A016130 -id:A016130 - cp's OEIS frontend

A036239 Number of 2-element intersecting families of an n-element set; number of 2-way interactions when 2 subsets of power set on {1..n} are chosen at random.

0, 2, 15, 80, 375, 1652, 7035, 29360, 120975, 494252, 2007555, 8120840, 32753175, 131818052, 529680075, 2125927520, 8525298975, 34165897052, 136857560595, 548011897400, 2193792030375, 8780400395252, 35137296305115, 140596265198480

Offset: 1

N. J. A. Sloane

Let P(A) be the power set of an n-element set A. Then a(n) = the number of pairs of elements {x,y} of P(A) for which either 0) x and y are intersecting but for which x is not a subset of y and y is not a subset of x, or 1) x and y are intersecting and for which either x is a proper subset of y or y is a proper subset of x. - Ross La Haye, Jan 10 2008

Graph theory formulation. Let P(A) be the power set of an n-element set A. Then a(n) = the number of edges in the intersection graph G of P(A). The vertices of G are the elements of P(A) and the edges of G are the pairs of elements {x,y} of P(A) such that x and y are intersecting (and x <> y). - Ross La Haye, Dec 23 2017

W. W. Kokko, "Interactions", manuscript, 1983.

T. D. Noe, Table of n, a(n) for n = 1..200
Taylor Brysiewicz, Holger Eble, and Lukas Kühne, Enumerating chambers of hyperplane arrangements with symmetry, arXiv:2105.14542 [math.CO], 2021.
V. Jovovic and G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, in Russian, Diskretnaya Matematika, 11 (1999), no. 4, 127-138.
V. Jovovic and G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, English translation, in Discrete Mathematics and Applications, 9, (1999), no. 6.
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
Thomas Wieder, The number of certain k-combinations of an n-set, Applied Mathematics Electronic Notes, vol. 8 (2008).
Index entries for linear recurrences with constant coefficients, signature (10,-35,50,-24).

Cf. A036240, A051180-A051185, A028243, A032263.

Cf. A003462, A006516, A016127, A016129, A016130, A016311, A016316, A016321, A016325. - Zerinvary Lajos, Jun 05 2009

LinearRecurrence[{10,-35,50,-24},{0,2,15,80},40] (* or *) With[{c=1/2!}, Table[ c(4^n-3^n-2^n+1),{n,40}]] (* Harvey P. Dale, May 11 2011 *)

a(n)=(4^n-3^n-2^n+1)/2 \\ Charles R Greathouse IV, Jul 25 2011

[(4^n - 2^n)/2-(3^n - 1)/2 for n in range(1,24)] # Zerinvary Lajos, Jun 05 2009

a(n) = (1/2) * (4^n - 3^n - 2^n + 1).
a(n) = 3*Stirling2(n+1,4) + 2*Stirling2(n+1,3). - Ross La Haye, Jan 10 2008
a(n) = A006516(n) - A003462(n). - Zerinvary Lajos, Jun 05 2009
From Harvey P. Dale, May 11 2011: (Start)
a(n) = 10*a(n-1) - 35*a(n-2) + 50*a(n-3) - 24*a(n-4); a(0)=0, a(1)=2, a(2)=15, a(3)=80.
G.f.: x^2*(2-5*x)/(1 - 10*x + 35*x^2 - 50*x^3 + 24*x^4). (End)
E.g.f.: exp(x)*(exp(x) - 1)^2*(exp(x) + 1)/2. - Stefano Spezia, Jun 26 2022

A367300 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 3 + 2x, p(n,x) = up(n-1,x) + vp(n-2,x) for n >= 3, where u = p(2,x), v = 1 - 2x - x^2.

Original entry on oeis.org

1, 3, 2, 10, 10, 3, 33, 46, 22, 4, 109, 194, 131, 40, 5, 360, 780, 678, 296, 65, 6, 1189, 3036, 3228, 1828, 581, 98, 7, 3927, 11546, 14514, 10100, 4194, 1036, 140, 8, 12970, 43150, 62601, 51664, 26479, 8604, 1722, 192, 9, 42837, 159082, 261598, 249720, 152245, 61318, 16248, 2712, 255, 10

Offset: 1

Clark Kimberling, Dec 23 2023

Because (p(n,x)) is a strong divisibility sequence, for each integer k, the sequence (p(n,k)) is a strong divisibility sequence of integers.

