A017137 -id:A017137 - cp's OEIS frontend

A098502 a(n) = 16*n - 4.

12, 28, 44, 60, 76, 92, 108, 124, 140, 156, 172, 188, 204, 220, 236, 252, 268, 284, 300, 316, 332, 348, 364, 380, 396, 412, 428, 444, 460, 476, 492, 508, 524, 540, 556, 572, 588, 604, 620, 636, 652, 668, 684, 700, 716, 732, 748, 764, 780, 796, 812, 828, 844

Offset: 1

Ralf Stephan, Sep 15 2004

For n > 3, the number of squares on the infinite 4-column chessboard at <= n knight moves from any fixed start point.

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1).
Index entries for sequences which agree for a long time but are different.

Cf. A004767, A008590, A017113, A017137, A017629, A098498, A158953.

[16*n - 4: n in [1..60]]; // Vincenzo Librandi, Jul 24 2011

Range[12, 1000, 16] (* Vladimir Joseph Stephan Orlovsky, May 31 2011 *)

a(n)=16*n-4 \\ Charles R Greathouse IV, Jul 10 2016

G.f.: 4*x*(3+x)/(1-x)^2. - Colin Barker, Jan 09 2012
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi + log(3 - 2*sqrt(2)))/(16*sqrt(2)). - Amiram Eldar, Sep 01 2024
From Elmo R. Oliveira, Apr 03 2025: (Start)
E.g.f.: 4*(exp(x)*(4*x - 1) + 1).
a(n) = 2*a(n-1) - a(n-2) for n > 2.
a(n) = 4*A004767(n-1) = 2*A017137(n-1) = A017113(2*n-1). (End)

A062316 Neither the sum or difference of 2 squares.

Original entry on oeis.org

6, 14, 22, 30, 38, 42, 46, 54, 62, 66, 70, 78, 86, 94, 102, 110, 114, 118, 126, 134, 138, 142, 150, 154, 158, 166, 174, 182, 186, 190, 198, 206, 210, 214, 222, 230, 238, 246, 254, 258, 262, 266, 270, 278, 282, 286, 294, 302, 310, 318, 322, 326, 330, 334, 342, 350, 354, 358

Offset: 1

Michel ten Voorde, Jul 05 2001

nonn

Elements of A022544 congruent to 2 (mod 4).
Union of numbers congruent to 6 mod 8 (A017137) with numbers of the form 2 * A084109(n). - Franklin T. Adams-Watters, Jan 21 2007
Explanation: odd numbers are equal to the difference between two successive squares and among even numbers, multiples of 4 are of the form (k+2)^2-k^2, thus odd numbers and multiples of 4 are not in the sequence. Conversely, a difference of 2 squares cannot equal 2 (mod 4), thus this sequence contains the integers of the form 4k+2 that are in A022544 (not the sum of two squares); among integers of form 4k+2, this sequence contains all the integers of the form 8n+6 (A017137) that are not the sum of 2 squares because they have at least one prime factor congruent to 3 (mod 4) to an odd power; it also contains integers of the form 8n+2 = 2(4n+1) with 4n+1 not the sum of two squares, which is sequence A084109. - Jean-Christophe Hervé, Oct 24 2015

			From _Jean-Christophe Hervé_, Oct 24 2015: (Start)
6, 14, 22, 30, 38, 46, ... are in the sequence because they equal 6 (mod 8).
42 = 2*3*7, 66 = 2*3*11, 114 = 2*7*11 are also in the sequence: of the form 2*(4n+1) with 4n+1 not the sum of 2 squares.
(End)

Jean-Christophe Hervé, Table of n, a(n) for n = 1..2507

Cf. A022544, A016825, union of A017137 and 2*A084109, complement of A263715.

Cf. A097271, A079299.

N:= 1000: # to get all terms <= N
S:= {seq(4*i+2,i=0..floor((N-2)/4))}
  minus {seq(seq(x^2 + y^2, y = x .. floor(sqrt(N-x^2)),2),x=1..floor(sqrt(N)))}:
sort(convert(S,list)); # Robert Israel, Oct 25 2015

Select[Range@ 360, SquaresR[2, #] == 0 && Mod[#, 4] == 2 &] (* Michael De Vlieger, Oct 26 2015, after Harvey P. Dale at A022544 *)

a(n) == 2 (mod 4). Subsequence of A016825 (non-differences of squares). All first differences are either 4 or 8, each of which occurs infinitely often. - David W. Wilson, Mar 09 2005

Lim_{n->inf} a(n)/n = 4.

