A329544 -id:A329544 - cp's OEIS frontend

A329796 Running totals in A329544.

1, 4, 2, 7, 3, 22, 33, 11, 5, 22, 8, 0, 7, 22, 6, 33, 9, 22, 4, 33, 7, 44, 77, 33, 5, 44, 8, 33, 3, 44, 6, 55, 9, 44, 4, 55, 7, 66, 111, 101, 33, 1, 22, 2, 11, 66, 8, 55, 5, 66, 6, 77, 11, 88, 111, 99, 11, 202, 303, 414, 505, 393, 424, 505, 626, 757, 898, 828, 696, 616, 494, 404, 262, 88, 131, 77, 5

Offset: 1

N. J. A. Sloane, Dec 10 2019

nonn

By construction this is always >= 0 and a palindrome.
For n <= 50000 only two 0 terms have appeared, at n = 12 and 1002.
a(n) = a(n-1) +- A329544(n), using - if A329544(n) is even and otherwise +.

N. J. A. Sloane, Table of n, a(n) for n = 1..50000

Cf. A329544.

A329797 Record high water marks in A329544.

Original entry on oeis.org

1, 3, 5, 19, 22, 27, 29, 37, 44, 49, 51, 59, 68, 71, 77, 88, 191, 195, 197, 199, 201, 205, 207, 211, 303, 313, 323, 324, 333, 343, 344, 348, 353, 354, 363, 364, 373, 374, 475, 485, 516, 526, 536, 548, 617, 1609, 1782, 1829, 1921, 2031, 2079, 2195, 2211, 2321

Offset: 1

N. J. A. Sloane, Dec 10 2019

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..9019 (n = 1..361 from N. J. A. Sloane)

Cf. A329544, A329798, A330311.

A330311 Where n appears in A329544, or -1 if n never appears.

Original entry on oeis.org

1, 3, 2, 5, 4, 9, 13, 12, 45, 40, 7, 56, 18, 11, 14, 15, 10, 19, 6, 44, 43, 8, 55, 17, 28, 21, 16, 25, 20, 29, 63, 42, 23, 175, 34, 27, 22, 31, 26, 35, 30, 80, 75, 24, 39, 33, 48, 37, 32, 49, 36, 85, 79, 76, 46, 187, 130, 47, 38, 51, 50, 89, 84, 110, 109

Offset: 1

N. J. A. Sloane, Dec 10 2019

nonn

a(919) is unknown. If it is not -1, it is greater than 10^6 (see A329544).

N. J. A. Sloane, Table of n, a(n) for n = 1..918

Cf. A104444, A329544, A329797, A328798.

a(A104444(n)) = -1. - Rémy Sigrist, Dec 11 2019

A329798 Where record high water marks appear in A329544.

Original entry on oeis.org

1, 2, 4, 6, 8, 16, 20, 22, 24, 32, 36, 38, 41, 52, 54, 57, 58, 82, 87, 96, 104, 120, 128, 132, 135, 140, 160, 164, 191, 207, 211, 239, 252, 254, 260, 262, 264, 266, 267, 298, 321, 332, 337, 347, 429, 473, 495, 568, 642, 659, 1058, 1064, 1123, 1132, 1137

Offset: 1

N. J. A. Sloane, Dec 10 2019

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..6948 (n = 1..361 from N. J. A. Sloane)

Cf. A329544, A329797, A330311.

A330313 Add the odd terms and subtract the even ones, the result must always be a palindrome in base 3. This is the lexicographically earliest sequence of distinct positive integers with this property.

Original entry on oeis.org

1, 3, 2, 11, 7, 4, 6, 8, 21, 5, 12, 14, 89, 9, 18, 26, 16, 20, 10, 13, 17, 24, 75, 39, 30, 32, 23, 51, 31, 22, 43, 34, 48, 44, 28, 36, 19, 29, 42, 81, 68, 33, 35, 69, 60, 72, 63, 73, 52, 56, 40, 105, 61, 70, 84, 93, 91, 82, 50, 41, 98, 45, 53, 103, 64, 78, 54, 123, 128, 57, 71, 129

Offset: 1

N. J. A. Sloane, Dec 11 2019

A base 3 analog of A329544. The latter has an exceptionally irregular graph, so it is natural to ask if the graph is more understandable in a smaller base (and base 2 does not work).

