alonso del Arte, Apr 27 2014 *)

A106456 Natural numbers mapped to Dyck path encodings of the rooted plane trees obtained by recursing on the exponents of the GF(2)[X] factorization of n.

Original entry on oeis.org

0, 10, 1010, 1100, 110010, 101100, 101010, 110100, 10110010, 11001100, 10101010, 10110100, 1010101010, 10101100, 11010010, 111000, 11100010, 1011001100, 101010101010, 1100110100, 11001010, 1010101100, 101010110010

Offset: 1

Antti Karttunen, May 09 2005

Note that we recurse on the exponent + 1 for all other irreducible polynomials except the largest one in the GF(2)[X] factorization. Thus for 6 = A048723(3,1) X A048723(2,1) we construct a tree by joining trees 1 and 2 with a new root node, for 7 = A048723(7,1) X A048723(3,0) X A048723(2,0) we join three 1-trees (single leaves) with a new root node, for 8 = A048273(2,3) we add a single edge below tree 3 and for 9 = A048723(7,1) X A048723(3,1) X A048273(2,0) we connect the trees 1 and 2 and 1 with a new root node.

			The rooted plane trees encoded here are:
.....................o....o..........o.........o...o....o.....
.....................|....|..........|..........\./.....|.....
.......o....o...o....o....o...o..o...o..o.o.o....o....o.o.o...
.......|.....\./.....|.....\./....\./....\|/.....|.....\|/....
*......*......*......*......*......*......*......*......*.....
1......2......3......4......5......6......7......8......9.....

A. Karttunen, Scheme-program for computing this sequence.

a(n) = A007088(A106455(n)) = A075166(A106443(n)). GF(2)[X]-analog of A075166. Permutation of A063171. Same sequence shown in decimal: A106455. The digital length of each term / 2 (the number of o-nodes in the corresponding trees) is given by A106457. Cf. A106451-A106454.

A268669 a(n) = polynomial quotient (computed over GF(2), result is its binary encoding) that is left after all instances of polynomial (X+1) have been factored out of the polynomial that is encoded by the binary expansion of n. (See comments for details).

Original entry on oeis.org

1, 2, 1, 4, 1, 2, 7, 8, 7, 2, 11, 4, 13, 14, 1, 16, 1, 14, 19, 4, 21, 22, 13, 8, 25, 26, 7, 28, 11, 2, 31, 32, 31, 2, 35, 28, 37, 38, 11, 8, 41, 42, 25, 44, 7, 26, 47, 16, 49, 50, 1, 52, 19, 14, 55, 56, 13, 22, 59, 4, 61, 62, 21, 64, 21, 62, 67, 4, 69, 70, 61, 56, 73, 74, 13, 76, 59, 22, 79, 16, 81, 82, 49, 84, 1

Offset: 1

Antti Karttunen, Feb 10 2016

When polynomials over GF(2) are encoded in the binary representation of n in a natural way where each polynomial b(n)*X^n+...+b(0)*X^0 over GF(2) is represented by the binary number b(n)*2^n+...+b(0)*2^0 in N (each coefficient b(k) is either 0 or 1), then a(n) = representation of the polynomial that is left as a quotient when all X+1 polynomials (encoded by 3, "11" in binary) have been divided out.
Each a(n) is one of the "Garden of Eden" patterns of Rule-60 one-dimensional cellular automaton, a seed pattern which after A268389(n) generations yields the configuration encoded in binary expansion of n.
No terms of A001969 occur so all terms are odious (in A000069). Each odious number occurs an infinitely many times.

			For n = 5 ("101" in binary) which encodes polynomial x^2 + 1, we observe that it can be factored in ring GF(2)[X] as (X+1)(X+1), and thus a(5) = 1, because after dividing both instances of (X+1) off, we are left with the quotient polynomial 1 which is encoded by 1.
For n = 8 ("1000" in binary) which encodes polynomial x^3, we observe that it is not divisible in ring GF(2)[X] by polynomial X+1, thus a(8) = 8.
For n = 9 ("1001" in binary) which encodes polynomial x^3 + 1, we observe that it can be factored over GF(2) as (X+1)(X^2 + X + 1), and thus a(9) = 7, because the quotient polynomial X^2 + X + 1 is encoded by 7 ("111" in binary).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Eric Weisstein's World of Mathematics, Rule 60
Index entries for sequences related to cellular automata
Index entries for sequences operating on GF(2)[X]-polynomials

Cf. A001317 (positions of ones).
Cf. A000069, A001969, A003188, A003987, A010060, A048720, A048723, A048724, A268384, A268670.
Cf. A268389 (the highest exponent for (X+1)).
Cf. also A136386.