			First eight rows:
     1
     3      2
    10     10      3
    33     46     22      4
   109    194    131     40     5
   360    780    678    296    65     6
  1189   3036   3228   1828   581    98    7
  3927  11546  14514  10100  4194  1036  140  8
Row 4 represents the polynomial p(4,x) = 33 + 46*x + 22*x^2 + 4*x^3, so (T(4,k)) = (33,46,22,4), k=0..3.

Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), Paper No. A14.

Cf. A006190 (column 1); A000027 (p(n,n-1)); A107839 (row sums, p(n,1)); A001045 (alternating row sums, p(n,-1)); A030240 (p(n,2)); A039834 (signed Fibonacci numbers, p(n,-2)); A016130 (p(n,3)); A225883 (p(n,-3)); A099450 (p(n,-4)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367299.

p[1, x_] := 1; p[2, x_] := 3 + 2 x; u[x_] := p[2, x]; v[x_] := 1 - 2 x - x^2;
p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]

p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where p(1,x) = 1, p(2,x) = 3 + 2*x, u = p(2,x), and v = 1 - 2*x - x^2.

p(n,x) = k*(b^n - c^n), where k = -(1/sqrt(13 + 4*x)), b = (1/2) (2*x + 3 + 1/k), c = (1/2) (2*x + 3 - 1/k).

A016311 Expansion of 1/((1-2x)(1-7x)(1-8*x)).

Original entry on oeis.org

1, 17, 203, 2101, 20163, 184821, 1643251, 14298917, 122461955, 1036190485, 8684988819, 72248167173, 597363137827, 4914549713909, 40265910006707, 328773866154469, 2676717032006979, 21739418975585493

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (17,-86,112).

Cf. A016130, A016131. - Zerinvary Lajos, Jun 05 2009

[(160*8^n-147*7^n+2*2^n)/15: n in [0..20]]; // Vincenzo Librandi, Sep 02 2011

CoefficientList[Series[1/((1-2x)(1-7x)(1-8x)),{x,0,30}],x] (* or *) LinearRecurrence[{17,-86,112},{1,17,203},30] (* Harvey P. Dale, Jul 12 2012 *)

[(8^n - 2^n)/6-(7^n - 2^n)/5 for n in range(2,21)] # Zerinvary Lajos, Jun 05 2009

a(n) = A016131(n+1) - A016130(n+1). - Zerinvary Lajos, Jun 05 2009
a(n) = 4*8^(n+1)/3 - 7^(n+2)/5 + 2^(n+1)/15. - R. J. Mathar, Mar 14 2011
From Vincenzo Librandi, Sep 02 2011: (Start)
a(n) = (160*8^n - 147*7^n + 2*2^n)/15;
a(n) = 15*a(n-1) - 56*a(n-2) + 2^n. (End)
a(n) = 17*a(n-1) - 86*a(n-2) + 112*a(n-3), with a(0)=1, a(1)=17, a(2)=203. - Harvey P. Dale, Jul 12 2012

A074602 a(n) = 2^n + 7^n.

Original entry on oeis.org

2, 9, 53, 351, 2417, 16839, 117713, 823671, 5765057, 40354119, 282476273, 1977328791, 13841291297, 96889018599, 678223089233, 4747561542711, 33232930635137, 232630514118279, 1628413598172593, 11398895185897431

Offset: 0

Robert G. Wilson v, Aug 25 2002

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-14).

Cf. A000051, A034472, A052539, A034474, A062394, A034491, A062395, A062396, A007689, A063376, A063481, A074600 - A074624.

[2^n + 7^n: n in [0..35]]; // Vincenzo Librandi, Apr 30 2011

Table[2^n + 7^n, {n, 0, 25}]
LinearRecurrence[{9,-14},{2,9},20] (* Harvey P. Dale, May 10 2025 *)

a(n) = 7*a(n-1) - 2^n = 9*a(n-1) - 14*a(n-2).
From Mohammad K. Azarian, Jan 02 2009: (Start)
G.f.: 1/(1-2*x) + 1/(1-7*x).
E.g.f.: e^(2*x) + e^(7*x). (End)
a(n) = 2*A016130(n)-9*A016130(n-1). - R. J. Mathar, Mar 10 2022

A255242 Calculate the aliquot parts of a number n and take their sum. Then repeat the process calculating the aliquot parts of all the previous aliquot parts and add their sum to the previous one. Repeat the process until the sum to be added is zero. Sequence lists these sums.