More terms from David W. Wilson, Feb 11 2003

A156676 a(n) = 81n^2 - 44n + 6.

Original entry on oeis.org

6, 43, 242, 603, 1126, 1811, 2658, 3667, 4838, 6171, 7666, 9323, 11142, 13123, 15266, 17571, 20038, 22667, 25458, 28411, 31526, 34803, 38242, 41843, 45606, 49531, 53618, 57867, 62278, 66851, 71586, 76483, 81542, 86763, 92146, 97691, 103398, 109267, 115298, 121491

Offset: 0

Vincenzo Librandi, Feb 15 2009

The identity (6561*n^2 - 3564*n + 485)^2 - (81*n^2 - 44*n + 6)*(729*n - 198)^2 = 1 can be written as A156774(n)^2 - a(n)*A156772(n)^2 = 1 for n > 0.

For n >= 1, the continued fraction expansion of sqrt(a(n)) is [9n-3; {1, 1, 3, 1, 9n-4, 1, 3, 1, 1, 18n-6}]. - Magus K. Chu, Sep 13 2022

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000290, A017137, A017245, A156772, A156774.

Magma
```
[81*n^2 - 44*n + 6: n in [0..40] ];
```

A156676:=n->81*n^2-44*n+6: seq(A156676(n), n=0..100); # Wesley Ivan Hurt, Apr 26 2017

LinearRecurrence[{3,-3,1},{6,43,242},40]
Table[81n^2-44n+6,{n,0,40}] (* Harvey P. Dale, Oct 29 2019 *)

a(n)=81*n^2-44*n+6 \\ Charles R Greathouse IV, Dec 23 2011

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: (6 + 25*x + 131*x^2)/(1-x)^3.
a(n) = A000290(A017245(n-1)) - A017137(n-1). - Reinhard Zumkeller, Jul 13 2010
E.g.f.: (6 + 37*x + 81*x^2)*exp(x). - Elmo R. Oliveira, Oct 19 2024

Edited by Charles R Greathouse IV, Jul 25 2010

A225975 Square root of A226008(n).

Original entry on oeis.org

0, 2, 2, 6, 1, 10, 6, 14, 4, 18, 10, 22, 3, 26, 14, 30, 8, 34, 18, 38, 5, 42, 22, 46, 12, 50, 26, 54, 7, 58, 30, 62, 16, 66, 34, 70, 9, 74, 38, 78, 20, 82, 42, 86, 11, 90, 46, 94, 24, 98, 50, 102, 13, 106, 54, 110, 28, 114, 58

Offset: 0

Paul Curtz, May 22 2013

Repeated terms of A016825 are in the positions 1,2,3,6,5,10,... (A043547).
From Wolfdieter Lang, Dec 04 2013: (Start)
This sequence a(n), n>=1, appears in the formula 2*sin(2*Pi/n) = R(p(n), x) modulo C(a(n), x), with x = rho(a(n)) = 2*cos(Pi/a(n)), the R-polynomials given in A127672 and the minimal C-polynomials of rho given in A187360. This follows from the identity 2*sin(2*Pi/n) = 2*cos(Pi*p(n)/a(n)) with gcd(p(n), a(n)) = 1. For p(n) see a comment on A106609,
Because R is an integer polynomial it shows that 2*sin(2*Pi/n) is an integer in the algebraic number field Q(rho(a(n))) of degree delta(a(n)) (the degree of C(a(n), x)), with delta(k) = A055034(k). This degree is given in A093819. For the coefficients of 2*sin(2*Pi/n) in the power basis of Q(rho(a(n))) see A231189 . (End)

			For the first formula: a(0)=-1+1=0, a(1)=-3+5=2, a(2)=-1+3=2, a(3)=-1+7=6, a(4)=0+1=1.

