cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A106442 Exponent-recursed cross-domain bijection from N to GF(2)[X]. Position of A075166(n) in A106456.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 6, 11, 8, 5, 14, 13, 12, 19, 22, 9, 16, 25, 10, 31, 28, 29, 26, 37, 24, 21, 38, 15, 44, 41, 18, 47, 128, 23, 50, 49, 20, 55, 62, 53, 56, 59, 58, 61, 52, 27, 74, 67, 192, 69, 42, 43, 76, 73, 30, 35, 88, 33, 82, 87, 36, 91, 94, 39, 64, 121, 46, 97, 100, 111, 98
Offset: 0

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Author

Antti Karttunen, May 09 2005

Keywords

Comments

This map from the multiplicative domain of N to that of GF(2)[X] preserves Catalan-family structures, e.g. A106454(n) = a(A075164(n)), A075163(n) = A106453(a(n)), A075165(n) = A106455(a(n)), A075166(n) = A106456(a(n)), A075167(n) = A106457(a(n)). Shares with A091202 and A106444 the property that maps A000040(n) to A014580(n). Differs from the former for the first time at n=32, where A091202(32)=32, while a(32)=128. Differs from the latter for the first time at n=48, where A106444(48)=48, while a(48)=192.

Examples

			a(5) = 7, as 5 is the 3rd prime and the third irreducible GF(2)[X] polynomial x^2+x+1 is encoded as A014580(3) = 7. a(32) = a(2^5) = A048723(A014580(1),a(5)) = A048723(2,7) = 128. a(48) = a(3 * 2^4) = 3 X A048723(2,a(4+1)-1) = 3 X A048723(2,7-1) = 3 X 64 = 192.

Links

Crossrefs

Inverse: A106443. a(n) = A106454(A075163(n)).

Formula

a(0)=0, a(1)=1, a(p_i) = A014580(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(1+e_j)-1) X A048723(a(p_k), a(1+e_k)-1) 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. Here p_i is the most significant prime in the factorization of n; its exponent e_i is not incremented before the recursion step, while the exponents of less significant primes e_j, e_k, ... are incremented by one before recursing and the result of the recursion is decremented by one before use.

A106443 Exponent-recursed cross-domain bijection from GF(2)[X] to N. Position of A106456(n) in A075166.

Original entry on oeis.org

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, 768, 35, 34, 19683, 44, 39, 90, 37, 40, 99, 42, 41, 108, 43, 38, 75, 64, 225, 114, 47
Offset: 0

Views

Author

Antti Karttunen, May 09 2005

Keywords

Comments

This map from the multiplicative domain of GF(2)[X] to that of N preserves Catalan-family structures, e.g. A075164(n) = a(A106454(n)), A106453(n) = A075163(a(n)), A106455(n) = A075165(a(n)), A106456(n) = A075166(a(n)), A106457(n) = A075167(a(n)). Shares with A091203 and A106445 the property that maps A014580(n) to A000040(n). Differs from the former for the first time at n=32, where A091203(32)=32, while a(32)=512. Differs from the latter for the first time at n=48, where A106445(48)=48, while a(48)=768.

Examples

			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+1)-1) = 3 * 2^(9-1) = 3 * 256 = 768.

Links

Crossrefs

Inverse: A106442. a(n) = A075164(A106453(n)).

Formula

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(1+e_j)-1) * a(ir_k)^(a(1+e_k)-1) * ... = A000040(i)^a(e_i) * A000040(j)^(a(1+e_j)-1) * A000040(k)^(a(1+e_k)-1), 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. Here ir_i is the most significant (largest) irreducible polynomial in the factorization of n; its exponent e_i is not incremented before the recursion step, while the exponents of less significant factors e_j, e_k, ... are incremented by one before recursing and the result of the recursion is decremented by one before use.

A106191 Expansion of sqrt(1-4x)/(1-x).

