A014695 -id:A014695 - cp's OEIS frontend

A141479 a(n) = A000111(n) + A014695(n).

2, 3, 3, 3, 6, 18, 63, 273, 1386, 7938, 50523, 353793, 2702766, 22368258, 199360983, 1903757313, 19391512146, 209865342978, 2404879675443, 29088885112833, 370371188237526, 4951498053124098, 69348874393137903, 1015423886506852353

Offset: 0

Paul Curtz, Aug 09 2008

a(n) is a multiple of 3 for n > 0.

A000111 := proc(n) local x; n!*coeftayl( sec(x)+tan(x),x=0,n) ; end: A014695 := proc(n) local x; coeftayl( (1+2*x+2*x^2+x^3)/(1-x^4),x=0,n) ; end: A141479 := proc(n) A000111(n)+A014695(n) ; end: for n from 0 to 30 do printf("%a,",A141479(n)) ; od; # R. J. Mathar, Sep 12 2008

from itertools import count, islice, accumulate
def A141479_gen(): # generator of terms
    yield from (2,3)
    blist = (0,1)
    for n in count(0):
        yield (blist := tuple(accumulate(reversed(blist),initial=0)))[-1] + (2,1,1,2)[n & 3]
A141479_list = list(islice(A141479_gen(),40)) # Chai Wah Wu, Jun 09-11 2022

Extended by R. J. Mathar, Sep 12 2008

A257942 a(n) = (n+1)*(n+2)/A014695(n+1), where A014695 is repeat (1, 2, 2, 1).

Original entry on oeis.org

1, 3, 12, 20, 15, 21, 56, 72, 45, 55, 132, 156, 91, 105, 240, 272, 153, 171, 380, 420, 231, 253, 552, 600, 325, 351, 756, 812, 435, 465, 992, 1056, 561, 595, 1260, 1332, 703, 741, 1560, 1640, 861, 903, 1892, 1980, 1035, 1081, 2256, 2352, 1225, 1275, 2652

Offset: 0

Paul Curtz, Jul 14 2015

Consider, for n >= 0, a sequence s(n). A useful transform is wi(n) = s(0), s(2), s(3), ..., i.e., s(n) without s(1).
For s(n) = 1/(n+1), wi(n)= 1, 1/3, 1/4, 1/5, ..., whose inverse binomial transform is f(n) = 1, -2/3, 7/12, -11/20, 8/15, -11/21, 29/56, -37/72, 23/45, -28/55, 67/132, -79/156, 46/91, -53/105, 121/240, -137/272, ...
The denominator of f(n) is a(n), for n >= 0.
If the numerator of f(n) is b(n), then it can be seen that b(n+1) = -(-1)^n* A226089(n).
Alternating a(n) - b(n) with a(n) + b(n) yields 0, 1, 5, 9, ... = A160050(n+1).
a(4n+1) is linked to the Rydberg-Ritz spectra of hydrogen.
h(n) = 0, 0, 1, 1, 4, 4, 3, 3, 8, 8, 5, 5, ... = duplicated A022998(n).
A022998(n) is linked to the Balmer series (see A246943(n)).
With an initial 0 and offset=0, a(-n) = a(n). Then (a(n+10) - a(n-10))/10 = 1, 6, 10, 7, 9, 22, 26, ... = g(n). a(n) mod 9 is of period 20.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-6,10,-12,12,-10,6,-3,1).

Cf. A002378(n+1), A014695(n+1)=A130658(n+2), A014634, A033567(n+1), A104188(n+1), 4*A007742(n+1), A160050 (in A226089), A022998, A109613, A000217(n+1), A246943.

A257942:=n->(n+1)*(n+2)/(3/2+(-1)^((2*n+7+(-1)^n)/4)/2): seq(A257942(n), n=0..100); # Wesley Ivan Hurt, Jul 18 2015

CoefficientList[Series[-(x^6 + 9 x^4 - 8 x^3 + 9 x^2 + 1)/((x - 1)^3 (x^2 + 1)^3), {x, 0, 50}], x] (* Michael De Vlieger, Jul 14 2015 *)
(* Using inverse binomial transform *) s[0]=1; s[n_] := 1/(n+2); f[n_] := Sum[(-1)^(n-k)*Binomial[n, k]*s[k], {k, 0, n}]; Table[f[n]//Denominator, {n, 0, 50}] (* Jean-François Alcover, Jul 14 2015 *)
LinearRecurrence[{3, -6, 10, -12, 12, -10, 6, -3, 1}, {1, 3, 12, 20, 15, 21, 56, 72, 45}, 55] (* Vincenzo Librandi, Jul 15 2015 *)

