A375336 -id:A375336 - cp's OEIS frontend

A080782 a(1)=1, a(n)=a(n-1)-1 if n is already in the sequence, a(n)=a(n-1)+2 otherwise.

1, 3, 2, 4, 6, 5, 7, 9, 8, 10, 12, 11, 13, 15, 14, 16, 18, 17, 19, 21, 20, 22, 24, 23, 25, 27, 26, 28, 30, 29, 31, 33, 32, 34, 36, 35, 37, 39, 38, 40, 42, 41, 43, 45, 44, 46, 48, 47, 49, 51, 50, 52, 54, 53, 55, 57, 56, 58, 60, 59, 61, 63, 62, 64, 66, 65, 67, 69, 68

Offset: 1

Benoit Cloitre, Mar 07 2003

Permutation of the integers: exchange trisections starting with 2 and 3.

a(a(n)) = n. - Reinhard Zumkeller, Oct 29 2004

Guenther Schrack, Table of n, a(n) for n = 1..10000
Robbert Fokkink and Gandhar Joshi, On Cloitre's hiccup sequences, arXiv:2507.16956 [math.CO], 2025. See p. 2.
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).
Index entries for sequences that are permutations of the natural numbers.

Cf. A064429, A079357, A079354, A080783, A064437.

Cf. A004442, A080412.

Array[#+Mod[#+1,3]&,70,0] (* or *) LinearRecurrence[{1,0,1,-1},{1,3,2,4},70] (* Harvey P. Dale, Mar 29 2013 *)
{#,#+1,#-1}[[Mod[#,3,1]]]&/@Range[99] (* Federico Provvedi, May 15 2021 *)

a(n) = A064429(n-1) + 1.
a(n) - n is periodic with period 3.
G.f.: x*(1+2*x-x^2+x^3)/(1-x-x^3+x^4). - Jaume Oliver Lafont, Mar 24 2009
a(0)=1, a(1)=3, a(2)=2, a(3)=4, a(n)=a(n-1)+0*a(n-2)+a(n-3)-a(n-4). - Harvey P. Dale, Mar 29 2013
a(n) = n + (2/sqrt(3))*sin(2*(n+2)*Pi/3). - Wesley Ivan Hurt, Sep 26 2017
From Guenther Schrack, Oct 23 2019: (Start)
a(n) = a(n-3) + 3 with a(1) = 1, a(2) = 3, a(3) = 2 for n > 3.
a(n) = n - (w^(2*n)*(2 + w) + w^n*(1 - w))/3 where w = (-1 + sqrt(-3))/2. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*Pi/(3*sqrt(3)) - log(2)/3. - Amiram Eldar, Jan 31 2023
From Charles L. Hohn, Sep 03 2024: (Start)
a(n) = n-1+n%3.
a(n) = A375336(n-2, 1) for n >= 6. (End)

A374602 Array of successive integer solutions to sqrt((d-c)b^2 + c(b+1)^2) for nonsquare integers d >= 2 (d=A000037(n) for n >= 1), where b and c are positive integers and c < d, read by antidiagonals.

Original entry on oeis.org

5, 29, 3, 169, 11, 5, 985, 41, 13, 3, 5741, 153, 34, 7, 4, 33461, 571, 89, 18, 5, 10, 195025, 2131, 233, 29, 11, 11, 4, 1136689, 7953, 610, 69, 28, 23, 5, 7, 6625109, 29681, 1597, 178, 62, 58, 13, 8, 6, 38613965, 110771, 4181, 287, 79, 338, 14, 13, 22, 4

Offset: 1

Charles L. Hohn, Jul 13 2024

T(n,k) is the diagonal lengths of increasingly nearly regular d-dimensional Pythagorean hyperrectangles.
Each row n divides into equal length, geometrically periodic subsequences, each with its own subsequence period length (A377290) and geometric growth factor (A377291); it is conjectured that this is the case for all n, and that all solutions conform as such and that there are no solutions that do not, but these are not proven.
It is also not known if there is an algorithm for generating values for all rows other than testing all possible values for a row until a subsequence pattern emerges.
Square d produce solutions following a different pattern, shown as A375336.

