A010722 -id:A010722 - cp's OEIS frontend

A008588 Nonnegative multiples of 6.

0, 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96, 102, 108, 114, 120, 126, 132, 138, 144, 150, 156, 162, 168, 174, 180, 186, 192, 198, 204, 210, 216, 222, 228, 234, 240, 246, 252, 258, 264, 270, 276, 282, 288, 294, 300, 306, 312, 318, 324, 330, 336, 342, 348

Offset: 0

N. J. A. Sloane

For n > 3, the number of squares on the infinite 3-column half-strip chessboard at <= n knight moves from any fixed point on the short edge.
Second differences of A000578. - Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 15 2004
A008615(a(n)) = n. - Reinhard Zumkeller, Feb 27 2008
A157176(a(n)) = A001018(n). - Reinhard Zumkeller, Feb 24 2009
These numbers can be written as the sum of four cubes (i.e., 6*n = (n+1)^3 + (n-1)^3 + (-n)^3 + (-n)^3). - Arkadiusz Wesolowski, Aug 09 2013
A122841(a(n)) > 0 for n > 0. - Reinhard Zumkeller, Nov 10 2013
Surface area of a cube with side sqrt(n). - Wesley Ivan Hurt, Aug 24 2014
a(n) is representable as a sum of three but not two consecutive nonnegative integers, e.g., 6 = 1 + 2 + 3, 12 = 3 + 4 + 5, 18 = 5 + 6 + 7, etc. (see A138591). - Martin Renner, Mar 14 2016 (Corrected by David A. Corneth, Aug 12 2016)
Numbers with three consecutive divisors: for some k, each of k, k+1, and k+2 divide n. - Charles R Greathouse IV, May 16 2016
Numbers k for which {phi(k),phi(2k),phi(3k)} is an arithmetic progression. - Ivan Neretin, Aug 12 2016

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 81.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 318
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Essentially the same as A008458.
Cf. A016921, A016933, A016945, A016957, A016969, A138591.
Cf. A044102 (subsequence).

a008588 = (* 6)
a008588_list = [0, 6 ..]  -- Reinhard Zumkeller, Nov 10 2013

[6*n: n in [0..60] ]; // Vincenzo Librandi, Jul 16 2011

Maple
```
[ seq(6*n,n=0..45) ];
```

Range[0, 500, 6] (* Vladimir Joseph Stephan Orlovsky, May 26 2011 *)
NestList[6+#&,0,60] (* Harvey P. Dale, Nov 17 2024 *)

makelist(6*n,n,0,30); /* Martin Ettl, Nov 12 2012 */

a(n)=6*n \\ Charles R Greathouse IV, Feb 08 2012

[6*n for n in range(59)] # Stefano Spezia, Mar 09 2025

From Vincenzo Librandi, Dec 24 2010: (Start)
a(n) = 6*n = 2*a(n-1) - a(n-2).
G.f.: 6*x/(1-x)^2. (End)
a(n) = Sum_{k>=0} A030308(n,k)*6*2^k. - Philippe Deléham, Oct 24 2011
a(n) = Sum_{k=2n-1..2n+1} k. - Wesley Ivan Hurt, Nov 22 2015
From Ilya Gutkovskiy, Aug 12 2016: (Start)
E.g.f.: 6*x*exp(x).
Convolution of A010722 and A057427.
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2)/6 = A002162*A020793. (End)
a(n) = 6 * A001477(n). - David A. Corneth, Aug 12 2016

A144223 Number of ways of placing n labeled balls into n unlabeled (but 6-colored) boxes.

