A080782 -id:A080782 - cp's OEIS frontend

A164765 Partial sums of [A080782^2].

1, 10, 14, 30, 66, 91, 140, 221, 285, 385, 529, 650, 819, 1044, 1240, 1496, 1820, 2109, 2470, 2911, 3311, 3795, 4371, 4900, 5525, 6254, 6930, 7714, 8614, 9455, 10416, 11505, 12529, 13685, 14981, 16206, 17575, 19096, 20540, 22140, 23904, 25585

Offset: 1

Carl R. White, Aug 25 2009

Yet another plausible solution to A115603.
The first differences of A115603 are all squares (assuming a prior term of 0), meaning that any sequence beginning 1,3,2,4 is sufficient to account for them; This solution chooses the permutation of integers A080782 = {1,3,2,4,6,5,7,9,8,...}
Ultimately that means this sequence is equal to A000330 for every two out of three consecutive terms, and is greater by 2n+1 where different.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,2,-4,2,-1,2,-1).

Original puzzle: A115603; Used in this solution: A080782, A000330; Other solutions: A115391, A116955, A162899

Accumulate[Array[#+Mod[#+1,3]&,70,0]^2] (* Harvey P. Dale, Mar 29 2013 *)

Vec(x*(1 + 8*x - 5*x^2 + 10*x^3 + 4*x^4 - x^5 + x^6) / ((1 - x)^4*(1 + x + x^2)^2) + O(x^40)) \\ Colin Barker, Aug 03 2020

a(n) = ( n(n+1) + 6 - 8*sin^2(Pi*(n+1)/3) )*(2n+1)/6.
a(n) = Sum_{k=0..n} A080782(k)^2.
From Colin Barker, Aug 03 2020: (Start)
G.f.: x*(1 + 8*x - 5*x^2 + 10*x^3 + 4*x^4 - x^5 + x^6) / ((1 - x)^4*(1 + x + x^2)^2).
a(n) = 2*a(n-1) - a(n-2) + 2*a(n-3) - 4*a(n-4) + 2*a(n-5) - a(n-6) + 2*a(n-7) - a(n-8) for n>8.
(End)

A064429 a(n) = floor(n / 3) * 3 + sign(n mod 3) * (3 - n mod 3).

Original entry on oeis.org

0, 2, 1, 3, 5, 4, 6, 8, 7, 9, 11, 10, 12, 14, 13, 15, 17, 16, 18, 20, 19, 21, 23, 22, 24, 26, 25, 27, 29, 28, 30, 32, 31, 33, 35, 34, 36, 38, 37, 39, 41, 40, 42, 44, 43, 45, 47, 46, 48, 50, 49, 51, 53, 52, 54, 56, 55, 57, 59, 58, 60, 62, 61, 63, 65, 64, 66, 68, 67, 69, 71, 70

Offset: 0

Reinhard Zumkeller, Oct 15 2001

a(a(n)) = n (a self-inverse permutation).
Take natural numbers, exchange trisections starting with 1 and 2.
Lodumo_3 of A080425. - Philippe Deléham, Apr 26 2009
From Franck Maminirina Ramaharo, Jul 27 2018: (Start)
The sequence is A008585 interleaved with A016789 and A016777.
a(n) is also obtained as follows: write n in base 3; if the rightmost digit is '1', then replace it with '2' and vice versa; convert back to decimal. For example a(14) = a('11'2') = '11'1' = 13 and a(10) = a('10'1') = '10'2' = 11. (End)
A permutation of the nonnegative integers partitioned into triples [3*k-3, 3*k-1, 3*k-2] for k > 0. - Guenther Schrack, Feb 05 2020

			From _Franck Maminirina Ramaharo_, Jul 27 2018: (Start)
Interleave 3 sequences:
A008585: 0.....3.....6.....9.......12.......15........
A016789: ..2.....5.....8.....11.......14.......17.....
A016777: ....1.....4.....7......10.......13.......16..
(End)

Muniru A Asiru, Table of n, a(n) for n = 0..3000
Eric Weisstein's World of Mathematics, Alternating Permutations.
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. A001477, A004442, A074066, A080425, A080782, A102283, A143097.

