A005652 -id:A005652 - cp's OEIS frontend

A190436 a(n) = [(bn+c)r] - b[nr] - [c*r], where (r,b,c)=(golden ratio,3,2) and []=floor.

2, 0, 2, 1, 0, 2, 1, 3, 1, 0, 2, 1, 0, 2, 1, 2, 1, 0, 2, 1, 3, 2, 0, 2, 1, 0, 2, 1, 3, 1, 0, 2, 1, 0, 2, 0, 2, 1, 0, 2, 1, 3, 1, 0, 2, 1, 0, 2, 1, 2, 1, 0, 2, 1, 3, 2, 0, 2, 1, 0, 2, 1, 3, 1, 0, 2, 1, 0, 2, 1, 2, 1, 0, 2, 1, 3, 2, 0, 2, 1, 0, 2, 1, 2, 1, 0, 2, 1, 0, 2, 0, 2, 1, 0, 2, 1, 3, 1, 0, 2, 1, 0, 2, 1, 2, 1, 0, 2, 1, 3, 2, 0, 2, 1, 0

Offset: 1

Clark Kimberling, May 10 2011

nonn

Write a(n)=[(bn+c)r]-b[nr]-[cr].  If r>0 and b and c are integers satisfying b>=2 and 0<=c<=b-1, then 0<=a(n)<=b.  The positions of 0 in the sequence a are of interest, as are the position sequences for 1,2,...,b.  These b+1 position sequences comprise a partition of the positive integers.
Examples:
(golden ratio,2,0):  A078588, A005653, A005652
(golden ratio,2,1):  A190427 - A190430
(golden ratio,3,0):  A140397 - A190400
(golden ratio,3,1):  A140431 - A190435
(golden ratio,3,2):  A140436 - A190439
(golden ratio,4,c):  A140440 - A190461

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A190437, A190438, A190439, A190440.

r = GoldenRatio; b = 3; c = 2;
f[n_] := Floor[(b*n + c)*r] - b*Floor[n*r] - Floor[c*r];
t = Table[f[n], {n, 1, 320}]
Flatten[Position[t, 0]] (* A190437 *)
Flatten[Position[t, 1]] (* A190438 *)
Flatten[Position[t, 2]] (* A190439 *)
Flatten[Position[t, 3]] (* A302253 *)

A190445 [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(golden ratio,4,1) and []=floor.

Original entry on oeis.org

3, 1, 4, 2, 0, 3, 1, 4, 2, 1, 3, 2, 0, 3, 1, 4, 2, 1, 3, 2, 4, 3, 1, 3, 2, 0, 3, 1, 4, 2, 1, 3, 2, 0, 3, 1, 4, 2, 1, 3, 1, 4, 2, 1, 3, 2, 0, 3, 1, 4, 2, 1, 3, 2, 4, 3, 1, 4, 2, 0, 3, 1, 4, 2, 1, 3, 2, 0, 3, 1, 4, 2, 1, 3, 2, 4, 2, 1, 3, 2, 0, 3, 1, 4, 2, 1, 3, 2, 0, 3, 1, 4, 2, 0, 3, 1, 4, 2, 1, 3, 2, 0, 3, 1, 4, 2, 1, 3, 2, 4, 3, 1, 3, 2, 0, 3, 1, 4, 2, 1, 3, 2, 0, 3, 1, 4, 2, 1, 3, 1, 4, 2, 1, 3, 2, 0

Offset: 1

Clark Kimberling, May 10 2011

nonn

Write a(n)=[(bn+c)r]-b[nr]-[cr].  If r>0 and b and c are integers satisfying b>=2 and 0<=c<=b-1, then 0<=a(n)<=b.  The positions of 0 in the sequence a are of interest, as are the position sequences for 1,2,...,b.  These b+1 position sequences comprise a partition of the positive integers.
Examples:
(golden ratio,2,0):  A078588, A005653, A005652
(golden ratio,2,1):  A190427-A190430
(golden ratio,3,0):  A140397-A190400
(golden ratio,3,1):  A140431-A190435
(golden ratio,3,2):  A140436-A190439
(golden ratio,4,c):  A190440-A190461

Cf. A190446-A190450, A190427.

r = GoldenRatio; b = 4; c = 1;
f[n_] := Floor[(b*n + c)*r] - b*Floor[n*r] - Floor[c*r];
t = Table[f[n], {n, 1, 320}]
Flatten[Position[t, 0]]
Flatten[Position[t, 1]]
Flatten[Position[t, 2]]
Flatten[Position[t, 3]]
Flatten[Position[t, 4]]

A190457 a(n) = [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(golden ratio,4,3) and []=floor.

