A352760 -id:A352760 - cp's OEIS frontend

A352756 Positive numbers k such that the centered cube number k^3 + (k+1)^3 is equal to the difference of two positive cubes and to A352755(n).

3, 46, 197, 528, 1111, 2018, 3321, 5092, 7403, 10326, 13933, 18296, 23487, 29578, 36641, 44748, 53971, 64382, 76053, 89056, 103463, 119346, 136777, 155828, 176571, 199078, 223421, 249672, 277903, 308186, 340593, 375196, 412067, 451278, 492901, 537008, 583671, 632962, 684953, 739716, 797323, 857846, 921357

Offset: 1

Vladimir Pletser, Apr 02 2022

Numbers B > 0 such that the centered cube number B^3 + (B+1)^3 is equal to the difference of two positive cubes, i.e., A = B^3 + (B+1)^3 = C^3 - D^3 and such that C - D = 2n - 1, with C > D > B > 0, and A > 0, A = t*(3*t^2 + 4)*(t^2*(3*t^2 + 4)^2 + 3)/4 with t = 2*n-1, and where A = A352755(n), B = a(n) (this sequence), C = A352757(n) and D = A352758(n).
There are infinitely many such numbers a(n) = B in this sequence.
Subsequence of A352134 and of A352221.

			a(1) = 3 is a term because 3^3 + 4^3 = 6^3 - 5^3 and 6 - 5 = 1 = 2*1 - 1.
a(2) = 46 is a term because 46^3 + 47^3 = 151^3 - 148^3 and 151 - 148 = 3 = 2*2 - 1.
a(3) = ((2*3 - 1)*(3*(2*3 - 1)^2 + 4) - 1)/2 = 197.
a(4) = 3*197 - 3*46 + 3 + 72 = 528.

Vladimir Pletser, Table of n, a(n) for n = 1..10000
A. Grinstein, Ramanujan and 1729, University of Melbourne Dept. of Math and Statistics Newsletter: Issue 3, 1998.
Vladimir Pletser, Euler's and the Taxi-Cab relations and other numbers that can be written twice as sums of two cubed integers, submitted. Preprint available on ResearchGate, 2022.
Eric Weisstein's World of Mathematics, Centered Cube Number
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A005898, A001235, A272885, A352133, A352134, A352135, A352136, A352220, A352222, A352223, A352224, A352225, A352755, A352757, A352758, A352759, A352760, A352761, A352762.

restart; for n to 20 do (1/2)* ((2*n - 1)*(3*(2*n - 1)^2 + 4) - 1);  end do;

a(n)^3 + (a(n)+1)^3 = A352757(n)^3 - A352758(n)^3 and A352757(n) - A352758(n) = 2*n - 1.
a(n) = ((2*n - 1)*(3*(2*n - 1)^2 + 4) - 1)/2.
For n > 3, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 72, with a(1) = 3, a(2) = 46 and a(3) = 197.
a(n) can be extended for negative n such that a(-n) = -a(n+1) - 1.
G.f.: x*(3 + 34*x + 31*x^2 + 4*x^3)/(1 - x)^4. - Stefano Spezia, Apr 08 2022

A352757 a(n) = (3(2n - 1)^2((2n - 1)^2 + 2) + 2*n + 1)/2 for n > 0.

Original entry on oeis.org

6, 151, 1016, 3753, 10090, 22331, 43356, 76621, 126158, 196575, 293056, 421361, 587826, 799363, 1063460, 1388181, 1782166, 2254631, 2815368, 3474745, 4243706, 5133771, 6157036, 7326173, 8654430, 10155631, 11844176, 13735041, 15843778, 18186515, 20779956, 23641381, 26788646, 30240183, 34015000, 38132681

Offset: 1

Vladimir Pletser, Apr 02 2022

Numbers C > 0 such that A = B^3 + (B+1)^3 = C^3 - D^3 such that the difference C - D is odd, C - D = 2*n - 1, and the difference between the positive cubes C^3 - D^3 is equal to a centered cube number, C^3 - D^3 = B^3 + (B+1)^3, with C > D > B > 0, and A > 0, A = t*(3*t^2 + 4)*(t^2*(3*t^2 + 4)^2 + 3)/4 with t = 2*n-1, and where A = A352755(n), B = A352756(n), C = a(n) (this sequence), and D = A352758(n).

There are infinitely many such numbers a(n) = C in this sequence.

			a(1) = 6 belongs to the sequence as 6^3 - 5^3 = 3^3 + 4^3 = 91 and 6 - 5 = 1 = 2*1 - 1.
a(2) = 151 belongs to the sequence as 151^3 - 148^3 = 46^3 + 47^3 = 201159 and 151 - 148 = 3 = 2*2 - 1.
a(3) = (3(2*3 - 1)^2*((2*3 - 1)^2 + 2) + 2*3 + 1)/2 = 1016.
a(4) = 3*a(3) - 3*a(2) + a(1) + 576*2 = 3*1016 - 3*151 + 6 + 576*2 = 3753.