Original entry on oeis.org

0, 1, 1, 4, 1, 8, 1, 12, 5, 10, 1, 30, 1, 12, 11, 32, 1, 36, 1, 38, 13, 16, 1, 92, 7, 18, 19, 46, 1, 74, 1, 80, 17, 22, 15, 140, 1, 24, 19, 116, 1, 90, 1, 62, 51, 28, 1, 256, 9, 62, 23, 70, 1, 136, 19, 140, 25, 34, 1, 286, 1, 36, 61, 192, 21, 122, 1, 86, 29, 114

Offset: 1

Paolo P. Lava, Feb 19 2015

nonn

a(n) = 1 if n is prime.

			The aliquot parts of 8 are 1, 2, 4 and their sum is 7.
Now, let us calculate the aliquot parts of 1, 2 and 4:
1 => 0;  2 => 1;  4 => 1, 2.  Their sum is 0 + 1 + 1 + 2 = 4.
Let us calculate the aliquot parts of 1, 1, 2:
1 => 0;  1 = > 0; 2 => 1. Their sum is 1.
We have left 1: 1 => 0.
Finally, 7 + 4 + 1 = 12. Therefore a(8) = 12.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Paolo P. Lava)
Jon Maiga, Computer-generated formulas for A255242, Sequence Machine.

Cf. A001047, A001065, A001787, A006516, A016127, A016130, A016135, A255243, A330575.

Sequences that appear in the convolution formulas: A000203, A001065, A051953, A062790, A067824, A074206, A157658, A174725, A174726, A191161, A211779, A245211, A323910, A253249.

with(numtheory): P:=proc(q) local a,b,c,k,n,t,v;
for n from 1 to q do b:=0; a:=sort([op(divisors(n))]); t:=nops(a)-1;
while add(a[k],k=1..t)>0 do b:=b+add(a[k],k=1..t); v:=[];
for k from 2 to t do c:=sort([op(divisors(a[k]))]); v:=[op(v),op(c[1..nops(c)-1])]; od;
a:=v; t:=nops(a); od; print(b); od; end: P(10^3);

f[s_] := Flatten[Most[Divisors[#]] & /@ s]; a[n_] := Total@Flatten[FixedPointList[ f, {n}]] - n; Array[a, 100] (* Amiram Eldar, Apr 06 2019 *)

ali(n) = setminus(divisors(n), Set(n));
a(n) = my(list = List(), v = [n]); while (#v, my(w = []); for (i=1, #v, my(s=ali(v[i])); for (j=1, #s, w = concat(w, s[j]); listput(list, s[j]));); v = w;); vecsum(Vec(list)); \\ Michel Marcus, Jul 15 2023

a(1) = 0.
a(2^k) = k*2^(k-1) = A001787(k), for k>=1.
a(n^k) = (n^k-2^k)/(n-2), for n odd prime and k>=1.
In particular:
a(3^k)  = A001047(k-1);
a(5^k)  = A016127(k-1);
a(7^k)  = A016130(k-1);
a(11^k) = A016135(k-1).
From Antti Karttunen, Nov 22 2024: (Start)
a(n) = A330575(n) - n.
Also, following formulas were conjectured by Sequence Machine:
a(n) = (A191161(n)-n)/2.
a(n) = Sum_{d|n} A001065(d)*A074206(n/d). [Compare to David A. Corneth's Apr 13 2020 formula for A330575]
a(n) = Sum_{d|n} A051953(d)*A067824(n/d).
a(n) = Sum_{d|n} A000203(d)*A174726(n/d).
a(n) = Sum_{d|n} A062790(d)*A253249(n/d).
a(n) = Sum_{d|n} A157658(d)*A191161(n/d).
a(n) = Sum_{d|n} A174725(d)*A211779(n/d).
a(n) = Sum_{d|n} A245211(d)*A323910(n/d).
(End)

A190540 a(n) = 7^n - 2^n.

Original entry on oeis.org

0, 5, 45, 335, 2385, 16775, 117585, 823415, 5764545, 40353095, 282474225, 1977324695, 13841283105, 96889002215, 678223056465, 4747561477175, 33232930504065, 232630513856135, 1628413597648305, 11398895184848855, 79792266296563425, 558545864081186855, 3909821048578793745

Offset: 0

Vincenzo Librandi, Jun 02 2011

Length-n words from letters {1,2,...,7} with at least one letter >2. [Joerg Arndt, Jun 02 2011]

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (9,-14).