Original entry on oeis.org

1, 6, 42, 330, 2850, 26682, 268098, 2869242, 32510850, 388109562, 4861622850, 63682081530, 869725707522, 12352785293562, 182049635623362, 2778394592545530, 43833623157604482, 713738052924821754

Offset: 0

Philippe Deléham, Sep 14 2008

nonn

a(n) is also the exp transform of A010722. - Alois P. Heinz, Oct 09 2008

The number of ways of putting n labeled balls into a set of bags and then putting the bags into 6 labeled boxes. - Peter Bala, Mar 23 2013

Alois P. Heinz, Table of n, a(n) for n = 0..200
N. J. A. Sloane, Transforms

Cf. A000110, A001861, A027710, A078944, A144180. A144263, A189233, A221159, A221176.

a:= proc(n) option remember; `if`(n=0, 1,
      (1+add(binomial(n-1, k-1)*a(n-k), k=1..n-1))*6)
    end:
seq(a(n), n=0..25); # Alois P. Heinz, Oct 09 2008

Table[BellB[n,6],{n,0,20}] (* Vaclav Kotesovec, Mar 12 2014 *)

expnums(18, 6) # Zerinvary Lajos, May 15 2009

a(n) = Sum_{k=0..n} 6^k*A048993(n,k); A048993: Stirling2 numbers.
G.f.: 6*(x/(1-x))*A(x/(1-x)) = A(x)-1; six times the binomial transform equals this sequence shifted one place left.
E.g.f.: exp(6(e^x-1)).
G.f.: T(0)/(1-6*x), where T(k) = 1 - 6*x^2*(k+1)/(6*x^2*(k+1) - (1-6*x-x*k)*(1-7*x-x*k)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Dec 04 2013
a(n) ~ n^n * exp(n/LambertW(n/6)-6-n) / (sqrt(1+LambertW(n/6)) * LambertW(n/6)^n). - Vaclav Kotesovec, Mar 12 2014
G.f.: Sum_{j>=0} 6^j*x^j / Product_{k=1..j} (1 - k*x). - Ilya Gutkovskiy, Apr 07 2019

More terms from Alois P. Heinz, Oct 09 2008

A176398 Decimal expansion of 3+sqrt(10).

Original entry on oeis.org

6, 1, 6, 2, 2, 7, 7, 6, 6, 0, 1, 6, 8, 3, 7, 9, 3, 3, 1, 9, 9, 8, 8, 9, 3, 5, 4, 4, 4, 3, 2, 7, 1, 8, 5, 3, 3, 7, 1, 9, 5, 5, 5, 1, 3, 9, 3, 2, 5, 2, 1, 6, 8, 2, 6, 8, 5, 7, 5, 0, 4, 8, 5, 2, 7, 9, 2, 5, 9, 4, 4, 3, 8, 6, 3, 9, 2, 3, 8, 2, 2, 1, 3, 4, 4, 2, 4, 8, 1, 0, 8, 3, 7, 9, 3, 0, 0, 2, 9, 5, 1, 8, 7, 3, 4

Offset: 1

Klaus Brockhaus, Apr 16 2010

Continued fraction expansion of 3+sqrt(10) is A010722.
This is the shape of a 6-extension rectangle; see A188640 for definitions. - Clark Kimberling, Apr 09 2011
c^n = c*A005668(n) + A005668(n-1). - Gary W. Adamson, Apr 04 2024

			6.16227766016837933199...

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000
Wikipedia, Metallic mean.
Index entries for algebraic numbers, degree 2

Cf. A010467 (decimal expansion of sqrt(10)), A010722 (all 6's sequence).

Cf. A049310.

Digits:=100; evalf(3+sqrt(10)); # Wesley Ivan Hurt, Mar 07 2014

r=6; t=(r+(4+r^2)^(1/2))/2; RealDigits[N[t,130]][[1]] (* Clark Kimberling, Apr 09 2011 *)

3+sqrt(10) \\ Charles R Greathouse IV, Jul 24 2013

a(n) = A010467(n) for n >= 2.
Equals exp(arcsinh(3)), since arcsinh(x) = log(x+sqrt(x^2+1)). - Stanislav Sykora, Nov 01 2013
Equals lim_{n->oo} S(n, 2*sqrt(10))/ S(n-1, 2*sqrt(10)), with the S-Chebyshev polynomials (see A049310). - Wolfdieter Lang, Nov 15 2023

A020793 Decimal expansion of 1/6.