a:=[0,2,1,3];; for n in [5..100] do a[n]:=a[n-1]+a[n-3]-a[n-4]; od; a; # Muniru A Asiru, Jul 27 2018

[2*n - 3 - 3*((n-2) div 3): n in [0..80]]; // Vincenzo Librandi, Aug 05 2018

A064429:=n->2*n-3-3*floor((n-2)/3): seq(A064429(n), n=0..100); # Wesley Ivan Hurt, Nov 30 2013

Table[2 n - 3 - 3 Floor[(n - 2)/3], {n, 0, 100}] (* Wesley Ivan Hurt, Nov 30 2013 *)
{#+1,#-1,#}[[Mod[#,3,1]]]&/@Range[0, 100] (* Federico Provvedi, May 11 2021 *)
LinearRecurrence[{1,0,1,-1},{0,2,1,3},80] (* or *) {#[[1]],#[[3]],#[[2]]}&/@Partition[Range[0,80],3]//Flatten (* Harvey P. Dale, Mar 28 2025 *)

a(n) = 2*n-3-3*((n-2)\3); \\ Altug Alkan, Oct 06 2017

a(n) = A080782(n+1) - 1.
a(n) = n - 2*sin(4*Pi*n/3)/sqrt(3). - Jaume Oliver Lafont, Dec 05 2008
a(n) = A001477(n) + A102283(n). - Jaume Oliver Lafont, Dec 05 2008
a(n) = lod_3(A080425(n)). - Philippe Deléham, Apr 26 2009
G.f.: x*(2 - x + 2*x^2)/((1 + x + x^2)*(1 - x)^2 ). - R. J. Mathar, Feb 20 2011
a(n) = 2*n - 3 - 3*floor((n-2)/3). - Wesley Ivan Hurt, Nov 30 2013
a(n) = a(n-1) + a(n-3) - a(n-4) for n > 3. - Wesley Ivan Hurt, Oct 06 2017
E.g.f.: x*exp(x) + (2*sin((sqrt(3)*x)/2))/(exp(x/2)*sqrt(3)). - Franck Maminirina Ramaharo, Jul 27 2018
From Guenther Schrack, Feb 05 2020: (Start)
a(n) = a(n-3) + 3 with a(0)=0, a(1)=2, a(2)=1 for n > 2;
a(n) = n + (w^(2*n) - w^n)*(1 + 2*w)/3 where w = (-1 + sqrt(-3))/2. (End)
Sum_{n>=1} (-1)^n/a(n) = log(2)/3. - Amiram Eldar, Jan 31 2023

A092486 Take natural numbers, exchange first and third quadrisection.

Original entry on oeis.org

3, 2, 1, 4, 7, 6, 5, 8, 11, 10, 9, 12, 15, 14, 13, 16, 19, 18, 17, 20, 23, 22, 21, 24, 27, 26, 25, 28, 31, 30, 29, 32, 35, 34, 33, 36, 39, 38, 37, 40, 43, 42, 41, 44, 47, 46, 45, 48, 51, 50, 49, 52, 55, 54, 53, 56, 59, 58, 57, 60, 63, 62, 61, 64, 67, 66, 65, 68, 71, 70, 69, 72

Offset: 0

Ralf Stephan, Apr 04 2004

Harry J. Smith, Table of n, a(n) for n = 0..20003
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).
Index entries for sequences that are permutations of the natural numbers.

Cf. A002162, A080412, A064429, A074066, A080782, A159966.

Flatten[Partition[Range[80],4]/.{a_,b_,c_,d_}->{c,b,a,d}] (* Harvey P. Dale, Aug 12 2012 *)

{ f="b092486.txt"; for (n=0, 5000, a0=4*n + 3; a1=a0 - 1; a2=a1 - 1; a3=a0 + 1; write(f, 4*n, " ", a0); write(f, 4*n+1, " ", a1); write(f, 4*n+2, " ", a2); write(f, 4*n+3, " ", a3); ); } \\ Harry J. Smith, Jun 21 2009

