A007310 -id:A007310 - cp's OEIS frontend

A047235 Numbers that are congruent to {2, 4} mod 6.

2, 4, 8, 10, 14, 16, 20, 22, 26, 28, 32, 34, 38, 40, 44, 46, 50, 52, 56, 58, 62, 64, 68, 70, 74, 76, 80, 82, 86, 88, 92, 94, 98, 100, 104, 106, 110, 112, 116, 118, 122, 124, 128, 130, 134, 136, 140, 142, 146, 148, 152, 154, 158, 160, 164, 166, 170, 172, 176, 178, 182, 184, 188, 190, 194, 196, 200, 202, 206

Offset: 1

N. J. A. Sloane

Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0( 19 ).
Complement of A047273; A093719(a(n)) = 0. - Reinhard Zumkeller, Oct 01 2008
One could prefix an initial term "1" (or not) and define this sequence through a(n+1) = a(n) + (a(n) mod 6). See A001651 for the analog with 3, A235700 (with 5), A047350 (with 7), A007612 (with 9) and A102039 (with 10). Using 4 or 8 yields a constant sequence from that term on. - M. F. Hasler, Jan 14 2014
Nonnegative m such that m^2/6 + 1/3 is an integer. - Bruno Berselli, Apr 13 2017
Numbers divisible by 2 but not by 3. - David James Sycamore, Apr 04 2018
Numbers k for which A276086(k) is of the form 6m+3. - Antti Karttunen, Dec 03 2022

David A. Corneth, Table of n, a(n) for n = 1..10000
Chunhui Lai, A note on potentially K_4-e graphical sequences, arXiv:math/0308105 [math.CO], 2003.
William A. Stein, The modular forms database.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A020760, A020832, A093719, A047273 (complement), A120325 (characteristic function).
Equals 2*A001651.
Cf. A007310 ((6*n+(-1)^n-3)/2). - Bruno Berselli, Jun 24 2010
Positions of 3's in A053669 and in A358840.

[ n eq 1 select 2 else Self(n-1)+2*(1+n mod 2): n in [1..70] ]; // Klaus Brockhaus, Dec 13 2008

seq(6*floor((n+1)/2) + 3 + (-1)^n, n=1..67); # Gary Detlefs, Mar 02 2010

Flatten[Table[{6n - 4, 6n - 2}, {n, 40}]] (* Alonso del Arte, Oct 27 2014 *)

a(n)=(n-1)\2*6+3+(-1)^n \\ Charles R Greathouse IV, Jul 01 2013

first(n) = my(v = vector(n, i, 3*i - 1)); forstep(i = 2, n, 2, v[i]--); v \\ David A. Corneth, Oct 20 2017

a(n) = 2*A001651(n).
n such that phi(3*n) = phi(2*n). - Benoit Cloitre, Aug 06 2003
G.f.: 2*x*(1 + x + x^2)/((1 + x)*(1 - x)^2). a(n) = 3*n - 3/2 - (-1)^n/2. - R. J. Mathar, Nov 22 2008
a(n) = 3*n + 5..n odd, 3*n + 4..n even a(n) = 6*floor((n+1)/2) + 3 + (-1)^n. - Gary Detlefs, Mar 02 2010
a(n) = 6*n - a(n-1) - 6 (with a(1) = 2). - Vincenzo Librandi, Aug 05 2010
a(n+1) = a(n) + (a(n) mod 6). - M. F. Hasler, Jan 14 2014
Sum_{n>=1} 1/a(n)^2 = Pi^2/27. - Dimitris Valianatos, Oct 10 2017
a(n) = (6*n - (-1)^n - 3)/2. - Ammar Khatab, Aug 23 2020
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(6*sqrt(3)). - Amiram Eldar, Dec 11 2021
E.g.f.: 2 + ((6*x - 3)*exp(x) - exp(-x))/2. - David Lovler, Aug 25 2022
From Amiram Eldar, Nov 22 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = 2/sqrt(3) (10 * A020832).
Product_{n>=1} (1 + (-1)^n/a(n)) = 1/sqrt(3) (A020760). (End)

A065331 Largest 3-smooth divisor of n.

