A010722 -id:A010722 - cp's OEIS frontend

A278545 Number of neighbors of the n-th term in a full square array read by antidiagonals.

3, 5, 5, 5, 8, 5, 5, 8, 8, 5, 5, 8, 8, 8, 5, 5, 8, 8, 8, 8, 5, 5, 8, 8, 8, 8, 8, 5, 5, 8, 8, 8, 8, 8, 8, 5, 5, 8, 8, 8, 8, 8, 8, 8, 5, 5, 8, 8, 8, 8, 8, 8, 8, 8, 5, 5, 8, 8, 8, 8, 8, 8, 8, 8, 8, 5, 5, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 5, 5, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 5, 5, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 5

Offset: 1

Omar E. Pol, Nov 23 2016

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

For the same idea but for a right triangle see A278480; for an isosceles triangle see A278481; for a square spiral see A010731; and for a hexagonal spiral see A010722.

			The corner of the square array begins:
3,5,5,5,5,5,5,5,5,5,...
5,8,8,8,8,8,8,8,8,...
5,8,8,8,8,8,8,8,...
5,8,8,8,8,8,8,...
5,8,8,8,8,8,...
5,8,8,8,8,...
5,8,8,8,...
5,8,8,...
5,8,...
5,...
...

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

Antidiagonal sums give 3 together with the elements > 2 of A017089.

Cf. A010722, A010731, A274912, A274913, A278317, A278290, A278480, A278481.

3, seq(op([5,8$i,5]),i=0..20); # Robert Israel, Dec 04 2016

G.f. 3+x+8*x/(1-x)-3*(1+x)*Theta_2(0,sqrt(x))/(2*x^(1/8)) where Theta_2 is a Jacobi Theta function. - Robert Israel, Dec 04 2016

A349577 Decimal expansion of the volume of the solid formed by the intersection of 4 right circular unit-diameter cylinders whose axes pass through the diagonals of a cube.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Nov 22 2021

Equivalently, the axes of the cylinders can be placed along the lines joining the vertices of a regular tetrahedron with the centers of the faces on the opposite sides.
This constant was first calculated by Moore (1974).
The corresponding volumes in the analogous cases of 2 and 3 mutually orthogonal cylinders are 2/3 (A010722) and 2 - sqrt(2) (A101465), respectively.

			0.56840607294451799910914006057025714776009440514583...

Paul Bourke, Intersecting cylinders, 2003-2016.
Moreton Moore, Symmetrical Intersections of Right Circular Cylinders, The Mathematical Gazette, Vol. 58, No. 405 (1974), pp. 181-185.
Eric Weisstein's World of Mathematics, Steinmetz Solid.
Wikipedia, Steinmetz solid.

Cf. A010722, A101465, A349578, A349579, A349580.

RealDigits[(3/2) * Sqrt[2] * (2 - Sqrt[3]), 10, 100][[1]]

Equals (3/2) * sqrt(2) * (2 - sqrt(3)).

A145429 Decimal expansion of Sum_{n > 0} n*(n!)^2/(2n)!.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Feb 08 2009

Also, decimal expansion of Sum_{n >= 0} n/binomial(2*n, n). - Bruno Berselli, Sep 14 2015

			1.069733192052..

Alexander Apelblat, Tables of Integrals and Series, Harri Deutsch, (1996), 4.1.35.

Cf. A010722 (decimal expansion of Sum_{n >= 0} n/binomial(2*n+1, n)).

Cf. A000796, A002194, A021139.

Maple
```
2/3+2/27*Pi*3^(1/2) ;
```

RealDigits[2/3 + 2*Pi/(9*Sqrt[3]), 10, 100][[1]] (* Amiram Eldar, Nov 16 2021 *)

Equals 10*A021139*(9+A000796*A002194).
From Amiram Eldar, Nov 16 2021: (Start)
Equals 2/3 + 2*Pi/(9*sqrt(3)).
Equals 1 + Integral_{x>=1} 1/(x^2 + x + 1)^2 dx. (End)

A162594 Differences of cubes: T(n,n) = n^3, T(n,k) = T(n,k+1) - T(n-1,k), 0 <= k < n, triangle read by rows.