Original entry on oeis.org

1, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6

Offset: 0

N. J. A. Sloane

Except for the first term identical to A010722, A040006 and A021019. Except for the first terms the same as A021028, A021100, A021388, A071279, A101272, A168608, A177057,... - M. F. Hasler, Oct 24 2011

Decimal expansion of gamma(1) = 5/3 (with offset 1), where gamma(n) = Cp(n)/Cv(n) = is the n-th Poisson's constant. For the definition of Cp and Cv see A272002. - Natan Arie Consigli, Jul 10 2016

Calvin C. Clawson, Mathematical Mysteries, The Beauty and Magic of Numbers, Springer, 2013, see p. 224.

Wikipedia, Poisson's constant.
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A010722, A021019, A021028, A021100, A021388, A040006, A070064, A071279, A081822, A101272, A168608, A177057, A272001, A272002.

RealDigits[1/6,10,120][[1]] (* or *) PadRight[{1},120,{6}] (* Harvey P. Dale, Dec 30 2018 *)

a(n)=6-5*!n  \\ M. F. Hasler, Oct 24 2011

a(n) = 6^n mod 10. - Zerinvary Lajos, Nov 26 2009
Equals Sum_{k>=1} 1/7^k. - Bruno Berselli, Jan 03 2014
10 * 1/6 = 5/3 = (5/2 R)/(3/2 R) = Cp(1)/Cv(1) = A272002/A272001, with R = A081822 (or A070064). - Natan Arie Consigli, Jul 10 2016
G.f.: (1 + 5*x)/(1 - x). - Ilya Gutkovskiy, Jul 10 2016
Equals Sum_{k>=1} 1/(k*Pi)^2. - Maciej Kaniewski, Sep 14 2017
Equals Sum_{k>=1} (zeta(2*k)-1)/4^k. - Amiram Eldar, Jun 08 2021
K_{n>=2} 2*n/(2*n - 3) = 5/3. (see Clawson at p. 224). - Stefano Spezia, Jul 01 2024
E.g.f.: 6*exp(x) - 5. - Elmo R. Oliveira, Aug 05 2024

A023005 Number of partitions of n into parts of 6 kinds.

Original entry on oeis.org

1, 6, 27, 98, 315, 918, 2492, 6372, 15525, 36280, 81816, 178794, 380051, 788004, 1597725, 3174210, 6190182, 11867310, 22395359, 41650050, 76413078, 138421358, 247783113, 438616728, 768291650, 1332444330, 2289213495, 3898064226, 6581591157, 11023247880

Offset: 0

David W. Wilson

nonn

a(n) is Euler transform of A010722. - Alois P. Heinz, Oct 17 2008

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Roland Bacher, P. De La Harpe, Conjugacy growth series of some infinitely generated groups. 2016, hal-01285685v2.
P. Nataf, M. Lajkó, A. Wietek, K. Penc, F. Mila, A. M. Läuchli, Chiral spin liquids in triangular lattice SU (N) fermionic Mott insulators with artificial gauge fields, arXiv preprint arXiv:1601.00958 [cond-mat.quant-gas], 2016.
N. J. A. Sloane, Transforms
Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Cf. 6th column of A144064. - Alois P. Heinz, Oct 17 2008

with(numtheory): a:=proc(n) option remember; `if`(n=0, 1, add(add(d*6, d=divisors(j)) *a(n-j), j=1..n)/n) end: seq(a(n), n=0..40); # Alois P. Heinz, Oct 17 2008

nmax=50; CoefficientList[Series[Product[1/(1-x^k)^6,{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Feb 28 2015 *)

G.f.: Product_{m>=1} 1/(1-x^m)^6.
a(n) ~ exp(2 * Pi * sqrt(n)) / (16 * n^(9/4)). - Vaclav Kotesovec, Feb 28 2015
a(0) = 1, a(n) = (6/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Mar 26 2017
G.f.: exp(6*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018

A141631 a(n) = 3n^2 - 4n + 3.