			n=row index; d=nonsquare integer of index n (A000037(n)):
 n    d   T(n,k)
---+----+-------------------------------------------------------------
 1 |  2 |  5, 29, 169, 985, 5741, 33461, 195025, 1136689, 6625109, ...
 2 |  3 |  3, 11,  41, 153,  571,  2131,   7953,   29681,  110771, ...
 3 |  5 |  5, 13,  34,  89,  233,   610,   1597,    4181,   10946, ...
 4 |  6 |  3,  7,  18,  29,   69,   178,    287,     683,    1762, ...
 5 |  7 |  4,  5,  11,  28,   62,    79,    175,     446,     988, ...
 6 |  8 | 10, 11,  23,  58,  338,   373,    781,    1970,   11482, ...
 7 | 10 |  4,  5,  13,  14,   25,    62,    111,     148,     185, ...
 8 | 11 |  7,  8,  13,  32,   57,   139,    158,     259,     638, ...
 9 | 12 |  6, 22,  39,  69,   82,   125,    306,     543,    1142, ...
10 | 13 |  4,  5,   7,  17,   30,    43,     53,      76,     185, ...
11 | 14 |  9, 11,  14,  19,   46,    81,    267,     329,     418, ...
12 | 15 |  6, 10,  21,  23,   30,    39,     94,     165,     362, ...
13 | 17 | 25, 27,  34,  41,   98,   171,    260,    1649,    1779, ...
14 | 18 |  6, 13,  15,  18,   21,    50,     87,     132,     198, ...
15 | 19 |  5,  7,   8,   9,   11,    31,     34,      37,      56, ...
16 | 20 | 10, 26,  68, 125,  159,   178,    197,     466,     807, ...
17 | 21 |  6,  9,  12,  13,   14,    33,     57,      86,     134, ...
18 | 22 |  5,  7,   8,  17,   18,    19,     31,      64,      77, ...
19 | 23 | 16, 19,  27,  28,   29,    68,    117,     176,     764, ...
20 | 24 |  6,  9,  11,  14,   36,    39,     57,      58,      59, ...
...
sqrt((2-1)*1^2 + 1*(1+1)^2) = sqrt(5) -> not an integer so not included.
sqrt((2-1)*3^2 + 1*(3+1)^2) = 5 -> T(1,1).
sqrt((2-1)*20^2 + 1*(20+1)^2) = 29 -> T(1,2).
sqrt((3-2)*1^2 + 2*(1+1)^2) = 3 -> T(2,1).
sqrt((6-2)*7^2 + 2*(7+1)^2) = 18 -> T(4,3).

Row 1 is A001653 starting at n=2.
Row 2 is A079935 starting at n=2.
Bisection of row 2 starting with the first term is A189356 starting at n=1.
Bisection of row 2 starting with the second term is A122769 starting at n=2.
Row 3 is A001519 starting at n=3.
Bisection of row 3 starting with the first term is A033889 starting at n=1.
Bisection of row 3 starting with the second term is A033891 starting at n=1.
Row 4 is A131093 starting at n=3.
Cf. A000037, A373666.
Cf. A377290, A377291, A375336.

row(n, c)=my(v=List(), d=n+floor(sqrt(n)+1/2) /* d=A000037(n) */, t=ceil(sqrt(d))); while(#v

T(n, 1) = A373666(A000037(n)).

A376005 For integers >=2, the number of integer solutions to sqrt((n^2-c)b^2 + c(b+1)^2) where b and c are positive integers and c < n^2.

Original entry on oeis.org

0, 0, 2, 2, 4, 6, 10, 8, 14, 16, 18, 22, 26, 22, 38, 36, 36, 44, 48, 42, 60, 62, 62, 64, 78, 68, 88, 90, 78, 102, 114, 92, 120, 104, 118, 134, 144, 122, 148, 156, 138, 168, 178, 142, 194, 194, 186, 192, 200, 188, 232, 230, 212, 218, 252, 224, 274, 272, 236

Offset: 2

Charles L. Hohn, Sep 05 2024

nonn

a(n) = count(k) of A375336(n, k) for each n >= 2.

Omitting n < 2, which yield no c values that meet the rubric.

			A375336 rows n = 2 and n = 3 are empty, so a(2) = 0 and a(3) = 0.
A375336 rows n = 4 and n = 5 each contain 2 terms, so a(4) = 2 and a(5) = 2.