Cf. A016130, A016169, A121213.

Magma
```
[7^n -2^n: n in [0..30]];
```

CoefficientList[Series[5 x/((1 - 2 x) (1 - 7 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 04 2014 *)

a(n)=7^n-1<Charles R Greathouse IV, Jun 08 2011

a(n) = 9*a(n-1) - 14*a(n-2).
G.f.: 5*x/((1-2*x)*(1-7*x)). - Vincenzo Librandi, Oct 04 2014
a(n) = 5*A016130(n-1). - R. J. Mathar, Mar 10 2022
E.g.f.: exp(2*x)*(exp(5*x) - 1). - Elmo R. Oliveira, Sep 10 2024

A016304 Expansion of 1/((1-2x)(1-6x)(1-7*x)).

Original entry on oeis.org

1, 15, 157, 1419, 11869, 94731, 733069, 5551323, 41378557, 304766187, 2224062061, 16112628987, 116053574365, 831966057483, 5941308640333, 42294437942811, 300292730428093, 2127439102098219, 15044413649559085

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (15,-68,84).

Cf. A016129, A016130, A016311, A016316, A016321, A016325. - Zerinvary Lajos, Jun 05 2009

[ n eq 1 select 1 else n eq 2 select 15 else n eq 3 select 157 else 15*Self(n-1)-68*Self(n-2) +84*Self(n-3): n in [1..20] ]; // Vincenzo Librandi, Aug 25 2011

CoefficientList[Series[1/((1-2x)(1-6x)(1-7x)), {x, 0, 30}], x] (* or *) LinearRecurrence[{15, -68, 84}, {1, 15, 157}, 30]

Vec(1/((1-2*x)*(1-6*x)*(1-7*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

[(7^n - 2^n)/5-(6^n - 2^n)/4 for n in range(2,21)] # Zerinvary Lajos, Jun 05 2009

a(n) = (7^(n+2) - 2^(n+2))/5-(6^(n+2) - 2^(n+2))/4. - Zerinvary Lajos, Jun 05 2009 [corrected by Joerg Arndt, Aug 25 2011]
From Vincenzo Librandi, Aug 25 2011: (Start)
a(n) = 15*a(n-1) - 68*a(n-2) + 84*a(n-3) for n > 2;
a(n) = 13*a(n-1) - 42*a(n-2) + 2^n for n > 1. (End)
E.g.f.: exp(2*x)*(1 - 45*exp(4*x) + 49*exp(5*x))/5. - Stefano Spezia, Aug 25 2025

A016282 Expansion of 1/((1-2x)(1-4x)(1-5*x)).

Original entry on oeis.org

1, 11, 83, 535, 3171, 17871, 97483, 520055, 2731091, 14179231, 72992283, 373347975, 1900290211, 9635660591, 48715157483, 245723238295, 1237206060531, 6220389909951, 31239388241083, 156746696495015, 785932504682051, 3938458614335311, 19727477439571083

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (11,-38,40).

Cf. A006516, A016127, A016129, A016130, A016311, A016316, A016321, A016325. - Zerinvary Lajos, Jun 05 2009

CoefficientList[ Series[ 1/((1 - 2x)(1 - 4x)(1 - 5x)), {x, 0, 20} ], x ]
LinearRecurrence[{11,-38,40},{1,11,83},30] (* Harvey P. Dale, Nov 29 2022 *)

Vec(1/((1-2*x)*(1-4*x)*(1-5*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

[(5^n - 2^n)/3-(4^n - 2^n)/2 for n in range(2,21)] # Zerinvary Lajos, Jun 05 2009

a(n) = (2/3)*2^n - 8*(4)^n + (25/3)*5^n. - Antonio Alberto Olivares, May 12 2012

A016295 Expansion of 1/((1-2x)(1-5x)(1-6x)).

Original entry on oeis.org

1, 13, 117, 905, 6461, 43953, 289717, 1868425, 11861421, 74423393, 462815717, 2858273145, 17556537181, 107373722833, 654414852117, 3977351721065, 24118423433741, 145982106270273, 882250466222917

Offset: 0

N. J. A. Sloane

nonn

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (13,-52,60).