G.f.: (3-4*x+3*x^2)/((1+x^2)*(1-x)^2).
a(4n) = 4n+3, a(4n+1) = 4n+2, a(4n+2) = 4n+1, a(4n+3) = 4n+4.
a(n) = n+1+i^n+(-i)^n, where i is the imaginary unit. - Bruno Berselli, Feb 08 2011
From Wesley Ivan Hurt, May 09 2021: (Start)
a(n) = 2*a(n-1)-2*a(n-2)+2*a(n-3)-a(n-4).
a(n) = 1 + n + 2*cos(n*Pi/2). (End)
Sum_{n>=0} (-1)^n/a(n) = log(2) (A002162). - Amiram Eldar, Nov 28 2023

A079357 a(1)=1; a(n)=a(n-1)-1 if n is already in the sequence, a(n)=a(n-1)+4 otherwise.

Original entry on oeis.org

1, 5, 9, 13, 12, 16, 20, 24, 23, 27, 31, 30, 29, 33, 37, 36, 40, 44, 48, 47, 51, 55, 54, 53, 57, 61, 60, 64, 63, 62, 61, 65, 64, 68, 72, 71, 70, 74, 78, 77, 81, 85, 89, 88, 92, 96, 95, 94, 98, 102, 101, 105, 104, 103, 102, 106, 105, 109, 113, 112, 111, 110, 109, 108, 107

Offset: 1

Benoit Cloitre, Feb 14 2003

If, in the defining recurrence, the rule a(n)=a(n-1)+4 when n is not already in the sequence is generalized to a(n)=a(n-1)+k, then the resulting sequence ultimately becomes periodic with period 1,3,10,35 for k=1,2,3,4, respectively. - John W. Layman, Apr 15 2003

Cf. A080782, A079354, A079355, A080783, A064437.

It appears that, for n >= 219, a(n)=n+b(n) where b(n) is the period-35 sequence (-1, 2, 5, 3, 6, 9, 7, 5, 8, 11, 9, 12, 10, 8, 6, 9, 7, 10, 13, 11, 9, 7, 5, 3, 1, -1, -3, -5, -7, -9, -11, -13, -10, -7, -4).

More terms from John W. Layman, Apr 15 2003

A375336 For n>=4, irregular triangular array of successive integer solutions to sqrt((d-c)b^2 + c(b+1)^2) for square integers d = n^2, where b and c are positive integers and c < d, read by rows.

Original entry on oeis.org

5, 7, 7, 8, 9, 13, 17, 27, 8, 10, 11, 13, 16, 19, 10, 11, 13, 14, 19, 21, 25, 31, 59, 61, 12, 15, 22, 23, 29, 34, 39, 42, 11, 13, 14, 16, 17, 19, 25, 33, 37, 41, 49, 103, 107, 125, 13, 14, 16, 17, 19, 20, 23, 27, 28, 32, 37, 40, 46, 53, 82, 83, 15, 18, 21, 26

Offset: 4

Charles L. Hohn, Aug 12 2024

Provable that every row n has a finite number of terms, with n < 4 producing no solutions, and T(n, k) never exceeding (n/2)^3.
This sequence excludes cases where c == 0, where all b produce integer solutions d*b.
Nonsquare d produce solutions following a different pattern, shown as A374602.

			4: {5, 7}
5: {7, 8}
6: {9, 13, 17, 27}
7: {8, 10, 11, 13, 16, 19}
8: {10, 11, 13, 14, 19, 21, 25, 31, 59, 61}
9: {12, 15, 22, 23, 29, 34, 39, 42}
10: {11, 13, 14, 16, 17, 19, 25, 33, 37, 41, 49, 103, 107, 125}
11: {13, 14, 16, 17, 19, 20, 23, 27, 28, 32, 37, 40, 46, 53, 82, 83}
12: {15, 18, 21, 26, 29, 31, 34, 41, 43, 51, 54, 57, 61, 71, 159, 165, 209, 211}
...
sqrt((2^2-1)*1^2 + 1*(1+1)^2) = sqrt(7) -> not an integer so not included.
sqrt((4^2-1)*1^2 + 1*(1+1)^2) = sqrt(19) -> not an integer so not included.
sqrt((4^2-3)*1^2 + 3*(1+1)^2) = 5 -> T(4,1).
sqrt((4^2-11)*1^2 + 11*(1+1)^2) = 7 -> T(4,2).
sqrt((5^2-8)*1^2 + 8*(1+1)^2) = 7 -> T(5,1).
sqrt((6^2-5)*2^2 + 5*(2+1)^2) = 13 -> T(6,2).