Original entry on oeis.org

3, 1, 4, 2, 1, 3, 2, 4, 3, 1, 4, 2, 0, 3, 1, 4, 2, 1, 3, 2, 4, 3, 1, 4, 2, 1, 3, 2, 4, 3, 1, 3, 2, 0, 3, 1, 4, 2, 1, 3, 2, 4, 3, 1, 4, 2, 1, 3, 1, 4, 2, 1, 3, 2, 4, 3, 1, 4, 2, 1, 3, 2, 4, 3, 1, 4, 2, 0, 3, 1, 4, 2, 1, 3, 2, 4, 3, 1, 4, 2, 1, 3, 2, 4, 2, 1, 3, 2, 0, 3, 1, 4, 2, 1, 3, 2, 4, 3, 1, 4, 2, 1, 3, 1, 4, 2, 1, 3, 2, 4, 3, 1, 4, 2, 1, 3, 2, 4, 3, 1, 3, 2, 0, 3, 1, 4, 2, 1, 3, 2, 4, 3

Offset: 1

Clark Kimberling, May 10 2011

nonn

Write a(n)=[(bn+c)r]-b[nr]-[cr].  If r>0 and b and c are integers satisfying b>=2 and 0<=c<=b-1, then 0<=a(n)<=b.  The positions of 0 in the sequence a are of interest, as are the position sequences for 1,2,...,b.  These b+1 position sequences comprise a partition of the positive integers.
Examples:
(golden ratio,2,0):  A078588, A005653, A005652
(golden ratio,2,1):  A190427-A190430
(golden ratio,3,0):  A140397-A190400
(golden ratio,3,1):  A140431-A190435
(golden ratio,3,2):  A140436-A190439
(golden ratio,4,c):  A190440-A190461

Jon Asier Bárcena-Petisco, Luis Martínez, María Merino, Juan Manuel Montoya, and Antonio Vera-López, Fibonacci-like partitions and their associated piecewise-defined permutations, arXiv:2503.19696 [math.CO], 2025. See p. 4.

Cf. A190458-A190461 and A190463.

r = GoldenRatio; b = 4; c = 3;
f[n_] := Floor[(b*n + c)*r] - b*Floor[n*r] - Floor[c*r];
t = Table[f[n], {n, 1, 320}]
Flatten[Position[t, 0]]
Flatten[Position[t, 1]]
Flatten[Position[t, 2]]
Flatten[Position[t, 3]]
Flatten[Position[t, 4]]

A191331 Positions of 2 in A191329.

Original entry on oeis.org

2, 6, 12, 16, 18, 22, 28, 32, 34, 38, 42, 44, 48, 54, 58, 60, 64, 70, 74, 80, 84, 86, 90, 96, 100, 102, 106, 110, 112, 116, 122, 126, 128, 132, 138, 142, 148, 152, 154, 158, 164, 168, 170, 174, 180, 184, 190, 194, 196, 200, 206, 210, 212, 216, 220, 222, 226, 232, 236, 238, 242, 248, 252, 258, 262, 264, 268, 274, 278, 280, 284, 288

Offset: 1

Clark Kimberling, May 31 2011

nonn

A191331=2*A005652.

			A191329=(1,2,1,0,1,2,1,0,1,0,1,2,...), so that
a(1)=2, a(2)=6, a(3)=12,...

Cf. A191329, A191330, A191332, A191333.

Mathematica
```
(See A191329.)
```

A302253 Positions of 3 in A190436.

Original entry on oeis.org

8, 21, 29, 42, 55, 63, 76, 97, 110, 118, 131, 144, 152, 165, 186, 199, 207, 220, 241, 254, 262, 275, 288, 296, 309, 330, 343, 351, 364, 377, 385, 398, 406, 419, 432, 440, 453, 474, 487, 495, 508, 521, 529, 542, 563, 576, 584, 597, 618, 631, 639, 652, 665, 673, 686, 707, 720, 728

Offset: 1

G. C. Greubel, Apr 04 2018

nonn

Write a(n) = [(bn+c)r] - b[nr] - [cr]. If r>0 and b and c are integers satisfying b >= 2 and 0 <= c <= b-1, then 0 <= a(n) <= b.  The positions of 0 in the sequence a are of interest, as are the position sequences for 1,2,...,b.  These b+1 position sequences comprise a partition of the positive integers.
Examples:
(golden ratio,2,0):  A078588, A005653, A005652
(golden ratio,2,1):  A190427-A190430
(golden ratio,3,0):  A140397-A190400
(golden ratio,3,1):  A140431-A190435
(golden ratio,3,2):  A140436-A190439
(golden ratio,4,c):  A140440-A190461

G. C. Greubel, Table of n, a(n) for n = 1..20000

Cf. A190436, A190437, A190438, A190439.

r = GoldenRatio; b = 3; c = 2;
f[n_] := Floor[(b*n + c)*r] - b*Floor[n*r] - Floor[c*r];
t = Table[f[n], {n, 1, 500}] (* A190436 *)
Flatten[Position[t, 0]] (* A190437 *)
Flatten[Position[t, 1]] (* A190438 *)
Flatten[Position[t, 2]] (* A190439 *)
Flatten[Position[t, 3]] (* A302253 *)

A190451 [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(golden ratio,4,2) and []=floor.