Rémy Sigrist, Table of n, a(n) for n = 1..50000 (first 1000 terms from N. J. A. Sloane)
Rémy Sigrist, Density plot of the first 5000000 terms
Rémy Sigrist, C++ program for A330313
N. J. A. Sloane, Table of n, a(n), A330314(n), A330314(n)/a(n) for n = 1..1000

Cf. A014190, A329544, A330312, A330314 (running totals).

A330314 Running totals in A330313.

Original entry on oeis.org

1, 4, 2, 13, 20, 16, 10, 2, 23, 28, 16, 2, 91, 100, 82, 56, 40, 20, 10, 23, 40, 16, 91, 130, 100, 68, 91, 142, 173, 151, 194, 160, 112, 68, 40, 4, 23, 52, 10, 91, 23, 56, 91, 160, 100, 28, 91, 164, 112, 56, 16, 121, 182, 112, 28, 121, 212, 130, 80, 121, 23, 68, 121, 224, 160, 82, 28, 151, 23

Offset: 1

N. J. A. Sloane, Dec 11 2019

By construction this is always >= 0 and a palindrome in base 3.

a(n) = a(n-1) +- A330313(n), using - if A330313(n) is even and otherwise +.

Rémy Sigrist, Table of n, a(n) for n = 1..50000 (first 1000 terms from N. J. A. Sloane)
Rémy Sigrist, Scatterplot of the first 1000000 terms

Cf. A330313; A329544, A329796.

A330401 Lexicographically earliest sequence of distinct positive terms with an associate sequence t such that t(0) = 0, and for any n > 0, at least one of t(n-1) - a(n) or t(n-1) + a(n) is a palindrome; in case t(n-1) - a(n) is a palindrome, set t(n) = t(n-1) - a(n), otherwise set t(n) = t(n-1) + a(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 7, 18, 11, 9, 20, 13, 24, 22, 10, 21, 14, 25, 26, 15, 16, 27, 28, 17, 19, 30, 29, 40, 33, 44, 46, 35, 36, 47, 48, 37, 38, 49, 50, 39, 41, 52, 51, 62, 45, 12, 32, 43, 23, 34, 54, 60, 31, 61, 71, 81, 63, 53, 42, 55, 57, 68, 66, 72, 70, 79

Offset: 1

Rémy Sigrist, Dec 13 2019

This sequence is a variant of A329544.

			The first terms, alongside t(n), are:
  n   a(n)  t(n)
  --  ----  ----
   0  N/A      0
   1     1     1
   2     2     3
   3     3     0
   4     4     4
   5     5     9
   6     6     3
   7     8    11
   8     7     4
   9    18    22
  10    11    11
  11     9     2
  12    20    22

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A330401

See A330402 and A330403 for similar sequences.

Cf. A002113, A329544, A330420 (running totals).

PARI
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A330385 Add the odd terms and subtract the even ones, the result must always be a square. This is the lexicographically earliest sequence of distinct positive integers with this property.

Original entry on oeis.org

1, 3, 4, 9, 7, 12, 5, 8, 15, 16, 25, 11, 13, 24, 39, 17, 19, 21, 23, 44, 36, 28, 20, 33, 40, 27, 32, 45, 48, 35, 85, 72, 51, 64, 133, 87, 60, 29, 31, 105, 123, 84, 41, 43, 141, 96, 47, 49, 159, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 164, 156

Offset: 1

Rémy Sigrist, Dec 12 2019

This sequence is a variant of A329544.

All terms belong to A042965.

			The first terms, alongside the corresponding running totals, are:
  n   a(n)  t(n)
  --  ----  --------
   1     1   1 = 1^2
   2     3   4 = 2^2
   3     4   0 = 0^2
   4     9   9 = 3^2
   5     7  16 = 4^2
   6    12   4 = 2^2
   7     5   9 = 3^2
   8     8   1 = 1^2
   9    15  16 = 4^2
  10    16   0 = 0^2
  11    25  25 = 5^2

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Scatterplot of the first 5000000 terms

Cf. A000290, A042965, A329544, A330386 (running totals).

s=t=0; for (n=1, 65, for (v=1, oo, if (!bittest(s,v) && issquare(u=t-v*(-1)^v), print1 (v", "); s+=2^v; t=u; break)))

A329545 After a(1) = 1, add the even terms and subtract the odd ones, the result must always be a palindrome. This is the lexicographically earliest sequence of distinct positive integers with this property.