Cf. A375336.

a(n)=my(d=n^2, t=n, an=0); while(t<=n^3/8, my(b=floor(sqrt(t^2/d)), r=t^2-d*b^2); if (r && r%(b*2+1)==0, an++); t++); an
for(n=2, 100, print(n, " ", a(n)))

A376007 For integers n>=4, greatest integer that can satisfy sqrt((n^2-c)b^2 + c(b+1)^2) where b and c are positive integers and c < n^2.

Original entry on oeis.org

7, 8, 27, 19, 61, 42, 125, 83, 211, 137, 343, 204, 505, 299, 729, 428, 991, 578, 1331, 749, 1717, 964, 2197, 1229, 2731, 1523, 3375, 1846, 4081, 2229, 4913, 2678, 5815, 3164, 6859, 3687, 7981, 4286, 9261, 4967, 10627, 5693, 12167, 6464, 13801, 7327, 15625

Offset: 4

Charles L. Hohn, Sep 05 2024

nonn

Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1,0,0,0,2,0,-4,0,2,0,0,0,-1,0,2,0,-1).

Cf. A375336, A376005.

a(n)=my(d=n^2, t=ceil(n^3/8)); while(t>=n, my(b=floor(sqrt(t^2/d)), r=t^2-d*b^2); if (r && r%(b*2+1)==0, return(t)); t--)
for(n=4, 100, print(n, " ", a(n)))

a(n)=if(n%4==2, 2*n^3, n%4==0, 2*n^3-8*n+16, abs(n%8-4)==1, n^3-n+8, n^3-9*n+24)/16
for(n=4, 100, print(n, " ", a(n)))

def A376007(n):
    if (m:=n&7)==0 or m==4:
        return n*(n**2-4)+8>>3
    elif m==1 or m==7:
        return n*(n**2-9)+24>>4
    elif m==2 or m==6:
        return n**3>>3
    else:
        return n*(n**2-1)+8>>4 # Chai Wah Wu, Sep 27 2024

a(n) = A375336(n, A376005(n)).
a(n) = (n^3-4*n+8)/8 for n%4==0.
a(n) = (n^3-9*n+24)/16 for n%8==1 or n%8==7.
a(n) = (n^3)/8 for n%4==2.
a(n) = (n^3-n+8)/16 for n%8==3 or n%8==5.
From Chai Wah Wu, Sep 27 2024: (Start)
a(n) = 2*a(n-2) - a(n-4) + 2*a(n-8) - 4*a(n-10) + 2*a(n-12) - a(n-16) + 2*a(n-18) - a(n-20) for n > 23.
G.f.: x^4*(-2*x^19 - x^18 + 3*x^17 + x^16 - 2*x^15 + 2*x^14 + 4*x^13 + 2*x^12 + 7*x^11 + 20*x^10 - 3*x^9 + 8*x^8 + 18*x^7 + 30*x^6 + 12*x^5 + 14*x^4 + 3*x^3 + 13*x^2 + 8*x + 7)/((x - 1)^4*(x + 1)^4*(x^2 + 1)^2*(x^4 + 1)^2). (End)

cp's OEIS Frontend

A080782 a(1)=1, a(n)=a(n-1)-1 if n is already in the sequence, a(n)=a(n-1)+2 otherwise.

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A374602 Array of successive integer solutions to sqrt((d-c)*b^2 + c*(b+1)^2) for nonsquare integers d >= 2 (d=A000037(n) for n >= 1), where b and c are positive integers and c < d, read by antidiagonals.

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A376005 For integers >=2, the number of integer solutions to sqrt((n^2-c)*b^2 + c*(b+1)^2) where b and c are positive integers and c < n^2.

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A376007 For integers n>=4, greatest integer that can satisfy sqrt((n^2-c)*b^2 + c*(b+1)^2) where b and c are positive integers and c < n^2.

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A374602 Array of successive integer solutions to sqrt((d-c)b^2 + c(b+1)^2) for nonsquare integers d >= 2 (d=A000037(n) for n >= 1), where b and c are positive integers and c < d, read by antidiagonals.

A376005 For integers >=2, the number of integer solutions to sqrt((n^2-c)b^2 + c(b+1)^2) where b and c are positive integers and c < n^2.

A376007 For integers n>=4, greatest integer that can satisfy sqrt((n^2-c)b^2 + c(b+1)^2) where b and c are positive integers and c < n^2.