Original entry on oeis.org

1, 2, 3, 4, 1, 6, 1, 8, 9, 2, 1, 12, 1, 2, 3, 16, 1, 18, 1, 4, 3, 2, 1, 24, 1, 2, 27, 4, 1, 6, 1, 32, 3, 2, 1, 36, 1, 2, 3, 8, 1, 6, 1, 4, 9, 2, 1, 48, 1, 2, 3, 4, 1, 54, 1, 8, 3, 2, 1, 12, 1, 2, 9, 64, 1, 6, 1, 4, 3, 2, 1, 72, 1, 2, 3, 4, 1, 6, 1, 16, 81, 2, 1, 12, 1, 2, 3, 8, 1, 18, 1, 4, 3, 2, 1, 96

Offset: 1

Reinhard Zumkeller, Oct 29 2001

Bennett, Filaseta, & Trifonov show that if n > 8 then a(n^2 + n) < n^0.715. - Charles R Greathouse IV, May 21 2014

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
M. A. Bennett, M. Filaseta, and O. Trifonov, On the factorization of consecutive integers, J. Reine Angew. Math. 629 (2009), pp. 171-200.
Jon Maiga, Computer-generated formulas for A065331, Sequence Machine.

Cf. A003586, A065330, A006519, A038500, A007814, A007949, A007310, A047229, A071521, A322026.
Related to A053165 via A225546.
Cf. A126760 (ordinal transform of this sequence, from its term a(1) = 1 onward).

a065331 = f 2 1 where
   f p y x | r == 0    = f p (y * p) x'
           | otherwise = if p == 2 then f 3 y x else y
           where (x', r) = divMod x p
-- Reinhard Zumkeller, Nov 19 2015

[Gcd(n,6^n): n in [1..100]]; // Vincenzo Librandi, Feb 09 2016

A065331 := proc(n) n/A065330(n) ; end: # R. J. Mathar, Jun 24 2009
seq(2^padic:-ordp(n,2)*3^padic:-ordp(n,3), n=1..100); # Robert Israel, Feb 08 2016

Table[GCD[n, 6^n], {n, 100}] (* Vincenzo Librandi, Feb 09 2016 *)
a[n_] := Times @@ ({2, 3}^IntegerExponent[n, {2, 3}]); Array[a, 100] (* Amiram Eldar, Sep 19 2020 *)

a(n)=3^valuation(n,3)<Charles R Greathouse IV, Aug 21 2011

a(n)=gcd(n,6^n) \\ Not very efficient, but simple. Stanislav Sykora, Feb 08 2016

a(n)=gcd(6^logint(n,2),n) \\ 'optimized' version of Sykora's script; Charles R Greathouse IV, Jul 23 2024

a(n) = n / A065330(n).
a(n) = A006519(n) * A038500(n).
a(n) = (2^A007814 (n)) * (3^A007949(n)).
Multiplicative with a(2^e)=2^e, a(3^e)=3^e, a(p^e)=1, p>3. - Vladeta Jovovic, Nov 05 2001
Dirichlet g.f.: zeta(s)*(1-2^(-s))*(1-3^(-s))/ ( (1-2^(1-s))*(1-3^(1-s)) ). - R. J. Mathar, Jul 04 2011
a(n) = gcd(n,6^n). - Stanislav Sykora, Feb 08 2016
a(A225546(n)) = A225546(A053165(n)). - Peter Munn, Jan 17 2020
Sum_{k=1..n} a(k) ~ n*(log(n)^2 + (2*gamma + 3*log(2) + 2*log(3) - 2)*log(n) + (2 + log(2)^2/6 + 3*log(2)*(log(3) - 1) - 2*log(3) + log(3)^2/6 + gamma*(3*log(2) + 2*log(3) - 2) - 2*sg1)) / (6*log(2)*log(3)), where gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). - Vaclav Kotesovec, Sep 19 2020
a(n) = A003586(A322026(n)), A322026(n) = A071521(a(n)). - Antti Karttunen, Sep 08 2024

A126760 a(0) = 0, a(2n) = a(n), a(3n) = a(n), a(6n+1) = 2n + 1, a(6n+5) = 2n + 2.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 3, 1, 1, 2, 4, 1, 5, 3, 2, 1, 6, 1, 7, 2, 3, 4, 8, 1, 9, 5, 1, 3, 10, 2, 11, 1, 4, 6, 12, 1, 13, 7, 5, 2, 14, 3, 15, 4, 2, 8, 16, 1, 17, 9, 6, 5, 18, 1, 19, 3, 7, 10, 20, 2, 21, 11, 3, 1, 22, 4, 23, 6, 8, 12, 24, 1, 25, 13, 9, 7, 26, 5, 27, 2, 1, 14, 28, 3, 29, 15, 10, 4, 30, 2