Vladimir Pletser, Table of n, a(n) for n = 1..10000
A. Grinstein, Ramanujan and 1729, University of Melbourne Dept. of Math and Statistics Newsletter: Issue 3, 1998.
Vladimir Pletser, Euler's and the Taxi-Cab relations and other numbers that can be written twice as sums of two cubed integers, submitted. Preprint available on ResearchGate, 2022.
Eric Weisstein's World of Mathematics, Centered Cube Number
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A005898, A001235, A272885, A352133, A352134, A352135, A352136, A352220, A352221, A352223, A352224, A352225, A352755, A352756, A352758, A352759, A352760, A352761, A352762.

restart; for n to 20 do (1/2)*(3*(2*n - 1)^2*((2*n - 1)^2 + 2) + 2*n + 1); end do;

def A352757(n): return n*(n*(n*(24*n - 48) + 48) - 23) + 5 # Chai Wah Wu, Jul 10 2022

a(n)^3 - A352758(n)^3 = A352756(n)^3 + (A352756(n) + 1)^3 = A352755(n) and a(n) - A352758(n) = 2*n - 1.
a(n) = (3*(2*n - 1)^2*((2*n - 1)^2 + 2) + 2*n + 1)/2.
For n > 3, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 576*(n - 2), with a(1) = 6, a(2) = 151 and a(3) = 1016.
a(n) can be extended for negative n such that a(-n) = a(n+1) - (2*n + 1).
G.f.: x*(6 + 121*x + 321*x^2 + 123*x^3 + 5*x^4)/(1 - x)^5. - Stefano Spezia, Apr 08 2022

A355504 Lexicographically earliest sequence of distinct nonnegative integers such that for any n >= 0, among the decimal digits of n and a(n) (counted with multiplicity) there are as many even digits as odd digits.

Original entry on oeis.org

1, 0, 3, 2, 5, 4, 7, 6, 9, 8, 10, 20, 12, 22, 14, 24, 16, 26, 18, 28, 11, 21, 13, 23, 15, 25, 17, 27, 19, 29, 30, 40, 32, 42, 34, 44, 36, 46, 38, 48, 31, 41, 33, 43, 35, 45, 37, 47, 39, 49, 50, 60, 52, 62, 54, 64, 56, 66, 58, 68, 51, 61, 53, 63, 55, 65, 57, 67

Offset: 0

Rémy Sigrist, Jul 05 2022

Leading zeros for positive integers are ignored.

This sequence is a self-inverse permutation of the nonnegative integers.

			Some terms alongside the corresponding even and odd digits are:
    n    a(n)  even   odd
    ---  ----  -----  -----
      0     1      0      1
      1     0      0      1
      2     3      2      3
      3     2      2      3
      4     5      4      5
      5     4      4      5
      6     7      6      7
      7     6      6      7
      8     9      8      9
      9     8      8      9
     10    10     00     11
     11    20     20     11
     12    12     22     11
     13    22     22     13
     14    14     44     11
    ...   ...    ...    ...
     90    90     00     99
     91  1000    000    911
     92    92     22     99
     93  1002    002    931
     94    94     44     99
     95  1004    004    951
     96    96     66     99
     97  1006    006    971
     98    98     88     99
     99  1008    008    991
    100   101    000    111

Rémy Sigrist, Table of n, a(n) for n = 0..9110
Rémy Sigrist, Log-log scatterplot of the sequence for n = 0..9111110
Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

Cf. A227870 (fixed points), A352546, A352547, A352760.

PARI
```
See Links section.
```

a(n) = n iff n belongs to A227870.

a(n) belongs to A352546 iff n belongs to A352547 and vice versa.

A357616 Lexicographically earliest sequence of distinct nonnegative integers such that for any n >= 0, the number of 1's in the ternary expansion of n equals the number of 2's in the ternary expansion of a(n) and vice versa.

Original entry on oeis.org

0, 2, 1, 6, 8, 5, 3, 7, 4, 18, 20, 11, 24, 26, 17, 15, 23, 14, 9, 19, 10, 21, 25, 16, 12, 22, 13, 54, 56, 29, 60, 62, 35, 33, 47, 32, 72, 74, 51, 78, 80, 53, 59, 71, 44, 45, 61, 34, 65, 77, 50, 38, 52, 41, 27, 55, 28, 57, 69, 42, 30, 46, 31, 63, 73, 48, 75, 79

Offset: 0

Rémy Sigrist, Oct 06 2022

This sequence is a self-inverse permutation of the nonnegative integers.

			The first terms, alongside their ternary expansions, are:
  n   a(n)  ter(n)  ter(a(n))
  --  ----  ------  ---------
   0     0       0          0
   1     2       1          2
   2     1       2          1
   3     6      10         20
   4     8      11         22
   5     5      12         12
   6     3      20         10
   7     7      21         21
   8     4      22         11
   9    18     100        200
  10    20     101        202
  11    11     102        102
  12    24     110        220

Rémy Sigrist, Table of n, a(n) for n = 0..6560
Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

Cf. A004488, A005823, A005836, A039001 (fixed points), A062756, A081603, A352760.

PARI
```
See Links section.
```

A081603(a(n)) = A062756(n).
A062756(a(n)) = A081603(n).
a(n) < 3^k iff n < 3^k.
a(n) = n iff n belongs to A039001.
Empirically:
- a(n) = n/2 iff n belongs to A005823,
- a(n) = 2*n iff n belongs to A005836.

cp's OEIS Frontend

A352756 Positive numbers k such that the centered cube number k^3 + (k+1)^3 is equal to the difference of two positive cubes and to A352755(n).

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A352757 a(n) = (3*(2*n - 1)^2*((2*n - 1)^2 + 2) + 2*n + 1)/2 for n > 0.

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A355504 Lexicographically earliest sequence of distinct nonnegative integers such that for any n >= 0, among the decimal digits of n and a(n) (counted with multiplicity) there are as many even digits as odd digits.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A357616 Lexicographically earliest sequence of distinct nonnegative integers such that for any n >= 0, the number of 1's in the ternary expansion of n equals the number of 2's in the ternary expansion of a(n) and vice versa.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A352757 a(n) = (3(2n - 1)^2((2n - 1)^2 + 2) + 2*n + 1)/2 for n > 0.