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

Cf. A016825, A017089, A017137, A022998, A106609, A145979, A226008.

a[0]=0; a[n_] := Sqrt[Denominator[1/4 - 4/n^2]]; Table[a[n], {n, 0, 58}] (* Jean-François Alcover, May 30 2013 *)
LinearRecurrence[{0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,-1},{0,2,2,6,1,10,6,14,4,18,10,22,3,26,14,30},60] (* Harvey P. Dale, Nov 21 2019 *)

a(n) = A106609(n-4) + A106609(n+4) with A106609(-4)=-1, A106609(-3)=-3, A106609(-2)=-1, A106609(-1)=-1.
a(n) = 2*a(n-8) -a(n-16).
a(2n+1) = A016825(n), a(2n) = A145979(n-2) for n>1, a(0)=0, a(2)=2.
a(4n)   = A022998(n).
a(4n+1) = A017089(n).
a(4n+2) = A016825(n).
a(4n+3) = A017137(n).
G.f.: x*(2 +2*x +6*x^2 +x^3 +10*x^4 +6*x^5 +14*x^6 +4*x^7 +14*x^8 +6*x^9 +10*x^10 +x^11 +6*x^12 +2*x^13 +2*x^14)/((1-x)^2*(1+x)^2*(1+x^2)^2*(1+x^4)^2). [Bruno Berselli, May 23 2013]
From Wolfdieter Lang, Dec 04 2013: (Start)
a(n) = 2*n if n is odd; if n is even then a(n) is n if n/2 == 1, 3, 5, 7 (mod 8), it is n/2 if n/2 == 0, 4 (mod 8) and it is n/4 if n/2 == 2, 6 (mod 8). This leads to the given G.f..
With c(n) = A178182(n), n>=1, a(n) = c(n)/2 if c(n) is even and c(n) if c(n) is odd. This leads to the preceding formula. (End)

Edited by Bruno Berselli, May 24 2013

A017138 a(n) = (8*n+6)^2.

Original entry on oeis.org

36, 196, 484, 900, 1444, 2116, 2916, 3844, 4900, 6084, 7396, 8836, 10404, 12100, 13924, 15876, 17956, 20164, 22500, 24964, 27556, 30276, 33124, 36100, 39204, 42436, 45796, 49284, 52900, 56644, 60516, 64516, 68644, 72900, 77284, 81796, 86436, 91204, 96100, 101124

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A006752, A016838, A017137.

[(8*n+6)^2: n in [0..35]]; // Vincenzo Librandi, Jul 22 2011

Table[(8*n + 6)^2, {n, 0, 30}] (* Amiram Eldar, Apr 26 2023 *)

a(n)=(8*n+6)^2 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: ( -36-88*x-4*x^2 ) / (x-1)^3. - R. J. Mathar, Aug 26 2015
From Amiram Eldar, Apr 26 2023: (Start)
a(n) = A017137(n)^2.
a(n) = 2^2*A016838(n).
Sum_{n>=0} 1/a(n) = Pi^2/64 - G/8, where G is Catalan's constant (A006752). (End)
E.g.f.: 4*exp(x)*(9 + 40*x + 16*x^2). - Stefano Spezia, Sep 02 2025

A177731 Numbers which can be written as a sum of consecutive numbers, where the largest term in the sum is an odd number >= 3.

Original entry on oeis.org

5, 6, 9, 12, 13, 14, 15, 17, 18, 21, 22, 24, 25, 27, 28, 29, 30, 33, 35, 36, 37, 38, 39, 41, 42, 44, 45, 46, 48, 49, 51, 53, 54, 55, 56, 57, 60, 61, 62, 63, 65, 66, 69, 70, 72, 73, 75, 76, 77, 78, 81, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 101, 102

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 12 2010

nonn

Numbers of the form sum_{i=j..2k+1} i where j>=1 and 2k+1>j and k>=1. Numbers of the form (2k+1+j)*(2k+2-j)/2, j>=1, k>=1, 2k+1>j. - R. J. Mathar, Dec 04 2011
Subsequences include the A000384 where >=6, the A014106 where >=5, A071355 where >=12, A130861 where >=9, A139577 where >=13, A139579 where >=17 etc. The sequence is the union of all odd-indexed rows of A141419, except its first column and numbers <=3: {5,6}, {9,12,14,15}, {13,18,22,25,27,28}, ... - R. J. Mathar, Dec 04 2011
Does this sequence have asymptotic density 1? - Robert Israel, Nov 27 2018

			5=2+3, 6=1+2+3, 9=4+5, 12=3+4+5,...