Original entry on oeis.org

2, 1, 3, 2, 0, 3, 1, 4, 2, 0, 3, 1, 0, 2, 1, 3, 2, 0, 3, 1, 4, 2, 1, 3, 2, 0, 2, 1, 3, 2, 0, 3, 1, 0, 2, 1, 3, 2, 0, 3, 1, 4, 2, 1, 3, 1, 0, 2, 1, 3, 2, 0, 3, 1, 4, 2, 1, 3, 2, 0, 3, 1, 3, 2, 0, 3, 1, 0, 2, 1, 3, 2, 0, 3, 1, 4, 2, 1, 3, 2, 0, 2, 1, 3, 2, 0, 3, 1, 0, 2, 1, 3, 2, 0, 3, 1, 4, 2, 0, 3, 1, 0, 2, 1, 3, 2, 0, 3, 1, 4, 2, 1, 3, 2, 0, 3, 1, 3, 2, 0, 3, 1, 0

Offset: 1

Clark Kimberling, May 10 2011

nonn

Write a(n)=[(bn+c)r]-b[nr]-[cr].  If r>0 and b and c are integers satisfying b>=2 and 0<=c<=b-1, then 0<=a(n)<=b.  The positions of 0 in the sequence a are of interest, as are the position sequences for 1,2,...,b.  These b+1 position sequences comprise a partition of the positive integers.
Examples:
(golden ratio,2,0):  A078588, A005653, A005652
(golden ratio,2,1):  A190427-A190430
(golden ratio,3,0):  A140397-A190400
(golden ratio,3,1):  A140431-A190435
(golden ratio,3,2):  A140436-A190439

Cf. A190428, A190453-A190455.

r = GoldenRatio; b = 4; c = 2;
f[n_] := Floor[(b*n + c)*r] - b*Floor[n*r] - Floor[c*r];
t = Table[f[n], {n, 1, 320}]
Flatten[Position[t, 0]]
Flatten[Position[t, 1]]
Flatten[Position[t, 2]]
Flatten[Position[t, 3]]
Flatten[Position[t, 4]]

A156353 A symmetrical powers triangle sequence: t(n,m) = (m^(n - m) + (n - m)^m).

Original entry on oeis.org

2, 3, 3, 4, 8, 4, 5, 17, 17, 5, 6, 32, 54, 32, 6, 7, 57, 145, 145, 57, 7, 8, 100, 368, 512, 368, 100, 8, 9, 177, 945, 1649, 1649, 945, 177, 9, 10, 320, 2530, 5392, 6250, 5392, 2530, 320, 10, 11, 593, 7073, 18785, 23401, 23401, 18785, 7073, 593, 11, 12, 1124, 20412

Offset: 1

Roger L. Bagula, Feb 08 2009

Equivalently, table by antidiagonals of n^m + m^n for n,m > 0.
Row sums are:
{2, 6, 16, 44, 130, 418, 1464, 5560, 22754, 99726, 465536,...}.

			{2},
{3, 3},
{4, 8, 4},
{5, 17, 17, 5},
{6, 32, 54, 32, 6},
{7, 57, 145, 145, 57, 7},
{8, 100, 368, 512, 368, 100, 8},
{9, 177, 945, 1649, 1649, 945, 177, 9},
{10, 320, 2530, 5392, 6250, 5392, 2530, 320, 10},
{11, 593, 7073, 18785, 23401, 23401, 18785, 7073, 593, 11},
{12, 1124, 20412, 69632, 94932, 93312, 94932, 69632, 20412, 1124, 12}

Boris Putievskiy, Rows n = 1..77 of triangle, flattened

Cf. A005652 is the same table with row 0 and column 0 included.

Clear[t, n, m];
t[n_, m_] = (m^(n - m) + (n - m)^m);
Table[Table[t[n, m], {m, 1, n - 1}], {n, 2, 12}];
Flatten[%]

t=int((math.sqrt(8*n-7) - 1)/ 2)
m=((t*t+3*t+4)/2-n)**(n-t*(t+1)/2)+(n-t*(t+1)/2)**((t*t+3*t+4)/2-n)
# Boris Putievskiy, Dec 14 2012

t(n,m) = (m^(n - m) + (n - m)^m).
a(n) = A004736(n)^A002260(n) + A002260(n)^A004736(n) or
((t*t+3*t+4)/2-n)^(n-(t*(t+1)/2))+ (n-(t*(t+1)/2))^((t*t+3*t+4)/2-n), where t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Dec 14 2012

Edited by Franklin T. Adams-Watters, Oct 26 2009

A203988 a(1)=1 and, for n>1, a(n) is the smallest positive integer greater than a(n-1) such that a(n)+a(k) is not a square for k=1,2,...,n-1.

