Hihi, all - I would love to help, and since i am retiring from the aerospace corporation at the end of this month, i will have more time for interesting problems More soon, Chris Personal e-mail topcycal at gmail d o t c o m I, phone
On Sep 24, 2019, at 07:40, Neil Sloane <njasloane@gmail.com> wrote:
Bill, 2 weeks ago you said Superseeker's reply was hard to understand, yet as Fred pointed out, one of the suggestions was that if you subtract 2 from your sequence you get the standard paper-folding sequence A014577. You never responded, so maybe you missed Fred's comment?
The OEIS is pretty useful on its own, but when combined with Superseeker it is sometimes amazing. I would like Superseeker to do more, perhaps adding some "AI" black box to it. Are there any folks on this list who could help?
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Thu, Sep 12, 2019 at 12:32 AM Fred Lunnon <fred.lunnon@gmail.com> wrote:
It did suggest http://oeis.org/A014577 , which under the circumstances seems quite perceptive! WFL
On Thu, Sep 12, 2019 at 3:49 AM Bill Gosper <billgosper@gmail.com> wrote:
I submitted the following trivalued sequence, based on the Dragon Curve. I don't see a connection with any of its suggestions. ? —rwg ---------- Forwarded message --------- From: <superseeker-reply@oeis.org> Date: Wed, Sep 11, 2019 at 7:14 PM Subject: Re: To: <billgosper@gmail.com>
Greetings from The On-Line Encyclopedia of Integer Sequences! https://oeis.org/
lookup 2 1 3 2 1 2 3 2 2 1 2 3 1 2 3 2 2 1 3 2 1 2 2 3 2 1 2 3 1
a(n) = 2, 1, 3, 2, 1, 2, 3, 2, 2, 1, 2, 3, 1, 2, 3, 2, 2, 1, 3, 2, 1, 2, 2, 3, 2, 1, 2, 3, 1
Note: Your sequence does not directly appear in the OEIS. If it is of general interest, please submit it at https://oeis.org/Submit.html.
# Transformations
These sequences match transformations of the original query.
T099 abs(a(n)), sorted with duplicates removed = 1, 2, 3
http://oeis.org/A000045 Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.
0, 1, <1, 2, 3>, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169, 63245986, 102334155
http://oeis.org/A000041 a(n) is the number of partitions of n (the partition numbers).
1, <1, 2, 3>, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1958, 2436, 3010, 3718, 4565, 5604, 6842, 8349, 10143, 12310, 14883, 17977, 21637, 26015, 31185, 37338, 44583, 53174, 63261, 75175, 89134, 105558, 124754, 147273, 173525
http://oeis.org/A000027 The positive integers. Also called the natural numbers, the whole numbers or the counting numbers, but these terms are ambiguous.
<1, 2, 3>, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77
http://oeis.org/A001222 Number of prime divisors of n counted with multiplicity (also called bigomega(n) or Omega(n)).
0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4, 1, 3, 1, 3, 2, 2, 1, 4, 2, 2, 3, 3, 1, 3, 1, 5, 2, 2, 2, 4, 1, 2, 2, 4, 1, 3, 1, 3, 3, 2, 1, 5, 2, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 4, <1, 2, 3>, 6, 2, 3, 1, 3, 2, 3, 1, 5, <1, 2, 3>, 3, 2, 3, 1, 5, 4, 2, 1, 4, 2, 2, 2, 4, 1, 4, 2, 3, 2, 2, 2, 6, 1, 3, 3, 4, 1, 3, 1, 4, 3, 2, 1, 5, 1, 3, 2
http://oeis.org/A001221 Number of distinct primes dividing n (also called omega(n)).
0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 3, 1, 2, 2, <1, 2, 3>, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 2, 3, 1, 2, 1, 2, 1, 3, 2, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 3, <1, 2, 3>, 2, 1, 2, 1, 3, 2
... 25666 total
# Transformations as Deltas
The deltas of these sequences match transformations of the original query.
T030 a(n+2) - a(n) = 1, 1, 2, 0, 2, 0, 1, 1, 0, 2, 1, 1, 2, 0, 1, 1, 1, 1, 2, 0, 1, 1, 0, 2, 0, 2, 1 (as deltas)
http://oeis.org/A301849 The Pagoda sequence: a sequence with isolated zeros in number-wall over finite fields.
-1, 0, <1, 0, -1, 1, 1, -1, -1, 0, 1, 1, -1, 0, 1, -1, -1, 0, 1, 0, -1, 1, 1, 0, -1, -1, 1, 1, -1, 0>, 1, -1, -1, 0, 1, 0, -1, 1, 1, -1, -1, 0, 1, 1, -1, 0, 1, 0, -1, -1, 1, 0, -1, 1, 1, 0, -1, -1, 1, 1, -1, 0, 1, -1, -1, 0, <1, 0, -1, 1, 1, -1, -1, 0, 1, 1, -1, 0, 1, -1, -1, 0, 1, 0, -1, 1, 1, 0, -1, -1, 1, 1, -1, 0>, 1, 0, -1, -1, 1, 0
http://oeis.org/A253414 G.f. satisfies (1+x^2)*g(x) = 1 + x*g(x^2).