Original entry on oeis.org

2, 7, 18, 35, 58, 87, 122, 163, 210, 263, 322, 387, 458, 535, 618, 707, 802, 903, 1010, 1123, 1242, 1367, 1498, 1635, 1778, 1927, 2082, 2243, 2410, 2583, 2762, 2947, 3138, 3335, 3538, 3747, 3962, 4183, 4410, 4643, 4882, 5127, 5378, 5635, 5898, 6167, 6442

Offset: 1

Paul Curtz, Aug 28 2008

First bisection of A133146.

Also first bisection of A271713. - Bruno Berselli, Mar 19 2021

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000004 (third differences), A010722 (second differences).

Cf. A000290, A016969, A067998, A133146, A271713.

Table[3 n^2 - 4 n + 3, {n, 50}] (* Harvey P. Dale, Oct 28 2012 *)

a(n)=3*n^2-4*n+3 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = A133146(2*n-2) = (n - 2)^2 + (n - 1)*(n + 1) + n^2.
First differences: a(n+1) - a(n) = A016969(n-1).
G.f.: x*(2 + x + 3*x^2)/(1 - x)^3. - R. J. Mathar, Oct 15 2008
a(n) = 6*n + a(n-1) - 7 for n > 1, a(1)=2. - Vincenzo Librandi, Nov 25 2010
a(n) = 2*A000290(n)^2 + A067998(n-1) = 2*n^2 + (n - 1)*(n - 3). - L. Edson Jeffery, Nov 30 2013
From Elmo R. Oliveira, Nov 13 2024: (Start)
E.g.f.: exp(x)*(3*x^2 - x + 3) - 3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

Edited and extended by R. J. Mathar, Oct 15 2008

A010726 Period 2: repeat (6,10).

Original entry on oeis.org

6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10

Offset: 0

N. J. A. Sloane

From Klaus Brockhaus, Dec 10 2009: (Start)
Interleaving of A010722 and A010692.
Also continued fraction expansion of 3 + 4*sqrt(15)/5.
Binomial transform of 6 followed by A122803 without initial terms 1,-2.
Inverse binomial transform of A171494. (End)

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

Equals 2*A010703. Cf. A010722 (all 6's sequence), A010692 (all 10's sequence), A122803 (powers of -2), A171494. - Klaus Brockhaus, Dec 10 2009

&cat[ [6, 10]: n in [1..42] ]; // Klaus Brockhaus, Dec 10 2009

LinearRecurrence[{0,1},{6,10},90] (* or *) PadRight[{},90,{6,10}] (* Harvey P. Dale, Mar 07 2015 *)

a(n)=6+n%2*4 \\ Charles R Greathouse IV, Dec 21 2011

a(n) = -2*(-1)^n + 8. - Paolo P. Lava, Oct 27 2006
From Klaus Brockhaus, Dec 10 2009: (Start)
a(n) = a(n-2) for n > 1; a(0) = 6, a(1) = 10.
G.f.: 2*(3+5*x)/((1-x)*(1+x)). (End)

A165680 Triangle of the divisors of the coefficients of triangles A138771 and A165675.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 6, 1, 1, 1, 2, 6, 24, 1, 1, 1, 2, 6, 24, 120, 1, 1, 1, 2, 6, 24, 120, 720, 1, 1, 1, 2, 6, 24, 120, 720, 5040, 1, 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 1, 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880

Offset: 1

Johannes W. Meijer, Oct 05 2009

			Triangle starts:
1,
1, 1,
1, 1, 1,
1, 1, 1, 2,
1, 1, 1, 2, 6,
1, 1, 1, 2, 6, 24,
1, 1, 1, 2, 6, 24, 120,
1, 1, 1, 2, 6, 24, 120, 720,
1, 1, 1, 2, 6, 24, 120, 720, 5040,
1, 1, 1, 2, 6, 24, 120, 720, 5040, 40320,
1, 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880,
...

Cf. A138771, A165675 and A000142.
A000012 (3x), A007395, A010722, A010863 equal the first six left hand columns.
A159333 equals, for n=>-1, all right hand columns.
A067078 equals the row sums.

nmax:=11: for n from 1 to nmax do a(n,1):=1 od: for n from 2 to nmax do for m from 2 to n do a(n,m):=(m-2)! od: od: for n from 1 to nmax do seq(a(n,m),m=1..n) od;

a(n) = A138771(n)/A165675(n-1).