Offset: 0

N. J. A. Sloane, Feb 19 2007

nonn

For further information see A126759, which provided the original motivation for this sequence.
From Antti Karttunen, Jan 28 2015: (Start)
The odd bisection of the sequence gives A253887, and the even bisection gives the sequence itself.
A254048 gives the sequence obtained when this sequence is restricted to A007494 (numbers congruent to 0 or 2 mod 3).
For all odd numbers k present in square array A135765, a(k) = the column index of k in that array. (End)
A322026 and this sequence (without the initial zero) are ordinal transforms of each other. - Antti Karttunen, Feb 09 2019
Also ordinal transform of A065331 (after the initial 0). - Antti Karttunen, Sep 08 2024

Antti Karttunen, Table of n, a(n) for n = 0..19683

One less than A126759.
Cf. A003586, A003602, A007310, A064989, A249746, A253887, A254048, A273669, A322317, A323881 (Dirichlet inverse), A323882.
Cf. A347233 (Möbius transform) and also A349390, A349393, A349395 for other Dirichlet convolutions.
Ordinal transform of A065331 and of A322026 (after the initial 0).
Related arrays: A135765, A254102.

f[n_] := Block[{a}, a[0] = 0; a[1] = a[2] = a[3] = 1; a[x_] := Which[EvenQ@ x, a[x/2], Mod[x, 3] == 0, a[x/3], Mod[x, 6] == 1, 2 (x - 1)/6 + 1, Mod[x, 6] == 5, 2 (x - 5)/6 + 2]; Table[a@ i, {i, 0, n}]] (* Michael De Vlieger, Feb 03 2015 *)

A126760(n)={n&&n\=3^valuation(n,3)<M. F. Hasler, Jan 19 2016

a(n) = A126759(n)-1. [The original definition.]
From Antti Karttunen, Jan 28 2015: (Start)
a(0) = 0, a(2n) = a(n), a(3n) = a(n), a(6n+1) = 2n + 1, a(6n+5) = 2n + 2.
Or with the last clause represented in another way:
a(0) = 0, a(2n) = a(n), a(3n) = a(n), a(6n+1) = 2n + 1, a(6n-1) = 2n.
Other identities. For all n >= 1:
a(n) = A253887(A003602(n)).
a(6n-3) = a(4n-2) = a(2n-1) = A253887(n).
(End)
a(n) = A249746(A003602(A064989(n))). - Antti Karttunen, Feb 04 2015
a(n) = A323882(4*n). - Antti Karttunen, Apr 18 2022

Name replaced with an independent recurrence and the old description moved to the Formula section - Antti Karttunen, Jan 28 2015

A047522 Numbers that are congruent to {1, 7} mod 8.

Original entry on oeis.org

1, 7, 9, 15, 17, 23, 25, 31, 33, 39, 41, 47, 49, 55, 57, 63, 65, 71, 73, 79, 81, 87, 89, 95, 97, 103, 105, 111, 113, 119, 121, 127, 129, 135, 137, 143, 145, 151, 153, 159, 161, 167, 169, 175, 177, 183, 185, 191, 193, 199, 201, 207, 209, 215, 217, 223, 225, 231, 233

Offset: 1

N. J. A. Sloane

Also n such that Kronecker(2,n) = mu(gcd(2,n)). - Jon Perry and T. D. Noe, Jun 13 2003
Also n such that x^2 == 2 (mod n) has a solution. The primes are given in sequence A001132. - T. D. Noe, Jun 13 2003
As indicated in the formula, a(n) is related to the even triangular numbers. - Frederick Magata (frederick.magata(AT)uni-muenster.de), Jun 17 2004
Cf. property described by Gary Detlefs in A113801: more generally, these a(n) are of the form (2*h*n + (h-4)*(-1)^n-h)/4 (h,n natural numbers). Therefore a(n)^2 - 1 == 0 (mod h); in this case, a(n)^2 - 1 == 0 (mod 8). Also a(n)^2 - 1 == 0 (mod 16). - Bruno Berselli, Nov 17 2010
A089911(3*a(n)) = 2. - Reinhard Zumkeller, Jul 05 2013
S(a(n+1)/2, 0) = (1/2)*(S(a(n+1), sqrt(2)) - S(a(n+1) - 2, sqrt(2)))  = T(a(n+1), sqrt(2)/2) = cos(a(n+1)*Pi/4) = sqrt(2)/2 = A010503, identically for n >= 0, where S is the Chebyshev polynomial (A049310) here extended to fractional n, evaluated at x = 0. (For T see A053120.) - Wolfdieter Lang, Jun 04 2023