1, 1, <-1, 0, 1, -1, -1, 1, 1, 0, -1, -1, 1, 0, -1, 1, 1, 0, -1, 0, 1, -1, -1, 0, 1, 1, -1, -1, 1, 0>, -1, 1, 1, 0, -1, 0, 1, -1, -1, 1, 1, 0, -1, -1, 1, 0, -1, 0, 1, 1, -1, 0, 1, -1, -1, 0, 1, 1, -1, -1, 1, 0, -1, 1, 1, 0, <-1, 0, 1, -1, -1, 1, 1, 0, -1, -1, 1, 0, -1, 1, 1, 0, -1, 0, 1, -1, -1, 0, 1, 1, -1, -1, 1, 0>, -1, 0, 1, 1, -1, 0, 1
T035 a(n) + 2 = 4, 3, 5, 4, 3, 4, 5, 4, 4, 3, 4, 5, 3, 4, 5, 4, 4, 3, 5, 4, 3, 4, 4, 5, 4, 3, 4, 5, 3 (as deltas)
http://oeis.org/A317331 Indices m for which A058304(m) = 1.
<4, 8, 11, 16, 20, 23, 27, 32, 36, 40, 43, 47, 52, 55, 59, 64, 68, 72, 75, 80, 84, 87, 91, 95, 100, 104, 107, 111, 116, 119>, 123, 128, 132, 136, 139, 144, 148, 151, 155, 160, 164, 168, 171, 175, 180, 183, 187, 191, 196, 200, 203, 208, 212, 215, 219, 223, 228, 232, 235, 239, 244, 247, 251
T038 a(n) - 2 = 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1 (as deltas)
http://oeis.org/A014577 The regular paper-folding sequence (or dragon curve sequence).
<1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0>, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, <1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0>, 0, 0, 1, 1, 0
http://oeis.org/A038189 Bit to left of least significant 1-bit in binary expansion of n.
0, <0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1>, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, <0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1>, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1
http://oeis.org/A014707 a(4n)=0, a(4n+2)=1, a(2n+1)=a(n).
<0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1>, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0
http://oeis.org/A082410 a(1)=0. Thereafter, the sequence is constructed using the rule: for any k >= 0, if a(1), a(2), ..., a(2^k+1) are known, the next 2^k terms are given as follows: a(2^k+1+i) = 1 - a(2^k+1-i) for 1 <= i <= 2^k.
0, <1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0>, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, <1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0>, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1
http://oeis.org/A014710 The regular paper-folding (or dragon curve) sequence.
<2, 2, 1, 2, 2, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 2, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1>, 1, 2, 2, 2, 1, 2, 2, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2, 2, 1, 2, 2, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 2
... 9 total
T099 abs(a(n)), sorted with duplicates removed = 1, 2, 3 (as deltas)
http://oeis.org/A000045 Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.
0, 1, 1, <2, 3, 5, 8>, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169, 63245986, 102334155
http://oeis.org/A000217 Triangular numbers: a(n) = binomial(n+1,2) = n(n+1)/2 = 0 + 1 + 2 + ... + n.
<0, 1, 3, 6>, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225, 1275, 1326, 1378, 1431
http://oeis.org/A001222 Number of prime divisors of n counted with multiplicity (also called bigomega(n) or Omega(n)).
0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4, 1, 3, 1, 3, 2, 2, 1, 4, 2, 2, 3, 3, 1, 3, 1, 5, 2, 2, 2, 4, 1, 2, 2, 4, 1, 3, 1, 3, 3, 2, 1, 5, 2, 3, <2, 3, 1, 4>, 2, 4, 2, 2, 1, 4, 1, 2, 3, 6, 2, 3, 1, 3, 2, 3, 1, 5, 1, 2, 3, 3, 2, 3, 1, 5, 4, 2, 1, 4, 2, 2, 2, 4, 1, 4, 2, 3, 2, 2, 2, 6, 1, 3, 3, 4, 1, 3, 1, 4, 3, 2, 1, 5, 1, 3, 2
http://oeis.org/A000700 Expansion of Product_{k>=0} (1 + x^(2k+1)); number of partitions of n into distinct odd parts; number of self-conjugate partitions; number of symmetric Ferrers graphs with n nodes.
1, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 5, 5, 5, 6, 7, 8, 8, 9, 11, 12, 12, 14, 16, <17, 18, 20, 23>, 25, 26, 29, 33, 35, 37, 41, 46, 49, 52, 57, 63, 68, 72, 78, 87, 93, 98, 107, 117, 125, 133, 144, 157, 168, 178, 192, 209, 223, 236, 255, 276, 294, 312, 335, 361, 385
http://oeis.org/A112798 Table where n-th row is factorization of n, with each prime p_i replaced by i.
1, 2, 1, 1, 3, <1, 2, 4, 1>, 1, 1, 2, 2, 1, 3, 5, 1, 1, 2, 6, 1, 4, 2, 3, 1, 1, 1, 1, 7, 1, 2, 2, 8, 1, 1, <3, 2, 4, 1>, 5, 9, 1, 1, 1, 2, 3, 3, 1, 6, 2, 2, 2, 1, 1, 4, 10, 1, 2, 3, 11, 1, 1, 1, 1, 1, 2, 5, 1, 7, 3, 4, 1, 1, 2, 2, 12, 1, 8, 2, 6, 1, 1, 1, 3, 13, 1, 2, 4, 14, 1, 1, 5, 2, 2, 3, 1, 9, 15, 1, 1, 1, 1
... 17444 total
In transformation descriptions, Sn(z) denotes the ordinary generating function with coefficients a(n), and En(z) denotes the exponential generating function with coefficients a(n).
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