A278480 Number of neighbors of the n-th term in a full right triangle read by rows.

Original entry on oeis.org

2, 4, 5, 5, 7, 5, 5, 8, 7, 5, 5, 8, 8, 7, 5, 5, 8, 8, 8, 7, 5, 5, 8, 8, 8, 8, 7, 5, 5, 8, 8, 8, 8, 8, 7, 5, 5, 8, 8, 8, 8, 8, 8, 7, 5, 5, 8, 8, 8, 8, 8, 8, 8, 7, 5, 5, 8, 8, 8, 8, 8, 8, 8, 8, 7, 5, 5, 8, 8, 8, 8, 8, 8, 8, 8, 8, 7, 5, 5, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 7, 5, 5, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 7, 5

Offset: 1

Omar E. Pol, Nov 23 2016

Apart from the first column and the first two diagonals, the rest of the elements are 8's.

For the same idea but for an isosceles triangle see A278481; for a square array see A278545, for a square spiral see A010731; and for a hexagonal spiral see A010722.

			The sequence written as a right triangle begins:
2;
4, 5;
5, 7, 5;
5, 8, 7, 5;
5, 8, 8, 7, 5;
5, 8, 8, 8, 7, 5;
5, 8, 8, 8, 8, 7, 5;
5, 8, 8, 8, 8, 8, 7, 5;
5, 8, 8, 8, 8, 8, 8, 7, 5;
5, 8, 8, 8, 8, 8, 8, 8, 7, 5;
...

Row sums give 2 together with the elements > 1 of A017077.
Also, row sums give 2 together with A004768.
Cf. A010722, A010731, A278317, A278481, A278545.

A278481 Number of neighbors of the n-th term in a full isosceles triangle read by rows.

Original entry on oeis.org

2, 4, 4, 4, 6, 4, 4, 6, 6, 4, 4, 6, 6, 6, 4, 4, 6, 6, 6, 6, 4, 4, 6, 6, 6, 6, 6, 4, 4, 6, 6, 6, 6, 6, 6, 4, 4, 6, 6, 6, 6, 6, 6, 6, 4, 4, 6, 6, 6, 6, 6, 6, 6, 6, 4, 4, 6, 6, 6, 6, 6, 6, 6, 6, 6, 4, 4, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 4, 4, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 4

Offset: 1

Omar E. Pol, Nov 23 2016

Apart from the left border and the right border, the rest of the elements are 6's.

For the same idea but for a right triangle see A278480; for a square array see A278545, for a square spiral see A010731; and for a hexagonal spiral see A010722.

			The sequence written as an isosceles triangle begins:
.
.                     2;
.                   4,  4;
.                 4,  6,  4;
.               4,  6,  6,  4;
.             4,  6,  6,  6,  4;
.           4,  6,  6,  6,  6,  4;
.         4,  6,  6,  6,  6,  6,  4;
.       4,  6,  6,  6,  6,  6,  6,  4;
.     4,  6,  6,  6,  6,  6,  6,  6,  4;
.   4,  6,  6,  6,  6,  6,  6,  6,  6,  4;
...

Row sums give A016933.
Left border gives A040002, the same as the right border.
Middle column gives the elements > 1 of A134201, also twice A122553.
Cf. A010722, A010731, A275015, A278317, A278480, A278545.

cp's OEIS Frontend

A008588 Nonnegative multiples of 6.

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A144223 Number of ways of placing n labeled balls into n unlabeled (but 6-colored) boxes.

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A176398 Decimal expansion of 3+sqrt(10).

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A020793 Decimal expansion of 1/6.

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A023005 Number of partitions of n into parts of 6 kinds.

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A141631 a(n) = 3*n^2 - 4*n + 3.

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A010726 Period 2: repeat (6,10).

A141631 a(n) = 3n^2 - 4n + 3.