Vec(-(x^6+9*x^4-8*x^3+9*x^2+1)/((x-1)^3*(x^2+1)^3) + O(x^100)) \\ Colin Barker, Jul 14 2015

a(n)=(n+1)*(n+2)/if(n%4<2,2,1) \\ Charles R Greathouse IV, Jul 14 2015

a(4n)    = (2*n+1)*(4*n+1).
a(4n+1)  = (2*n+1)*(4*n+3).
a(4n+2)  = (4*n+3)*(4*n+4).
a(4n+3)  = (4*n+4)*(4*n+5).
a(n) = A064038(n+2) * (period 4: repeat 1, 1, 4, 4).
From Colin Barker, Jul 14 2015: (Start)
a(n) = (-1/8+i/8)*(((-3-i*3)+i*(-i)^n+i^n)*(2+3*n+n^2)) where i=sqrt(-1).
G.f.: -(x^6+9*x^4-8*x^3+9*x^2+1) / ((x-1)^3*(x^2+1)^3). (End)
a(n) = h(n+2) * A109613(n+1).
a(n) = (n+1)*(n+2) * period 4:repeat (1, 1, 2, 2) /2.
From Wesley Ivan Hurt, Jul 18 2015: (Start)
a(n) = (n+1)*(n+2)/(3/2+(-1)^((2*n+7+(-1)^n)/4)/2).
a(n) = 3*a(n-1)-6*a(n-2)+10*a(n-3)-12*a(n-4)+12*a(n-5)-10*a(n-6)+6*a(n-7)-3*a(n-8)+a(n-9), n>9. (End)
Sum_{n>=0} 1/a(n) = Pi/4 + 1. - Amiram Eldar, Aug 14 2022

A064038 Numerator of average number of swaps needed to bubble sort a string of n distinct letters.

Original entry on oeis.org

0, 1, 3, 3, 5, 15, 21, 14, 18, 45, 55, 33, 39, 91, 105, 60, 68, 153, 171, 95, 105, 231, 253, 138, 150, 325, 351, 189, 203, 435, 465, 248, 264, 561, 595, 315, 333, 703, 741, 390, 410, 861, 903, 473, 495, 1035, 1081, 564, 588, 1225, 1275, 663, 689, 1431, 1485, 770

Offset: 1

Antti Karttunen, Aug 23 2001

Denominators are given by the simple periodic sequence [1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, ...] (= A014695) thus we get an average of 1/2, 3/2, 3, 5, 15/2, 21/2, 14, 18, etc. swappings required to bubble sort a string of 2, 3, 4, 5, 6, ... letters.

E. Reingold, J. Nievergelt and N. Deo, Combinatorial Algorithms, Prentice-Hall, 1977, section 7.1, p. 287.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Eric Weisstein's World of Mathematics, Simple Graph.
Index entries for linear recurrences with constant coefficients, signature (3,-6,10,-12,12,-10,6,-3,1).

Cf. A014695 (denominators), A190186, A190187, A350660.

Cf. A000217, A001809, A007742, A014634, A026741, A033567, A033991, A060819, A061041.

[Numerator(n*(n-1)/4): n in [1..100]]; // G. C. Greubel, Sep 21 2018

Maple
```
[seq(numer((n*(n-1))/4), n=1..120)];
```

f[n_] := Numerator[n (n - 1)/4]; Array[f, 56]
f[n_] := n/GCD[n, 4]; Array[f[#] f[# - 1] &, 56]
LinearRecurrence[{3,-6,10,-12,12,-10,6,-3,1},{0,1,3,3,5,15,21,14,18},80] (* Harvey P. Dale, Jan 23 2023 *)

vector(100, n, numerator(n*(n-1)/4)) \\ G. C. Greubel, Sep 21 2018

a(n) = numerator(A001809(n)/(n!)).
a(4n) = A033991(n).
a(4n+1) = A007742(n).
a(4n+2) = A014634(n).
a(4n+3) = A033567(n+1).
a(n+1) = A061041(8*n-4). - Paul Curtz, Jan 03 2011
G.f.: -x^2*(1+4*x^3+x^6) / ( (x-1)^3*(1+x^2)^3 ). - R. J. Mathar, Jan 03 2011
a(n+1) = A060819(n)*A060819(n+1).
a(n+1) = A000217(n)/(period 4:repeat 2,1,1,2=A014695(n+2)=A130658(n+3)).
a(n) = 3*a(n-4) -3*a(n-8) +a(n-12). - Paul Curtz, Mar 04 2011
a(n) = +3*a(n-1) -6*a(n-2) +10*a(n-3) -12*a(n-4) +12*a(n-5) -10*a(n-6) +6*a(n-7) -3*a(n-8) +1*a(n-9). - Joerg Arndt, Mar 04 2011
a(n+1) = A026741(A000217(n)). - Paul Curtz, Apr 04 2011
a(n) = numerator(Sum_{k=0..n-1} k/2). - Arkadiusz Wesolowski, Aug 09 2012
a(n) = n*(n-1)*(3-i^(n*(n-1)))/8, where i=sqrt(-1). - Bruno Berselli, Oct 01 2012, corrected by Vaclav Kotesovec, Aug 09 2022
Sum_{n>=2} 1/a(n) = 4 - Pi/2. - Amiram Eldar, Aug 09 2022
E.g.f.: x^2*(3*exp(x) + cos(x) + sin(x))/8. - Stefano Spezia, Aug 23 2025

A130658 Period 4: repeat [1, 1, 2, 2].