Original entry on oeis.org

0, 1, 1, 6, 7, 8, 6, 12, 19, 27, 0, 6, 18, 37, 64, 0, 0, 6, 24, 61, 125, 0, 0, 0, 6, 30, 91, 216, 0, 0, 0, 0, 6, 36, 127, 343, 0, 0, 0, 0, 0, 6, 42, 169, 512, 0, 0, 0, 0, 0, 0, 6, 48, 217, 729, 0, 0, 0, 0, 0, 0, 0, 6, 54, 271, 1000, 0, 0, 0, 0, 0, 0, 0, 0, 6, 60, 331, 1331

Offset: 0

Reinhard Zumkeller, Jul 07 2009

T(n,n) = A000578(n);
T(n,n-1) = A003215(n-1), n > 0;
T(n,n-2) = A008588(n-2), n > 1;
T(n,n-3) = A010722(n-3), n > 2;
T(n,n-j) = A000004(n-j), 4 <= j <= n;
for n > 2: sum of n-th row = (n+1)^3.

			Triangle begins:
  0,
  1,  1,
  6,  7,  8,
  6, 12, 19, 27,
  0,  6, 18, 37, 64,
  0,  0,  6, 24, 61, 125,
  ...

G. C. Greubel, Rows n = 0..99 of triangle, flattened

Cf. A162593 (differences of squares).

T[n_, n_] := n^3; T[n_, k_] := T[n, k] = T[n, k + 1] - T[n - 1, k]; Table[T[n, k], {n, 0, 15}, {k, 0, n}] // Flatten (* G. C. Greubel, Jul 04 2018 *)

T(n, k) = if (k==n, n^3, T(n, k+1) - T(n-1, k));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n,k), ", ")); print); \\ Michel Marcus, Jul 05 2018

A343052 Table read by ascending antidiagonals: T(k, n) is the minimum vertex sum in a perimeter-magic k-gon of order n.

Original entry on oeis.org

6, 12, 6, 15, 10, 6, 24, 15, 12, 6, 28, 21, 15, 10, 6, 40, 28, 24, 15, 12, 6, 45, 36, 28, 21, 15, 10, 6, 60, 45, 40, 28, 24, 15, 12, 6, 66, 55, 45, 36, 28, 21, 15, 10, 6, 84, 66, 60, 45, 40, 28, 24, 15, 12, 6, 91, 78, 66, 55, 45, 36, 28, 21, 15, 10, 6, 112, 91, 84, 66, 60, 45, 40, 28, 24, 15, 12, 6

Offset: 3

Stefano Spezia, Apr 03 2021

			The table begins:
k\n|   3   4   5   6   7 ...
---+--------------------
3  |   6   6   6   6   6 ...
4  |  12  10  12  10  12 ...
5  |  15  15  15  15  15 ...
6  |  24  21  24  21  24 ...
7  |  28  28  28  28  28 ...
...

Terrel Trotter, Perimeter-Magic Polygons, Journal of Recreational Mathematics Vol. 7, No. 1, 1974, pp. 14-20 (see equations 5 and 7).

Cf. A000217 (n = 4), A010722 (k = 3), A010854 (k = 5), A010867 (k = 7), A265225, A343053 (maximum).

T[k_,n_]:=k(1+k+Mod[n,2](1-Mod[k,2]))/2; Table[T[k+3-n,n],{k,3,14},{n,3,k}]//Flatten

O.g.f.: x*(1 + x^2 + y + x*(2 + 3*y))/((1 - x)^3*(1 + x)^2*(1 - y^2)).
E.g.f.: x*((5 + 2*x)*cosh(x + y) - cosh(x - y) + 2*(2 + x)*sinh(x + y))/4.
T(k, n) = k*(1 + k + (n mod 2)*(1 - (k mod 2)))/2.
T(k, 3) = A265225(k-1) (conjectured).

A021019 Decimal expansion of 1/15.

Original entry on oeis.org

0, 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

			0.06666666666666666666666666666666666666666666666666666...

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

Cf. A010722, A020793.

From Elmo R. Oliveira, Aug 05 2024: (Start)
G.f.: 6*x/(1-x).
E.g.f.: 6*(exp(x) - 1).
a(n) = 6 for n >= 1. (End)

A176355 Periodic sequence: Repeat 6, 1.