Original entry on oeis.org

1, 2, 4, 6, 9, 11, 13, 17, 18, 20, 22, 26, 28, 33, 37, 39, 41, 49, 50, 52, 54, 56, 57, 66, 73, 76, 81, 85, 86, 97, 100, 102, 106, 109, 114, 121, 129, 134, 137, 145, 148, 153, 161, 162, 164, 166, 172, 177, 181, 182, 191, 193, 196, 198, 201, 211, 220, 225, 226

Offset: 1

John W. Layman, Jan 09 2012

nonn

See A005652 for the case where the sum of two terms is never a Fibonacci number.

Matthew Conroy, Table of n, a(n) for n = 1..10000

Cf. A005652.

t = {1}; Do[k = t[[-1]] + 1; While[Length[Select[t, ! IntegerQ[Sqrt[# + k]] &]] < Length[t], k++]; AppendTo[t, k], {n, 2, 100}]; t (* T. D. Noe, Jan 10 2012 *)

seq(n)={my(a=vector(n)); a[1]=1; for(n=2, n, my(m=a[n-1], f=1); while(f, m++; f=0; for(k=1, n-1, f=issquare(a[k]+m); if(f, break))); a[n]=m); a} \\ Andrew Howroyd, Sep 19 2020

A279934 Positive integers k such that {(k-1)*r} > 1/2, where r = golden ratio = (1 + sqrt(5))/2 and { } = fractional part.

Original entry on oeis.org

2, 4, 7, 9, 10, 12, 15, 17, 18, 20, 22, 23, 25, 28, 30, 31, 33, 36, 38, 41, 43, 44, 46, 49, 51, 52, 54, 56, 57, 59, 62, 64, 65, 67, 70, 72, 75, 77, 78, 80, 83, 85, 86, 88, 91, 93, 96, 98, 99, 101, 104, 106, 107, 109, 111, 112, 114, 117, 119, 120

Offset: 1

Clark Kimberling, Dec 23 2016

Clark Kimberling, Table of n, a(n) for n = 1..5000

Cf. A005652, A279933 (complement).

r = GoldenRatio;
t = Table[If[FractionalPart[n r - r] < 1/2, 0, 1 ], {n, 1, 120}] (* {A078588(n-1)} *)
Flatten[Position[t, 0]]  (* A279933 *)
Flatten[Position[t, 1]]  (* A279934 *)

a(n) = 1 + A005652(n).

New name from Jianing Song, Sep 12 2019

A140401 Let S be the set of numbers formed from the sum of three distinct elements of A140398, or the sum of three distinct elements of A140399, or the sum of three distinct elements of A140400; sequence gives complement of S.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 18, 21, 23, 26, 29, 31, 34, 39, 42, 47, 55, 60, 68, 76, 81, 89, 102, 110, 123, 144, 157, 178, 199, 212, 233, 267, 288, 322, 377, 411, 466, 521, 555, 610, 699, 754, 843, 987

Offset: 1

Fred Lunnon, Jun 20 2008

Cf. A140398, A140399, A140400, A140397, A005652, A005653, A078588.

It appears that this consists of the following numbers: { F_{k}, F_{k} + F_{k-3}, F_{k} + F_{k-2}, F_{2k} + F_{2k-5}, F_{2k+1} - F_{2k-4}, F_{2k+1} + F_{2k-3} }, where F (A000045) are the Fibonacci numbers and k and other subscripts are restricted to positive values.

cp's OEIS Frontend

A190436 a(n) = [(b*n+c)*r] - b*[n*r] - [c*r], where (r,b,c)=(golden ratio,3,2) and []=floor.

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A190445 [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(golden ratio,4,1) and []=floor.

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A190457 a(n) = [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(golden ratio,4,3) and []=floor.

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A191331 Positions of 2 in A191329.

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A302253 Positions of 3 in A190436.

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A190451 [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(golden ratio,4,2) and []=floor.

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A156353 A symmetrical powers triangle sequence: t(n,m) = (m^(n - m) + (n - m)^m).

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A203988 a(1)=1 and, for n>1, a(n) is the smallest positive integer greater than a(n-1) such that a(n)+a(k) is not a square for k=1,2,...,n-1.

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A279934 Positive integers k such that {(k-1)*r} > 1/2, where r = golden ratio = (1 + sqrt(5))/2 and { } = fractional part.

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A190436 a(n) = [(bn+c)r] - b[nr] - [c*r], where (r,b,c)=(golden ratio,3,2) and []=floor.