L. B. W. Jolley, Summation of Series, Dover Publications, 1961, p. 16.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A001132, A014494, A056575, A010709, A074378, A047336, A056020, A005408, A047209, A007310, A090771, A175885, A091998, A175886, A175887, A058529, A047621, A179260, A352125.
Cf. A077221 (partial sums).
Cf. A000217, A089911, A113801.
Cf. A010503, A049310, A053120.

a047522 n = a047522_list !! (n-1)
a047522_list = 1 : 7 : map (+ 8) a047522_list
-- Reinhard Zumkeller, Jan 07 2012

Select[Range[1, 191, 2], JacobiSymbol[2, # ]==1&]

a(n)=4*n-2+(-1)^n \\ Charles R Greathouse IV, Sep 24 2015

a(n) = sqrt(8*A014494(n)+1) = sqrt(16*ceiling(n/2)*(2*n+1)+1) = sqrt(8*A056575(n)-8*(2n+1)*(-1)^n+1). - Frederick Magata (frederick.magata(AT)uni-muenster.de), Jun 17 2004
1 - 1/7 + 1/9 - 1/15 + 1/17 - ... = (Pi/8)*(1 + sqrt(2)). [Jolley] - Gary W. Adamson, Dec 16 2006
From R. J. Mathar, Feb 19 2009: (Start)
a(n) = 4n - 2 + (-1)^n = a(n-2) + 8.
G.f.: x(1+6x+x^2)/((1+x)(1-x)^2). (End)
a(n) = 8*n - a(n-1) - 8. - Vincenzo Librandi, Aug 06 2010
From Bruno Berselli, Nov 17 2010: (Start)
a(n) = -a(-n+1) = a(n-1) + a(n-2) - a(n-3).
a(n) = 8*A000217(n-1)+1 - 2*Sum_{i=1..n-1} a(i) for n > 1. (End)
E.g.f.: 1 + (4*x - 1)*cosh(x) + (4*x - 3)*sinh(x). - Stefano Spezia, May 13 2021
E.g.f.: 1 + (4*x - 3)*exp(x) + 2*cosh(x). - David Lovler, Jul 16 2022
From Amiram Eldar, Nov 22 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = sqrt(2+sqrt(2)) (A179260).
Product_{n>=2} (1 + (-1)^n/a(n)) = (Pi/8)*cosec(Pi/8) (A352125). (End)

A047241 Numbers that are congruent to {1, 3} mod 6.

Original entry on oeis.org

1, 3, 7, 9, 13, 15, 19, 21, 25, 27, 31, 33, 37, 39, 43, 45, 49, 51, 55, 57, 61, 63, 67, 69, 73, 75, 79, 81, 85, 87, 91, 93, 97, 99, 103, 105, 109, 111, 115, 117, 121, 123, 127, 129, 133, 135, 139, 141, 145, 147, 151, 153, 157, 159, 163, 165, 169, 171, 175, 177, 181, 183

Offset: 1

N. J. A. Sloane

Also the numbers k such that 10^p+k could possibly be prime. - Roderick MacPhee, Nov 20 2011 This statement can be written as follows. If 10^m + k = prime, for any m >= 1, then k is in this sequence. See the pink box comments by Roderick MacPhee from Dec 09 2014. - Wolfdieter Lang, Dec 09 2014
The odd-indexed terms are one more than the arithmetic mean of their neighbors; the even-indexed terms are one less than the arithmetic mean of their neighbors. - Amarnath Murthy, Jul 29 2003
Partial sums are A212959. - Philippe Deléham, Mar 16 2014
12*a(n) is conjectured to be the length of the boundary after n iterations of the hexagon and square expansion shown in the link. The squares and hexagons have side length 1 in some units. The pattern is supposed to become the planar Archimedean net 4.6.12 when n -> infinity. - Kival Ngaokrajang, Nov 30 2014
Positive numbers k for which 1/2 + k/3 + k^2/6 is an integer. - Bruno Berselli, Apr 12 2018