Original entry on oeis.org

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

Offset: 0

Klaus Brockhaus, Apr 15 2010

Interleaving of A010722 and A000012.
Also continued fraction expansion of 3+sqrt(15).
Also decimal expansion of 61/99.
Essentially first differences of A047335.
Binomial transform of 6 followed by A166577 without initial terms 1, 4.
Inverse binomial transform of A005009 preceded by 6.

			0.6161616161616161616161616161616161616161...

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

Cf. A010722 (all 6's sequence), A000012 (all 1's sequence), A092294 (decimal expansion of 3+sqrt(15)), A010687 (repeat 1, 6), A047335 (congruent to 0 or 6 mod 7), A166577, A005009 (7*2^n).

Magma
```
&cat[ [6, 1]: n in [0..52] ];
```
Magma
```
[(7+5*(-1)^n)/2: n in [0..104]];
```

PadRight[{},120,{6,1}] (* Harvey P. Dale, Apr 12 2018 *)

G.f.: (6 + x)/(1 - x^2).
a(n) = (7 + 5*(-1)^n)/2.
a(n) = a(n-2) for n>1, a(0)=6, a(1)=1.
a(n) = -a(n-1)+7 for n>0, a(0)=6.
a(n) = 6*((n+1) mod 2) + (n mod 2).
a(n) = A010687(n+1).
a(n) = 13^n mod 7. - Vincenzo Librandi, Jun 01 2016
From Amiram Eldar, Jan 01 2023: (Start)
Multiplicative with a(2^e) = 6, and a(p^e) = 1 for p >= 3.
Dirichlet g.f.: zeta(s)*(1+5/2^s). (End)
E.g.f.: 6*cosh(x) + sinh(x). - Stefano Spezia, Feb 09 2025

A347769 a(0) = 0; a(1) = 1; for n > 1, a(n) = A001065(a(n-1)) = sigma(a(n-1)) - a(n-1) (the sum of aliquot parts of a(n-1)) if this is not yet in the sequence; otherwise a(n) is the smallest number missing from the sequence.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 16, 15, 13, 14, 17, 18, 21, 19, 20, 22, 23, 24, 36, 55, 25, 26, 27, 28, 29, 30, 42, 54, 66, 78, 90, 144, 259, 45, 33, 31, 32, 34, 35, 37, 38, 39, 40, 50, 43, 41, 44, 46, 47, 48, 76, 64, 63, 49, 51, 52, 53, 56, 57, 58, 59, 60, 108, 172

Offset: 0

Eric Chen, Sep 13 2021

nonn

This sequence is a permutation of the nonnegative integers iff Catalan's aliquot sequence conjecture (also called Catalan-Dickson conjecture) is true.
a(563) = 276 is the smallest number whose aliquot sequence has not yet been fully determined.
As long as the aliquot sequence of 276 is not known to be finite or eventually periodic, a(563+k) = A008892(k).

			a(0) = 0, a(1) = 1;
since A001065(a(1)) = 0 has already appeared in this sequence, a(2) = 2;
since A001065(a(2)) = 1 has already appeared in this sequence, a(3) = 3;
...
a(11) = 11;
since A001065(a(11)) = 1 has already appeared in this sequence, a(12) = 12;
since A001065(a(12)) = 16 has not yet appeared in this sequence, a(13) = A001065(a(12)) = 16;
since A001065(a(13)) = 15 has not yet appeared in this sequence, a(14) = A001065(a(13)) = 15;
since A001065(a(14)) = 9 has already appeared in this sequence, a(15) = 13;
...

Eric Chen, Table of n, a(n) for n = 0..2703 (all currently known terms, after the b-file of A008892)
Eric Weisstein's World of Mathematics, Aliquot sequence
Eric Weisstein's World of Mathematics, Catalan's Aliquot Sequence Conjecture
Wikipedia, Aliquot sequence