Original entry on oeis.org

1, -1, -3, -7, -17, -45, -129, -393, -1251, -4111, -13835, -47427, -164999, -581023, -2066823, -7415703, -26805393, -97520733, -356810313, -1312087713, -4846614093, -17974854933, -66907388973, -249872516253, -935991743553, -3515800038201, -13239692841105
Offset: 0

Views

Author

Paul Barry, Apr 24 2005

Keywords

Comments

Row sums of number triangle A106190. Partial sums of A002420.
For n >= 1, the absolute values also give the iterates of A122237, starting from 0. (A122237(0), A122237(A122237(0)), A122237(A122237(A122237(0))), ...), this stems from the fact that the sequence gives the positions of terms with binary expansion 1(10){n-1}0 in A014486 (see A080675).

Crossrefs

|a(n)| = A080300(A080675(n)) = A075161(A001348(n)) (for n >= 1) = A075163(A000244(A008578(n-2))) = A014137(n-1)+A014138(n-2) = 2*A014137(n-1)-1, for n >= 2 (because binomial(2n+2, n+1)/(2n+1) = 2*A000108(n)).

Formula

a(n) = Sum_{k=0..n} binomial(2k, k)/(1-2k).
G.f.: (2/(1-x))/G(0), where G(k) = 1 + 1/(1 - 2*x*(2*k+1)/(2*x*(2*k+1) + (k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 24 2013
D-finite with recurrence: a(0)=1, a(1)=-1; for n>1, a(n) = (1/n)*((5*n-6)*a(n-1) - (4*n-6)*a(n-2)). - Tani Akinari, Aug 25 2013

Extensions

Barry's formula made more succinct, as well as comments regarding interpretation as absolute values added by Antti Karttunen, Sep 14 2006

A075161 Position of A075165(n+1) in A014486.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 9, 7, 6, 10, 23, 12, 65, 24, 11, 8, 197, 15, 626, 26, 25, 66, 2056, 13, 14, 198, 16, 68, 6918, 29, 23714, 17, 67, 627, 28, 40, 82500, 2057, 199, 27, 290512, 71, 1033412, 200, 30, 6919, 3707852, 31, 37, 38, 628, 629, 13402697, 43, 70, 69, 2058
Offset: 0

Views

Author

Antti Karttunen, Sep 13 2002

Keywords

Comments

See A075166.

Links

Crossrefs

Inverse of A075162. a(n) = A075163(n+1)-1. Cf. A075168.

A075164 Position of A014486(n-1) in A075165.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 8, 16, 7, 10, 15, 12, 24, 25, 18, 27, 32, 64, 81, 512, 256, 65536, 11, 14, 21, 20, 40, 35, 30, 45, 48, 96, 135, 768, 384, 98304, 49, 50, 75, 36, 72, 125, 54, 243, 128, 1024, 729, 32768, 4096, 16777216, 625, 162, 19683, 33554432, 262144
Offset: 1

Views

Author

Antti Karttunen, Sep 13 2002

Keywords

Comments

See A075166.

Links

Crossrefs

Inverse of A075163. a(n) = A075162(n-1)+1.

Formula

a(n) = A106443(A106454(n)). A000040(n) = a(1+A014137(n)). The powers of two are located at indices given by A057548 + 1, permuted by this same sequence, i.e. a(A057548(n)+1) = A000079(a(n)). - Antti Karttunen, May 09 2005

A106453 Position of A106455(n) in A014486 plus one.

Original entry on oeis.org

1, 2, 3, 4, 7, 6, 5, 8, 12, 16, 10, 13, 24, 11, 17, 9, 20, 30, 66, 41, 15, 25, 68, 14, 198, 67, 31, 27, 26, 44, 627, 21, 2059, 53, 71, 83, 2057, 199, 73, 42, 6919, 39, 629, 69, 34, 204, 23715, 35, 29, 628, 54, 201, 200, 86, 82501, 28, 205, 72, 290513, 125, 1033413
Offset: 1

Views

Author

Antti Karttunen, May 09 2005

Keywords

Comments

See A106456.

Links

Crossrefs

Inverse: A106454. a(n) = A075163(A106443(n)). a(n) = A106451(n-1)+1. GF(2)[X]-analog of A075163.
Showing 1-6 of 6 results.

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