L. Lovasz, J. Pelikan, K. Vesztergombi, Discrete Mathematics, Springer (2003); 14.4, p. 225.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
L. Lovász, J. Pelikán and K. Vesztergombi, Discrete Mathematics, Elementary and Beyond, Springer (2003); 14.4, p. 225.
Kival Ngaokrajang, Illustration of initial terms.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A047233, A056970, A007310, A047228, A047261, A047273, A212959.
Subsequence of A186422.
Union of A016921 and A016945. - Wesley Ivan Hurt, Sep 28 2013

a047241 n = a047241_list !! (n-1)
a047241_list = 1 : 3 : map (+ 6) a047241_list
-- Reinhard Zumkeller, Feb 19 2013

seq(3*k-2-((k+1) mod 2), k=1..100); # Wesley Ivan Hurt, Sep 28 2013

Table[{2, 4}, {30}] // Flatten // Prepend[#, 1]& // Accumulate (* Jean-François Alcover, Jun 10 2013 *)
Select[Range[200], MemberQ[{1, 3}, Mod[#, 6]]&] (* or *) LinearRecurrence[{1, 1, -1}, {1, 3, 7}, 70] (* Harvey P. Dale, Oct 01 2013 *)

a(n)=bitor(3*n-3,1) \\ Charles R Greathouse IV, Sep 28 2013

for n in range(1,10**5):print(3*n-2-((n+1)%2)) # Soumil Mandal, Apr 14 2016

From Paul Barry, Sep 04 2003: (Start)
O.g.f.: (1 + 2*x + 3*x^2)/((1 + x)*(1 - x)^2) = (1 + 2*x + 3*x^2)/((1 - x)*(1 - x^2)).
E.g.f.: (6*x + 1)*exp(x)/2 + exp(-x)/2;
a(n) = 3*n - 5/2 - (-1)^n/2. (End)
a(n) = 2*floor((n-1)/2) + 2*n - 1. - Gary Detlefs, Mar 18 2010
a(n) = 6*n - a(n-1) - 8 with n > 1, a(1)=1. - Vincenzo Librandi, Aug 05 2010
a(n) = 3*n - 2 - ((n+1) mod 2). - Wesley Ivan Hurt, Jun 29 2013
a(1)=1, a(2)=3, a(3)=7; for n>3, a(n) = a(n-1) + a(n-2) - a(n-3). - Harvey P. Dale, Oct 01 2013
From Benedict W. J. Irwin, Apr 13 2016: (Start)
A005408(a(n)+1) = A016813(A001651(n)),
A007310(a(n)) = A005408(A087444(n)-1),
A007310(A005408(a(n)+1)) = A017533(A001651(n)). (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(4*sqrt(3)) + log(3)/4. - Amiram Eldar, Dec 11 2021

Formula corrected by Bruno Berselli, Jun 24 2010

A056020 Numbers that are congruent to +-1 mod 9.

Original entry on oeis.org

1, 8, 10, 17, 19, 26, 28, 35, 37, 44, 46, 53, 55, 62, 64, 71, 73, 80, 82, 89, 91, 98, 100, 107, 109, 116, 118, 125, 127, 134, 136, 143, 145, 152, 154, 161, 163, 170, 172, 179, 181, 188, 190, 197, 199, 206, 208, 215, 217, 224, 226, 233, 235, 242, 244, 251, 253

Offset: 1

Robert G. Wilson v, Jun 08 2000

Or, numbers k such that k^2 == 1 (mod 9).

Or, numbers k such that the iterative cycle j -> sum of digits of j^2 when started at k contains a 1. E.g., 8 -> 6+4 = 10 -> 1+0+0 = 1 and 17 -> 2+8+9 = 19 -> 3+6+1 = 10 -> 1+0+0 = 1. - Asher Auel, May 17 2001

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

Cf. A007953, A047522 (n=1 or 7 mod 8), A090771 (n=1 or 9 mod 10).
Cf. A129805 (primes), A195042 (partial sums).
Cf. A005408, A019676, A019968, A047209, A007310, A047336, A175885, A091998, A175886, A113801, A175887, A332437.
Cf. A381319 (general case mod n^2).

a056020 n = a056020_list !! (n-1)
a05602_list = 1 : 8 : map (+ 9) a056020_list
-- Reinhard Zumkeller, Jan 07 2012