Cf. A080782, A374602.

row(n)=my(d=n^2, t=n, v=List()); while(t

T(n, 1) = A080782(n+2).

A324080 a(n) is the rarest values appearing recorded in n-th pair in A324078.

Original entry on oeis.org

1, 1, 3, 3, 2, 2, 3, 2, 4, 4, 4, 4, 2, 6, 6, 6, 6, 6, 5, 5, 5, 5, 5, 6, 5, 7, 7, 7, 7, 7, 7, 7, 3, 9, 9, 9, 9, 9, 9, 9, 9, 8, 8, 8, 8, 8, 8, 8, 8, 9, 8, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 4, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 11, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 5

Offset: 1

Joerg Arndt, Feb 15 2019

nonn

Dropping all but the first occurrence of the terms appears to give A080782.

Joerg Arndt, Table of n, a(n) for n = 1..5000

See A324081 for the multiplicities.

a(n) = A324078(2*n-1).

a(n) = A324079(2*n).

A324081 a(n) is the multiplicity of the rarest values appearing recorded in n-th pair in A324078.

Original entry on oeis.org

1, 3, 1, 2, 1, 2, 3, 4, 1, 2, 3, 4, 6, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 9, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 1, 2, 3, 4, 5

Offset: 1

Joerg Arndt, Feb 15 2019

nonn

Dropping all but the first occurrence of the terms appears to give A080782.

Joerg Arndt, Table of n, a(n) for n = 1..5000

See A324080 for the corresponding rarest values.

a(n) = A324078(2*n).

a(n) = A324079(2*n-1).

A080783 a(1)=1, a(n)=a(n-1)-1 if n is already in the sequence, a(n)=a(n-1)+5 otherwise.

Original entry on oeis.org

1, 6, 11, 16, 21, 20, 25, 30, 35, 40, 39, 44, 49, 54, 59, 58, 63, 68, 73, 72, 71, 76, 81, 86, 85, 90, 95, 100, 105, 104, 109, 114, 119, 124, 123, 128, 133, 138, 137, 136, 141, 146, 151, 150, 155, 160, 165, 170, 169, 174, 179, 184, 189, 188, 193, 198

Offset: 1

Benoit Cloitre, Mar 07 2003

nonn

Cf. A079357, A079354, A080782, A064437.

lst={1};i=2;Do[If[MemberQ[lst,i],AppendTo[lst,Last[lst]-1], AppendTo[ lst,Last[lst]+5]];i++,{60}];lst (* Harvey P. Dale, Aug 20 2011 *)

Conjectured to be asymptotic to 3n as n -> infinity.

cp's OEIS Frontend

A164765 Partial sums of [A080782^2].

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A064429 a(n) = floor(n / 3) * 3 + sign(n mod 3) * (3 - n mod 3).

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A092486 Take natural numbers, exchange first and third quadrisection.

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A079357 a(1)=1; a(n)=a(n-1)-1 if n is already in the sequence, a(n)=a(n-1)+4 otherwise.

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A375336 For n>=4, irregular triangular array of successive integer solutions to sqrt((d-c)*b^2 + c*(b+1)^2) for square integers d = n^2, where b and c are positive integers and c < d, read by rows.

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A324080 a(n) is the rarest values appearing recorded in n-th pair in A324078.

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A324081 a(n) is the multiplicity of the rarest values appearing recorded in n-th pair in A324078.

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A080783 a(1)=1, a(n)=a(n-1)-1 if n is already in the sequence, a(n)=a(n-1)+5 otherwise.

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A375336 For n>=4, irregular triangular array of successive integer solutions to sqrt((d-c)b^2 + c(b+1)^2) for square integers d = n^2, where b and c are positive integers and c < d, read by rows.