Cf. A032451.
Cf. A001065 (sum of aliquot parts).
Cf. A003023, A044050, A098007, A098008: ("length" of aliquot sequences, four versions).
Cf. A007906.
Cf. A115060 (maximum term of aliquot sequences).
Cf. A115350 (termination of the aliquot sequences).
Cf. A098009, A098010 (records of "length" of aliquot sequences).
Cf. A290141, A290142 (records of maximum term of aliquot sequences).
Cf. A000396, A063990, A122726, A206708, A347770.
Cf. A080907, A063769, A121507, A121508, A131884, A126016.
Aliquot sequences starting at various numbers: A000004 (0), A000007 (1), A033322 (2), A010722 (6), A143090 (12), A143645 (24), A010867 (28), A008885 (30), A143721 (38), A008886 (42), A143722 (48), A143723 (52), A008887 (60), A143733 (62), A143737 (68), A143741 (72), A143754 (75), A143755 (80), A143756 (81), A143757 (82), A143758 (84), A143759 (86), A143767 (87), A143846 (88), A143847 (96), A143919 (100), A008888 (138), A008889 (150), A008890 (168), A008891 (180), A203777 (220), A008892 (276), A014360 (552), A014361 (564), A074907 (570), A014362 (660), A269542 (702), A045477 (840), A014363 (966), A014364 (1074), A014365 (1134), A074906 (1521), A143930 (3630), A072891 (12496), A072890 (14316), A171103 (46758), A072892 (1264460).

A347769_list(N)=print1(0, ", "); if(N>0, print1(1, ", ")); v=[0, 1]; b=1; for(n=2, N, if(setsearch(Set(v), sigma(b)-b), k=1; while(k<=n, if(!setsearch(Set(v), k), b=k; k=n+1, k++)), b=sigma(b)-b); print1(b, ", "); v=concat(v, b))

A387235 Decimal expansion of 2*log(2)/3.

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Aug 23 2025

Area enclosed by the curve of the equation x^6 + y^6 - x^3*y + x*y^3 = 0.

The asymptotic mean of A256232. - Amiram Eldar, Aug 23 2025

			0.46209812037329687294482141430545104538366675624...

Michael I. Shamos, A catalog of the real numbers, (2007). See p. 436.
Index entries for transcendental numbers.

Cf. A002162, A001504, A010701, A010722, A016627, A193535, A256232.

Mathematica
```
RealDigits[2Log[2]/3,10,100][[1]]
```

Equals A002162*A010722.
Equals log(4)/3 = A010701*A016627.
Equals Sum_{k>=0} (-1)^k/((3*k + 1)*(3*k + 2)) = Integral_{x=0..1} x^2*log(1 + 1/x^3) = -Integral_{x=0..1} log[1 - x^6]/x^4. [Shamos]
Equals A016627/3 = 2*A193535. - Hugo Pfoertner, Aug 23 2025

A257936 Decimal expansion of 11/18.

Original entry on oeis.org

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

Offset: 0

Bruno Berselli, May 13 2015

Decimal expansion of Sum_{i>=1} 1/A028552(i).

Also, continued fraction expansion of 5+A001622.

			.6111111111111111111111111111111111111111111111111111111111111111...

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

Cf. A028552, A030880.

Cf. A010716 (decimal expansion of 5/9 = 10/18), A010722 (decimal expansion of 2/3 = 12/18).

Magma
```
[6,1^^90];
```
Mathematica
```
RealDigits[11/18, 10, 90][[1]]
```

11/18. \\ Charles R Greathouse IV, Apr 18 2016

Equals A020773 + A142464.
From Elmo R. Oliveira, Aug 05 2024: (Start)
G.f.: (6-5*x)/(1-x).
E.g.f.: exp(x) + 5.
a(n) = 1, n >= 1. (End)

cp's OEIS Frontend

A278545 Number of neighbors of the n-th term in a full square array read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

A349577 Decimal expansion of the volume of the solid formed by the intersection of 4 right circular unit-diameter cylinders whose axes pass through the diagonals of a cube.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A145429 Decimal expansion of Sum_{n > 0} n*(n!)^2/(2n)!.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Maple

Mathematica

Formula

A162594 Differences of cubes: T(n,n) = n^3, T(n,k) = T(n,k+1) - T(n-1,k), 0 <= k < n, triangle read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A343052 Table read by ascending antidiagonals: T(k, n) is the minimum vertex sum in a perimeter-magic k-gon of order n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A021019 Decimal expansion of 1/15.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A176355 Periodic sequence: Repeat 6, 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Magma

Mathematica

Formula

A347769 a(0) = 0; a(1) = 1; for n > 1, a(n) = A001065(a(n-1)) = sigma(a(n-1)) - a(n-1) (the sum of aliquot parts of a(n-1)) if this is not yet in the sequence; otherwise a(n) is the smallest number missing from the sequence.

Views

Author