Select[ Range[ 300 ], PowerMod[ #, 2, 3^2 ]==1& ]
(* or *)
LinearRecurrence[{1, 1, -1}, {1, 8, 10}, 67] (* Mike Sheppard, Feb 18 2025 *)

a(n)=9*(n>>1)+if(n%2,1,-1) \\ Charles R Greathouse IV, Jun 29 2011

for(n=1,40,print1(9*n-8,", ",9*n-1,", ")) \\ Charles R Greathouse IV, Jun 29 2011

a(1) = 1; a(n) = 9(n-1) - a(n-1). - Rolf Pleisch, Jan 31 2008 [Offset corrected by Jon E. Schoenfield, Dec 22 2008]
From R. J. Mathar, Feb 10 2008: (Start)
O.g.f.: 1 + 5/(4(x+1)) + 27/(4(-1+x)) + 9/(2(-1+x)^2).
a(n+1) - a(n) = A010697(n). (End)
a(n) = (9*A132355(n) + 1)^(1/2). - Gary Detlefs, Feb 22 2010
From Bruno Berselli, Nov 17 2010: (Start)
a(n) = a(n-2) + 9, for n > 2.
a(n) = 9*A000217(n-1) + 1 - 2*Sum_{i=1..n-1} a(i), n > 1. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi/9)*cot(Pi/9) = A019676 * A019968. - Amiram Eldar, Dec 04 2021
E.g.f.: 1 + ((18*x - 9)*exp(x) + 5*exp(-x))/4. - David Lovler, Sep 04 2022
From Amiram Eldar, Nov 22 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = 2*cos(Pi/9) (A332437).
Product_{n>=2} (1 + (-1)^n/a(n)) = (Pi/9)*cosec(Pi/9). (End)
From Mike Sheppard, Feb 18 2025: (Start)
a(n) = a(n-1) + a(n-2) - a(n-3).
a(n) ~ (3^2/2)*n. (End)

A084967 Multiples of 5 whose GCD with 6 is 1.

Original entry on oeis.org

5, 25, 35, 55, 65, 85, 95, 115, 125, 145, 155, 175, 185, 205, 215, 235, 245, 265, 275, 295, 305, 325, 335, 355, 365, 385, 395, 415, 425, 445, 455, 475, 485, 505, 515, 535, 545, 565, 575, 595, 605, 625, 635, 655, 665, 685, 695, 715, 725, 745, 755, 775, 785

Offset: 1

Robert G. Wilson v, Jun 15 2003

Third row of A083140.

Positions of 5 in A020639. - Zak Seidov, Apr 29 2015

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A038110, A038111, A083140, A007310 (5-rough numbers), A273669.
Cf. A020639. - Zak Seidov, Apr 29 2015
Essentially the same as A063149.

5Select[ Range[160], GCD[ #, 2*3] == 1 & ]
Select[Range[5, 785, 10], Mod[#, 3] > 0 &] (* Zak Seidov, Apr 29 2015 *)
a[1] = 5; a[n_] := a[n] = a[n - 1] + 10*(2 - Mod[n, 2]); Table[a[n], {n, 50}] (* Zak Seidov, Apr 29 2015 *)

is(n)=n%5==0 && gcd(n,6)==1 \\ Charles R Greathouse IV, Nov 19 2014

list(lim)=5*select(k->gcd(n,6)==1, [1..lim\5]) \\ Charles R Greathouse IV, Nov 19 2014

Numbers of the form 5k for which gcd(5k, 6) = 1.
a(n) = 5*A007310(n). - Adriano Caroli, Oct 03 2010
From Colin Barker, Feb 24 2013: (Start)
a(n) = 5*(-3 + (-1)^n + 6*n)/2.
a(n) = a(n-1) + a(n-2) - a(n-3).
G.f.: 5*x*(x^2+4*x+1) / ((x-1)^2*(x+1)). (End)
Limit_{n->infinity} a(n)/n = A038111(3)/A038110(3) = 15. - Vladimir Shevelev, Jan 20 2015
For n > 2, a(n) = a(n-2) + 30. - Zak Seidov, Apr 29 2015
a(n) = A007310(A273669(n)). - Antti Karttunen, May 20 2017
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(10*sqrt(3)). - Amiram Eldar, Nov 03 2022

A057353 a(n) = floor(3n/4).

Original entry on oeis.org

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

Offset: 0

Mitch Harris

The cyclic pattern (and numerator of the gf) is computed using Euclid's algorithm for GCD.
For n >= 2, a(n) is the number of different integers that can be written as floor(k^2/n) for k = 1, 2, 3, ..., n-1. Generalization of the 1st problem proposed during the 15th Balkan Mathematical Olympiad in 1998 where the question was asked for n = 1998 with a(1998) = 1498. - Bernard Schott, Apr 22 2022
For n > 1, a(n) is also the Hadwiger number of the (n+1)-cycle complement graph (up to at least n = 16). - Eric W. Weisstein, Mar 10 2025

N. Dershowitz and E. M. Reingold, Calendrical Calculations, Cambridge University Press, 1997.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, NY, 1994.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Balkan Mathematical Olympiad, Problem 1, 15th Balkan Mathematical Olympiad 1998.
Eric Weisstein's World of Mathematics, Cycle Complement Graph.
Eric Weisstein's World of Mathematics, Hadwiger Number.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).
Index entries for sequences related to Beatty sequences.
Index to sequences related to Olympiads.