Original entry on oeis.org

1, 2, 3, 4, 18, 11, 5, 16, 13, 7, 6, 14, 15, 26, 22, 33, 17, 28, 25, 36, 35, 9, 8, 58, 55, 44, 46, 10, 20, 30, 73, 77, 66, 24, 40, 50, 103, 79, 68, 34, 23, 81, 48, 47, 80, 83, 72, 54, 43, 85, 52, 49, 38, 37, 70, 64, 53, 87, 32, 27, 60, 57, 90, 12, 45, 59, 92, 42, 75, 61, 94, 62, 95, 63, 74, 69, 194

Offset: 1

Eric Angelini and Jean-Marc Falcoz, Nov 16 2019

Negative palindromes are not allowed. After 50000 terms, the smallest unused integers are 964, 1020, 1029, 1031, 1038, 1041, 1047, 1051, ... and the largest used is 173410. The largest palindrome produced so far is 309903. Is the sequence a permutation of the integers > 0?

a(63411) = 964. Rémy Sigrist's comment in A329544 shows that terms in A104444 are not in the sequence. Conjecture: Sequence is a permutation of positive integers not in A104444. - Chai Wah Wu, Dec 11 2019

			The sequence starts with 1, smallest positive integer.
1 + 2 = 3 (palindrome)
1 + 2 - 3 = 0 (palindrome)
1 + 2 - 3 + 4 = 1 (palindrome)
1 + 2 - 3 + 4 + 18 = 22 (palindrome)
1 + 2 - 3 + 4 + 18 - 11 = 11 (palindrome)
1 + 2 - 3 + 4 + 18 - 11 - 5 = 6 (palindrome)
1 + 2 - 3 + 4 + 18 - 11 - 5 + 16 = 22 (palindrome), etc.

Jean-Marc Falcoz, Table of n, a(n) for n = 1..49999

Cf. A329544 (same idea, but where the odd integers are added and the even ones are subtracted).
Cf. A002113 (palindromes), A086862 (first differences of palindromes).
Cf. A104444.

A329545_vec(N, u=1, U, a, s=2, d)={vector(N, n, a=u; while(bittest(U, a-u)|| Vecrev(d=digits(s+(-1)^a*a))!=d|| (a>s&&bittest(a, 0)), a++); s+=(-1)^a*a; U+=1<<(a-u); while(bittest(U, 0), U>>=1; u++); a)} \\ M. F. Hasler, Nov 16 2019

A330325 Positions of 0's in A330314.

Original entry on oeis.org

103, 385, 2059, 2111, 2226, 2312, 2499, 3245, 15952, 16281, 16497, 16640, 18052, 18544, 18562, 18690, 18810, 19111, 19473, 19823, 19889, 19971, 20596, 20678, 21034, 21305, 21703, 21797, 23012, 23105, 23305, 23364, 25090, 25908, 26112, 26574, 27026, 27748

Offset: 1

N. J. A. Sloane, Dec 12 2019

The 0 terms in A329796 seem very scarce, but the present sequence suggests that maybe one needs to search further in that sequence.

Rémy Sigrist, Table of n, a(n) for n = 1..1000

Cf. A329544, A329796, A330313, A330314.

cp's OEIS Frontend

A329796 Running totals in A329544.

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A329797 Record high water marks in A329544.

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A330311 Where n appears in A329544, or -1 if n never appears.

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A329798 Where record high water marks appear in A329544.

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A330313 Add the odd terms and subtract the even ones, the result must always be a palindrome in base 3. This is the lexicographically earliest sequence of distinct positive integers with this property.

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A330314 Running totals in A330313.

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A330385 Add the odd terms and subtract the even ones, the result must always be a square. This is the lexicographically earliest sequence of distinct positive integers with this property.

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A329545 After a(1) = 1, add the even terms and subtract the odd ones, the result must always be a palindrome. This is the lexicographically earliest sequence of distinct positive integers with this property.

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A330325 Positions of 0's in A330314.

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