Cf. A016127, A016129, A016130, A016311, A016316, A016321, A016325. - Zerinvary Lajos, Jun 05 2009

LinearRecurrence[{13,-52,60},{1,13,117},20] (* Harvey P. Dale, Mar 26 2016 *)

[(6^n - 2^n)/4-(5^n - 2^n)/3 for n in range(2,21)] # Zerinvary Lajos, Jun 05 2009

a(n) = A016129(n+1) - A016127(n+1). - Zerinvary Lajos, Jun 05 2009
a(n) = 13*a(n-1) - 52*a(n-2) + 60*a(n-3), n >= 3.
a(n) = 11*a(n-1) - 30*a(n-2) + 2^n, n >= 2. - Vincenzo Librandi, Mar 16 2011
a(n) = 7*a(n-1) - 10*a(n-2) + 6^n, n >= 2. - Vincenzo Librandi, Mar 16 2011
a(n) = 8*a(n-1) - 12*a(n-2) + 5^n, n >= 2. - Vincenzo Librandi, Mar 16 2011
a(n) = -5^(n+2)/3 + 9*6^n + 2^n/3. - R. J. Mathar, Mar 18 2011

A016633 Expansion of g.f. 1/((1-2x)(1-11x)(1-12*x)).

Original entry on oeis.org

1, 25, 447, 6989, 101759, 1417941, 19180519, 253983853, 3309800367, 42599540357, 542895780791, 6863463633117, 86197420501375, 1076563471968373, 13382900349107463, 165700329729679181, 2044564737700501583, 25152545442794015589, 308625999807796411735, 3778261997130507936445

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..900
Index entries for linear recurrences with constant coefficients, signature (25,-178,264).

Cf. A006516, A016127, A016129, A016130, A016311, A016316, A016321, A016325. - Zerinvary Lajos, Jun 05 2009

Cf. A016135, A016136.

[(648*12^n +2^(n+1)-5*11^(n+2))/45 : n in [0..20]]; // Vincenzo Librandi, Oct 09 2011

CoefficientList[Series[1/((1 - 2 x) (1 - 11 x) (1 - 12 x)), {x, 0, 15}], x] (* Michael De Vlieger, Jan 31 2018 *)

Vec(1/((1-2*x)*(1-11*x)*(1-12*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

[(12^n - 2^n)/10-(11^n - 2^n)/9 for n in range(2,18)] # Zerinvary Lajos, Jun 05 2009

From Vincenzo Librandi, Oct 09 2011: (Start)
a(n) = (648*12^n + 2^(n+1) - 5*11^(n+2))/45.
a(n) = 23*a(n-1) - 132*a(n-2) + 2^n.
a(n) = 25*a(n-1) - 178*a(n-2) + 264*a(n-3), n >= 3. (End)
From Elmo R. Oliveira, Mar 26 2025: (Start)
E.g.f.: exp(2*x)*(648*exp(10*x) - 605*exp(9*x) + 2)/45.
a(n) = A016136(n+1) - A016135(n+1). (End)

cp's OEIS Frontend

A036239 Number of 2-element intersecting families of an n-element set; number of 2-way interactions when 2 subsets of power set on {1..n} are chosen at random.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Sage

Formula

A367300 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 3 + 2*x, p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where u = p(2,x), v = 1 - 2*x - x^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A016311 Expansion of 1/((1-2*x)*(1-7*x)*(1-8*x)).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

A074602 a(n) = 2^n + 7^n.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A255242 Calculate the aliquot parts of a number n and take their sum. Then repeat the process calculating the aliquot parts of all the previous aliquot parts and add their sum to the previous one. Repeat the process until the sum to be added is zero. Sequence lists these sums.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A190540 a(n) = 7^n - 2^n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A016304 Expansion of 1/((1-2*x)*(1-6*x)*(1-7*x)).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

A367300 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 3 + 2x, p(n,x) = up(n-1,x) + vp(n-2,x) for n >= 3, where u = p(2,x), v = 1 - 2x - x^2.

A016311 Expansion of 1/((1-2x)(1-7x)(1-8*x)).

A016304 Expansion of 1/((1-2x)(1-6x)(1-7*x)).

A016282 Expansion of 1/((1-2x)(1-4x)(1-5*x)).

A016633 Expansion of g.f. 1/((1-2x)(1-11x)(1-12*x)).