f[n_] := If[OddQ@ #, n, f[#/2]] &@ Fold[BitXor, n, Quotient[n, 2^Range[BitLength@ n - 1]]]; Array[f, {85}] (* Michael De Vlieger, Feb 12 2016, after Jan Mangaldan at A006068 *)

a(n) = {p = Pol(binary(n))*Mod(1,2); q = (x+1)*Mod(1,2); while (type(r = p/q) == "t_POL", p = r); np = Polrev(vector(poldegree(p)+1, k, k--; lift(polcoeff(p, k)))); subst(np, x, 2);} \\ Michel Marcus, Feb 12 2016

;; This employs the given recurrence and uses memoization-macro definec:
(definec (A268669 n) (if (odd? (A006068 n)) n (A268669 (/ (A006068 n) 2))))
(define (A268669 n) (let loop ((n n)) (let ((k (A006068 n))) (if (odd? k) n (loop (/ k 2)))))) ;; Computed in a loop, no memoization.

If A006068(n) is odd, then a(n) = n, otherwise a(n) = a(A006068(n)/2).
Other identities and observations. For all n >= 1:
a(n) = A003188(A268670(n)).
A010060(a(n)) = 1. [All terms are odious.]
a(n) <= n.
More precisely, a(A000069(n)) = A000069(n) and a(A001969(n)) < A001969(n).
The equivalence of the following two formulas stems from the additive nature of Rule-60 cellular automaton. Or more plainly, because carryless binary multiplication A048720 distributes over carryless binary sum, XOR A003987:
A048724^A268389(n) (a(n)) = n. [Starting from k = a(n), and iterating map k -> A048724(k) exactly A268389(n) times yields n back.]
A048720(a(n),A048723(3,A268389(n))) = A048720(a(n),A001317(A268389(n))) = n.

A037096 Periodic vertical binary vectors computed for powers of 3: a(n) = Sum_{k=0 .. (2^n)-1} (floor((3^k)/(2^n)) mod 2) * 2^k.

Original entry on oeis.org

1, 2, 0, 204, 30840, 3743473440, 400814250895866480, 192435610587299441243182587501623263200, 2911899996313975217187797869354128351340558818020188112521784134070351919360

Offset: 0

Antti Karttunen, Jan 29 1999

This sequence can be also computed with a recurrence that does not explicitly refer to 3^n. See the C program.

Conjecture: For n >= 3, each term a(n), when considered as a GF(2)[X] polynomial, is divisible by the GF(2)[X] polynomial (x + 1) ^ A055010(n-1). If this holds, then for n >= 3, a(n) = A048720(A136386(n), A048723(3,A055010(n-1))).

			When powers of 3 are written in binary (see A004656), under each other as:
  000000000001 (1)
  000000000011 (3)
  000000001001 (9)
  000000011011 (27)
  000001010001 (81)
  000011110011 (243)
  001011011001 (729)
  100010001011 (2187)
it can be seen that the bits in the n-th column from the right can be arranged in periods of 2^n: 1, 2, 4, 8, ... This sequence is formed from those bits: 1, is binary for 1, thus a(0) = 1. 01, reversed is 10, which is binary for 2, thus a(1) = 2, 0000 is binary for 0, thus a(2)=0, 000110011, reversed is 11001100 = A007088(204), thus a(3) = 204.

S. Wolfram, A New Kind of Science, Wolfram Media Inc., (2002), p. 119.

Antti Karttunen, Table of n, a(n) for n = 0..11
Antti Karttunen, C program for computing this sequence.
S. Wolfram, A New Kind of Science, Wolfram Media Inc., (2002), p. 119.

Cf. A036284, A037095, A037097, A136386 for related sequences.
Cf. A000079, A000215, A000225, A000244, A004656, A007088, A048720, A048723, A055010.
Cf. also A004642, A265209, A265210 (for 2^n written in base 3).

a(n) := sum( 'bit_n(3^i, n)*(2^i)', 'i'=0..(2^(n))-1);
bit_n := (x, n) -> `mod`(floor(x/(2^n)), 2);

a(n) = Sum_{k=0 .. A000225(n)} (floor(A000244(k)/(2^n)) mod 2) * 2^k.
Other identities and observations:
For n >= 2, a(n) = A000215(n-1)*A037097(n) = A048720(A037097(n), A048723(3, A000079(n-1))).