Floors of other ratios: A004526, A002264, A002265, A004523, A057354, A057355, A057356, A057357, A057358, A057359, A057360, A057361, A057362, A057363, A057364, A057365, A057366, A057367.

Cf. A002378, A007310, A057077, A073010, A173562, A182210.

[Floor(3*n/4): n in [0..90]]; // Vincenzo Librandi, Feb 12 2012

Table[Floor[3 n/4], {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, Jan 28 2012 *)
Floor[3 Range[0, 20]/4] (* Eric W. Weisstein, Mar 10 2025 *)
LinearRecurrence[{1, 0, 0, 1, -1}, {0, 1, 2, 3, 3}, {0, 20}] (* Eric W. Weisstein, Mar 10 2025 *)
CoefficientList[Series[x^2 (1 + x + x^2)/(1 - x - x^4 + x^5), {x, 0, 20}], x] (* Eric W. Weisstein, Mar 10 2025 *)

a(n)=3*n\4 \\ Charles R Greathouse IV, Sep 02 2015

G.f.: (1+x+x^2)*x^2/((1-x)*(1-x^4)). - Bruce Corrigan (scentman(AT)myfamily.com), Jul 03 2002
For all m>=0 a(4m)=0 mod 3; a(4m+1)=0 mod 3; a(4m+2)= 1 mod 3; a(4m+3) = 2 mod 3
a(n) = A002378(n) - A173562(n). - Reinhard Zumkeller, Feb 21 2010
a(n+1) = A140201(n) - A002265(n+1). - Reinhard Zumkeller, Jan 26 2011
a(n) = n-1 - A002265(n-1) = ( A007310(n) + A057077(n+1) )/4 for n>0. a(n) = a(n-1)+a(n-4)-a(n-5) for n>4. - Bruno Berselli, Jan 28 2011
a(n) = 1/8*(6*n + 2*cos((Pi*n)/2) + cos(Pi*n) - 2*sin((Pi*n)/2) - 3). - Ilya Gutkovskiy, Sep 18 2015
a(4n) = a(4n+1). - Altug Alkan, Sep 26 2015
Sum_{n>=2} (-1)^n/a(n) = Pi/(3*sqrt(3)) (A073010). - Amiram Eldar, Sep 29 2022

A090771 Numbers that are congruent to {1, 9} mod 10.

Original entry on oeis.org

1, 9, 11, 19, 21, 29, 31, 39, 41, 49, 51, 59, 61, 69, 71, 79, 81, 89, 91, 99, 101, 109, 111, 119, 121, 129, 131, 139, 141, 149, 151, 159, 161, 169, 171, 179, 181, 189, 191, 199, 201, 209, 211, 219, 221, 229, 231, 239, 241, 249, 251, 259, 261, 269, 271, 279, 281

Offset: 1

Giovanni Teofilatto, Feb 07 2004

Cf. property described by Gary Detlefs in A113801: more generally, these numbers are of the form (2*h*n + (h-4)*(-1)^n - h)/4 (h, n natural numbers), therefore ((2*h*n + (h-4)*(-1)^n - h)/4)^2-1 == 0 (mod h); in this case, a(n)^2 - 1 == 0 (mod 10). - Bruno Berselli, Nov 17 2010

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A000217, A000796, A090298, A005408, A047209, A007310, A019970, A047336, A047522, A091998, A094888, A113801, A175886, A175887, A188593.
Cf. A056020 (n = 1 or 8 mod 9), A175885 (n = 1 or 10 mod 11).
Cf. A045468 (primes), A195142 (partial sums).

a090771 n = a090771_list !! (n-1)
a090771_list = 1 : 9 : map (+ 10) a090771_list
-- Reinhard Zumkeller, Jan 07 2012