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A177712, A136724, A177713.
Cf. A000384, A014106, A071355, A130861, A139577, A139579, A141419.
Contains A004766, A017137 and nonzero terms of A008588.
Disjoint from A002145.
Subsequence of A138591.

f:= proc(n) local r,k;
  for r in select(t -> (2*t-1)^2 >= 1+8*n, numtheory:-divisors(2*n) minus {2*n}) do
    k:= (r + 2*n/r - 3)/4;
    if k::posint and r >= 2*k+2 then return true fi
  od:
  false
end proc:
select(f, [$1..1000]); # Robert Israel, Nov 27 2018

z=200;lst1={};Do[c=a;Do[c+=b;If[c<=2*z,AppendTo[lst1,c]],{b,a-1,1,-1}],{a,1,z,2}];Union@lst1

A189785 a(n) = n+floor(n*r/s)+floor(nt/s); r=Pi/2, s=arcsin(5/13), t=arcsin(12/13).

Original entry on oeis.org

6, 14, 22, 30, 38, 46, 54, 62, 70, 78, 86, 94, 102, 110, 118, 126, 134, 142, 150, 158, 166, 174, 182, 190, 198, 206, 214, 222, 230, 238, 246, 254, 262, 270, 278, 286, 294, 302, 310, 318, 326, 334, 342, 350, 358, 366, 374, 380, 388, 396, 404, 412, 420, 428, 436, 444, 452, 460, 468, 476, 484, 492, 500, 508, 516, 524, 532, 540, 548

Offset: 1

Clark Kimberling, Apr 27 2011

nonn

This is one of three sequences that partition the positive integers.  In general, suppose that r, s, t are positive real numbers for which the sets {i/r: i>=1}, {j/s: j>=1}, {k/t: k>=1} are pairwise disjoint.  Let a(n) be the rank of n/r when all the numbers in the three sets are jointly ranked.  Define b(n) and c(n) as the ranks of n/s and n/t.  It is easy to prove that
a(n)=n+[ns/r]+[nt/r],
b(n)=n+[nr/s]+[nt/s],
c(n)=n+[nr/t]+[ns/t], where []=floor.
Taking r=Pi/2, s=arcsin(5/13), t=arcsin(12/13) gives
a=A005408, b=A189785, c=A189786.  Note that r=s+t.
a(n) first differs from A017137(n-1) at n=48 (a(48)=380 but A017137(47)=382). - Nathaniel Johnston, May 16 2011

Cf. A189786, A189787, A004773.

r=Pi/2; s=ArcSin[5/13]; t=ArcSin[12/13];
a[n_] := n + Floor[n*s/r] + Floor[n*t/r];
b[n_] := n + Floor[n*r/s] + Floor[n*t/s];
c[n_] := n + Floor[n*r/t] + Floor[n*s/t];
Table[a[n], {n, 1, 120}]  (*A005408*)
Table[b[n], {n, 1, 120}]  (*A189785*)
Table[c[n], {n, 1, 120}]  (*A189786*)
Table[b[n]/2, {n, 1, 120}]  (*A189787*)
Table[c[n]/2, {n, 1, 120}]  (*A004773*)

A017139 a(n) = (8*n + 6)^3.

Original entry on oeis.org

216, 2744, 10648, 27000, 54872, 97336, 157464, 238328, 343000, 474552, 636056, 830584, 1061208, 1331000, 1643032, 2000376, 2406104, 2863288, 3375000, 3944312, 4574296, 5268024, 6028568, 6859000, 7762392, 8741816, 9800344, 10941048, 12167000, 13481272, 14886936

Offset: 0

N. J. A. Sloane

4*n + 3 = (8*n + 6) / 2 is never a square, as 3 is not a quadratic residue modulo 4. Using this, we can show that each term has an even square part and an even squarefree part, neither part being a power of 2. (Less than 2% of integers have this property - see A339245.) - Peter Munn, Dec 14 2020

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Quadratic Residue.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

A000578, A016839, A017137 are used in a formula defining this sequence.
Subsequence of A339245.
Cf. A002117, A017138, A017140.