Entry revised by Antti Karttunen, Dec 29 2007

Name changed and the example corrected by Antti Karttunen, Dec 05 2015

A106446 Doubly-recursed cross-domain bijection from N to GF(2)[X]. Variant of A091204 and A106444.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 6, 11, 8, 5, 14, 25, 12, 19, 22, 9, 16, 47, 10, 31, 28, 29, 50, 13, 24, 21, 38, 15, 44, 61, 18, 137, 128, 43, 94, 49, 20, 55, 62, 53, 56, 97, 58, 115, 100, 27, 26, 37, 48, 69, 42, 113, 76, 73, 30, 79, 88, 33, 122, 319, 36, 41, 274, 39, 64, 121, 86, 185

Offset: 0

Antti Karttunen, May 09 2005

nonn

Differs from A091204 for the first time at n=32, where A091204(32)=32, while a(32)=128. Differs from A106444 for the first time at n=11, where A106444(11)=13, while a(11)=25.

			a(5) = 7, as 5 is the 3rd prime, a(3)=3 and the third irreducible GF(2)[X] polynomial x^2+x+1 is encoded as A014580(3) = 7. a(11) = 25, as 11 is the 5th prime, a(5)=7 and the seventh irreducible GF(2)[X] polynomial x^4+x^3+1 is encoded as A014580(7) = 25. a(32) = a(2^5) = A048723(a(2),a(5)) = A048723(2,7) = 128.

Inverse: A106447. Variant: A091204.

a(0)=0, a(1)=1, a(p_i) = A014580(a(i)) for primes p_i with index i and for composites n = p_i^e_i * p_j^e_j * p_k^e_k * ..., a(n) = A048723(a(p_i), a(e_i)) X A048723(a(p_j), a(e_j)) X A048723(a(p_k), a(e_k)) X ..., where X stands for carryless multiplication of GF(2)[X] polynomials (A048720) and A048723(n, y) raises the n-th GF(2)[X] polynomial to the y:th power.

A106447 Doubly-recursed cross-domain bijection from GF(2)[X] to N. Variant of A091205 and A106445.

Original entry on oeis.org

0, 1, 2, 3, 4, 9, 6, 5, 8, 15, 18, 7, 12, 23, 10, 27, 16, 81, 30, 13, 36, 25, 14, 69, 24, 11, 46, 45, 20, 21, 54, 19, 512, 57, 162, 115, 60, 47, 26, 63, 72, 61, 50, 33, 28, 135, 138, 17, 48, 35, 22, 19683, 92, 39, 90, 37, 40, 207, 42, 83, 108, 29, 38, 75, 64, 225, 114

Offset: 0

Antti Karttunen, May 09 2005

nonn

Differs from A091205 for the first time at n=32, where A091205(32)=32, while a(32)=512. Differs from A106445 for the first time at n=13, where A106445(13)=11, while a(13)=23.

			a(5) = 9, as 5 encodes the GF(2)[X] polynomial x^2+1, which is the square of the second irreducible GF(2)[X] polynomial x+1 (encoded as 3), a(2)=2 and the square of the second prime is 3^2=9. a(13) = a(A014580(5)) = A000040(a(5)) = A000040(9) = 23. a(32) = a(A048723(2,5)) = a(2)^a(5) = 2^9 = 512. a(48) = a(3 X A048723(2,4)) = a(3) * a(2)^a(4) = 3 * 2^4 = 3 * 16 = 48.

Inverse: A106446. Variant: A091205.

a(0)=0, a(1)=1. For irreducible GF(2)[X] polynomials ir_i with index i (i.e. A014580(i)), a(ir_i) = A000040(a(i)) and for composite polynomials n = A048723(ir_i, e_i) X A048723(ir_j, e_j) X A048723(ir_k, e_k) X ..., a(n) = a(ir_i)^a(e_i) * a(ir_j)^a(e_j) * a(ir_k)^a(e_k) * ... = A000040(a(i))^a(e_i) * A000040(a(j))^a(e_j) * A000040(a(k))^a(e_k), where X stands for carryless multiplication of GF(2)[X] polynomials (A048720) and A048723(n, y) raises the n-th GF(2)[X] polynomial to the y:th power, while * is the ordinary multiplication and ^ is the ordinary exponentiation.