Flatten[Table[10n - {9, 1}, {n, 30}]] (* Alonso del Arte, Sep 02 2014 *)
LinearRecurrence[{1,1,-1},{1,9,11},60] (* Harvey P. Dale, Jul 05 2020 *)

a(n)=n\2*10-(-1)^n \\ Charles R Greathouse IV, Sep 24 2015

a(n) = sqrt(40*A057569(n) + 1). - Gary Detlefs, Feb 22 2010
From Bruno Berselli, Sep 16 2010 - Nov 17 2010: (Start)
  G.f.: x*(1 + 8*x + x^2)/((1 + x)*(1 - x)^2).
  a(n) = (10*n + 3*(-1)^n - 5)/2.
  a(n) = -a(-n + 1) = a(n-1) + a(n-2) - a(n-3) = a(n-2) + 10.
  a(n) = 10*A000217(n-1) + 1 - 2*Sum_{i=1..n-1} a(i) for n > 1. (End)
a(n) = 10*n - a(n-1) - 10 (with a(1) = 1). - Vincenzo Librandi, Nov 16 2010
a(n) = sqrt(10*A132356(n-1) + 1). - Ivan N. Ianakiev, Nov 09 2012
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi/10)*cot(Pi/10) = A000796 * A019970 / 10 = sqrt(5 + 2*sqrt(5))*Pi/10. - Amiram Eldar, Dec 04 2021
E.g.f.: 1 + ((10*x - 5)*exp(x) + 3*exp(-x))/2. - David Lovler, Sep 03 2022
From Amiram Eldar, Nov 23 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = sqrt(phi+2) (A188593).
Product_{n>=2} (1 + (-1)^n/a(n)) = Pi*phi/5 = A094888/10. (End)

Edited and extended by Ray Chandler, Feb 10 2004

A047336 Numbers that are congruent to {1, 6} mod 7.

Original entry on oeis.org

1, 6, 8, 13, 15, 20, 22, 27, 29, 34, 36, 41, 43, 48, 50, 55, 57, 62, 64, 69, 71, 76, 78, 83, 85, 90, 92, 97, 99, 104, 106, 111, 113, 118, 120, 125, 127, 132, 134, 139, 141, 146, 148, 153, 155, 160, 162, 167, 169, 174, 176, 181, 183, 188, 190, 195, 197, 202, 204, 209

Offset: 1

N. J. A. Sloane

Cf. property described by Gary Detlefs in A113801: more generally, these numbers are of the form (2*h*n+(h-4)*(-1)^n-h)/4 (h, n natural numbers), therefore ((2*h*n+(h-4)*(-1)^n-h)/4)^2-1 == 0 (mod h); in this case, a(n)^2-1 == 0 (mod 7). - Bruno Berselli, Nov 17 2010

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A007310, A019674, A047522, A045472 (primes), A195041 (partial sums), A005408, A047209, A056020, A090771, A091998, A113801, A160389, A175885, A175886, A175887, A178818, A371858.

a047336 n = a047336_list !! (n-1)
a047336_list = 1 : 6 : map (+ 7) a047336_list
-- Reinhard Zumkeller, Jan 07 2012

[n: n in [1..210]| n mod 7 in {1,6}]; // Bruno Berselli, Feb 22 2011

Rest[Flatten[Table[{7i-1,7i+1},{i,0,40}]]] (* Harvey P. Dale, Nov 20 2010 *)

a(n)=n\2*7-(-1)^n \\ Charles R Greathouse IV, May 02 2016

a(1) = 1; a(n) = 7(n-1) - a(n-1). - Rolf Pleisch, Jan 31 2008 (corrected by Jon E. Schoenfield, Dec 22 2008)
a(n) = (7/2)*(n-(1-(-1)^n)/2) - (-1)^n. - Rolf Pleisch, Nov 02 2010
From Bruno Berselli, Nov 17 2010: (Start)
G.f.: x*(1+5*x+x^2)/((1+x)*(1-x)^2).
a(n) = -a(-n+1) = a(n-1) + a(n-2) - a(n-3).
a(n) = a(n-2)+7.
a(n) = 7*A000217(n-1)+1 - 2*Sum_{i=1..n-1} a(i) for n > 1. (End)
a(n) = 7*floor(n/2)+(-1)^(n+1). - Gary Detlefs, Dec 29 2011
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi/7)*cot(Pi/7) = A019674 * A178818. - Amiram Eldar, Dec 04 2021
E.g.f.: 1 + ((14*x - 7)*exp(x) + 3*exp(-x))/4. - David Lovler, Sep 01 2022
From Amiram Eldar, Nov 22 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = 2*cos(Pi/7) (A160389).
Product_{n>=2} (1 + (-1)^n/a(n)) = (Pi/7) * cosec(Pi/7) (A371858). (End)

More terms from Jon E. Schoenfield, Jan 18 2009

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