[(8*n+6)^3: n in [0..35]]; // Vincenzo Librandi, Jul 22 2011

Table[(8*n+6)^3,{n,0,5!}] (* Vladimir Joseph Stephan Orlovsky, Mar 17 2010 *)
LinearRecurrence[{4,-6,4,-1},{216,2744,10648,27000},30] (* Harvey P. Dale, Dec 11 2012 *)

From R. J. Mathar, Mar 22 2010: (Start)
G.f.: 8*(27 + 235*x + 121*x^2 + x^3)/(x-1)^4.
a(n) = 8*A016839(n). (End)
a(0)=216, a(1)=2744, a(2)=10648, a(3)=27000, a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Harvey P. Dale, Dec 11 2012
a(n) = A017137(n)^3 = A000578(A017137(n)). - Peter Munn, Dec 20 2020
Sum_{n>=0} 1/a(n) = 7*zeta(3)/128 - Pi^2/512. - Amiram Eldar, Apr 26 2023

More terms from Vladimir Joseph Stephan Orlovsky, Mar 17 2010

A047595 Numbers that are congruent to {0, 1, 2, 3, 4, 5, 7} mod 8.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 39, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75

Offset: 1

N. J. A. Sloane

Complement of A017137. - Michel Marcus, Sep 15 2015

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

Cf. A017137 (8n+6).

[n-1+Floor(n/7) : n in [1..100]]; // Wesley Ivan Hurt, Sep 15 2015

I:=[0,1,2,3,4,5,7,8]; [n le 8 select I[n] else Self(n-1) + Self(n-7) - Self(n-8): n in [1..70]]; // Vincenzo Librandi, Sep 16 2015

A047595:=n->n-1+floor(n/7): seq(A047595(n), n=1..100); # Wesley Ivan Hurt, Sep 15 2015

Table[n - 1 + Floor[n/7], {n, 100}] (* Wesley Ivan Hurt, Sep 15 2015 *)
LinearRecurrence[{1, 0, 0, 0, 0, 0, 1, -1}, {0, 1, 2, 3, 4, 5, 7, 8}, 70] (* Vincenzo Librandi, Sep 16 2015 *)

vector(200, n, n-1+floor(n/7)) \\ Altug Alkan, Oct 23 2015

From Wesley Ivan Hurt, Sep 15 2015: (Start)
G.f.: x*(1+x+x^2+x^3+x^4+2*x^5+x^6)/((x-1)^2*(1+x+x^2+x^3+x^4+x^5+x^6)).
a(n) = a(n-1) + a(n-7) - a(n-8) for n>8.
a(n) = n - 1 + floor(n/7). (End)
From Wesley Ivan Hurt, Jul 21 2016: (Start)
a(n) = a(n-7) + 8 for n>7.
a(n) = (56*n - 70 - 6*(n mod 7) + ((n+1) mod 7) + ((n+2) mod 7) + ((n+3) mod 7) + ((n+4) mod 7) + ((n+5) mod 7) + ((n+6) mod 7))/49.
a(7*k) = 8*k-1, a(7*k-1) = 8*k-3, a(7*k-2) = 8*k-4, a(7*k-3) = 8*k-5, a(7*k-4) = 8*k-6, a(7*k-5) = 8*k-7, a(7*k-6) = 8*k-8. (End)

A092112 Where A092111 equals 2.

Original entry on oeis.org

14, 22, 26, 36, 38, 42, 54, 57, 62, 70, 78, 81, 90, 94, 110, 122, 132, 134, 138, 142, 147, 150, 158, 166, 168, 171, 172, 174, 178, 182, 190, 194, 198, 206, 210, 222, 238, 254, 285, 294, 312, 315, 318, 334, 336, 350, 366, 372, 382, 405, 414, 416, 432, 434, 446, 454

Offset: 1

Robert G. Wilson v, Feb 20 2004, corrected Nov 02 2006

nonn

Not as obvious as A092100, this sequence differs from multiples of 8 plus 6 (A017137).

Cf. A092111, A091935.

Run the second Mathematica line of A091938, then g[n_] := (n + 1 - Count[ IntegerDigits[f[n], 2], 1]); Select[ Range[100], g[ # ] == 2 &]

cp's OEIS Frontend

A098502 a(n) = 16*n - 4.

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A062316 Neither the sum or difference of 2 squares.

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A156676 a(n) = 81*n^2 - 44*n + 6.

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A225975 Square root of A226008(n).

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A017138 a(n) = (8*n+6)^2.

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A177731 Numbers which can be written as a sum of consecutive numbers, where the largest term in the sum is an odd number >= 3.

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A189785 a(n) = n+floor(n*r/s)+floor(nt/s); r=Pi/2, s=arcsin(5/13), t=arcsin(12/13).

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A156676 a(n) = 81n^2 - 44n + 6.