A106493 Total number of bases and exponents in GF(2)[X] Superfactorization of n, excluding the unity-exponents at the tips of branches.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 1, 2, 2, 3, 1, 3, 1, 2, 2, 3, 3, 3, 1, 4, 2, 2, 2, 3, 1, 2, 3, 3, 2, 3, 1, 3, 2, 4, 2, 4, 1, 2, 3, 4, 1, 3, 2, 3, 3, 3, 1, 4, 2, 2, 3, 3, 2, 4, 1, 3, 3, 3, 1, 4, 1, 2, 3, 3, 4, 3, 1, 5, 2, 3, 2, 4, 1, 2, 3, 3, 2, 4, 2, 5, 2, 2, 3, 4, 3, 3, 1, 3, 2, 4, 1, 4, 2, 2, 3, 4, 1, 3, 3, 3, 3, 4

Offset: 1

Antti Karttunen, May 09 2005

nonn

GF(2)[X] Superfactorization proceeds in a manner analogous to normal superfactorization explained in A106490, but using factorization in domain GF(2)[X], instead of normal integer factorization in N.

			a(64) = 3, as 64 = A048723(2,6) = A048723(2,(A048723(2,1) X A048723(3,1))) and there are three non-1 nodes in that superfactorization. Similarly, for 27 = 5x7 = A048723(3,2) X A048273(7,1) we get a(27) = 3. The operation X stands for GF(2)[X] multiplication defined in A048720, while A048723(n,y) raises the n-th GF(2)[X] polynomial to the y:th power.

A. Karttunen, Scheme-program for computing this sequence.

a(n) = A106490(A106445(n)). a(n) = A106494(n)-A106495(n).

A106494 Total number of bases and exponents in GF(2)[X] Superfactorization of n, including the unity-exponents at the tips of branches.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 2, 3, 4, 5, 2, 5, 2, 4, 3, 4, 4, 6, 2, 6, 3, 4, 4, 5, 2, 4, 5, 5, 4, 5, 2, 4, 4, 6, 4, 7, 2, 4, 5, 6, 2, 5, 4, 5, 5, 6, 2, 6, 4, 4, 4, 5, 4, 7, 2, 5, 5, 6, 2, 6, 2, 4, 5, 5, 6, 6, 2, 7, 3, 6, 4, 7, 2, 4, 5, 5, 4, 7, 4, 7, 3, 4, 6, 6, 5, 6, 2, 5, 4, 7, 2, 7, 4, 4, 5, 6, 2, 6, 5, 5, 6, 6

Offset: 1

Antti Karttunen, May 09 2005

nonn

See comments at A106493.

			a(64) = 5, as 64 = A048723(2,6) = A048723(2,(A048723(2,1) X A048723(3,1))) and there are five nodes in that superfactorization. Similarly, for 27 = 5x7 = A048723(3, A048723(2,1)) X A048273(7,1) we get a(27) = 5. The operation X stands for GF(2)[X] multiplication defined in A048720, while A048723(n,y) raises the n-th GF(2)[X] polynomial to the y:th power.

A. Karttunen, Scheme-program for computing this sequence.

a(n) = A106491(A106445(n)). a(n) = A106493(n)+A106495(n).

A048732 a(n) = Xpower(n,3).

Original entry on oeis.org

0, 1, 8, 15, 64, 85, 120, 107, 512, 585, 680, 743, 960, 925, 856, 771, 4096, 4369, 4680, 4959, 5440, 5189, 5944, 5691, 7680, 8025, 7400, 7607, 6848, 7053, 6168, 6483, 32768, 33825, 34952, 36015, 37440

Offset: 0

Antti Karttunen, Apr 26 1999

nonn

Column 3 of A048723.

cp's OEIS Frontend

A106445 Exponent-recursed cross-domain bijection from GF(2)[X] to N. Variant of A091203 and A106443.

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A048757 Sum_{i=0..2n} (C(2n,i) mod 2)*Fibonacci(i+2) = Sum_{i=0..n} (C(n,i) mod 2)*Fibonacci(2i+2).

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A106456 Natural numbers mapped to Dyck path encodings of the rooted plane trees obtained by recursing on the exponents of the GF(2)[X] factorization of n.

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A268669 a(n) = polynomial quotient (computed over GF(2), result is its binary encoding) that is left after all instances of polynomial (X+1) have been factored out of the polynomial that is encoded by the binary expansion of n. (See comments for details).

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PARI

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A037096 Periodic vertical binary vectors computed for powers of 3: a(n) = Sum_{k=0 .. (2^n)-1} (floor((3^k)/(2^n)) mod 2) * 2^k.

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A106446 Doubly-recursed cross-domain bijection from N to GF(2)[X]. Variant of A091204 and A106444.

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A106447 Doubly-recursed cross-domain bijection from GF(2)[X] to N. Variant of A091205 and A106445.

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A106493 Total number of bases and exponents in GF(2)[X] Superfactorization of n, excluding the unity-exponents at the tips of branches.

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A048757 Sum_{i=0..2n} (C(2n,i) mod 2)Fibonacci(i+2) = Sum_{i=0..n} (C(n,i) mod 2)Fibonacci(2i+2).