Original entry on oeis.org

1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1

Offset: 0

Paul Curtz, Jun 21 2007

Continued fraction expansion of (9+sqrt(221))/14. - Klaus Brockhaus, May 03 2010
From Klaus Brockhaus, May 14 2010: (Start)
Decimal expansion of 34/303.
a(n) = A014695(n+3). (End)
Lengths of runs in A214090. - Reinhard Zumkeller, Jul 06 2012

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

Cf. A014695, A021913, A177156, A214090.

a130658 = (+ 1) . (`div` 2) . (`mod` 4)
a130658_list = cycle [1,1,2,2]  -- Reinhard Zumkeller, Jul 06 2012

&cat [[1, 1, 2, 2]^^30]; // Wesley Ivan Hurt, Jul 11 2016

seq(op([1, 1, 2, 2]), n=0..50); # Wesley Ivan Hurt, Jul 11 2016

PadRight[{}, 100, {1, 1, 2, 2}] (* Wesley Ivan Hurt, Jul 11 2016 *)

a(n)=(n%4>1)+1 \\ Charles R Greathouse IV, Jul 11 2016

G.f.: ( 1+2*x^2 ) / ( (1-x)*(1+x^2) ). - R. J. Mathar, Jan 18 2011
a(n) = (n^3 mod 4 + (n+1)^3 mod 4 + 1)/2. - Gary Detlefs, Apr 15 2011
a(n) = -1/2*cos(1/2*Pi*n)-1/2*sin(1/2*Pi*n)+3/2. - Leonid Bedratyuk, May 13 2012
From Wesley Ivan Hurt, May 30 2015: (Start)
a(n) = a(n-1) - a(n-2) + a(n-3), n>3.
a(n) = (3+(-1)^((2*n+3+(-1)^n)/4))/2. (End)
From Wesley Ivan Hurt, Jul 11 2016: (Start)
a(n) = a(n-4) for n>3.
a(n) = A021913(n) + 1. (End)

More terms from Klaus Brockhaus, May 14 2010

A176126 Numerator of -A127276(n)/A001788(n).

Original entry on oeis.org

-1, -1, 1, 2, 4, 13, 19, 13, 17, 43, 53, 32, 38, 89, 103, 59, 67, 151, 169, 94, 104, 229, 251, 137, 149, 323, 349, 188, 202, 433, 463, 247, 263, 559, 593, 314, 332, 701, 739, 389, 409, 859, 901, 472, 494, 1033, 1079, 563, 587, 1223, 1273, 662, 688, 1429, 1483, 769, 797, 1651, 1709, 884, 914, 1889, 1951, 1007, 1039, 2143, 2209, 1138, 1172, 2413, 2483, 1277, 1313, 2699, 2773, 1424, 1462, 3001, 3079, 1579

Offset: 0

Paul Curtz, Dec 07 2010

The sequence of fractions starts -1/0, -1/1, 1/3, 2/3, 4/5, 13/15, 19/21, 13/14, 17/18, 43/45, 53/55, 32/33, 38/39, ...
The denominators are apparently A064038(n+1) = A061041(4+8*n) (i.e., specified as numerators in A061041).
The difference between denominator and numerator is A014695(n), n > 0.

Cf. A001788, A127276.

Cf. A061041, A064038, A014695.

A001788 := proc(n) n*(n+1)*2^(n-2) ; end proc:
A127276 := proc(n) 2^n-A001788(n) ; end proc:
A176126 := proc(n) if n = 0 then -1 else 2^n/A001788(n)-1 ; numer(-%) ; end if; end proc:
seq(A176126(n),n=0..40) ;

Conjecture: a(n) = +3*a(n-1) -6*a(n-2) +10*a(n-3) -12*a(n-4) +12*a(n-5) -10*a(n-6) +6*a(n-7) -3*a(n-8) +a(n-9) with g.f. (x^4-x^3+3*x^2-x+1)*(x^4-x^3-2*x^2-x+1) / ( (x-1)^3*(x^2+1)^3 ). - R. J. Mathar, Dec 12 2010

a(n) = 3*a(n-4) -3*a(n-8) +a(n-12).

A178242 Numerator of n(5+n)/((n+1)(n+4)).

Original entry on oeis.org

0, 3, 7, 6, 9, 25, 33, 21, 26, 63, 75, 44, 51, 117, 133, 75, 84, 187, 207, 114, 125, 273, 297, 161, 174, 375, 403, 216, 231, 493, 525, 279, 296, 627, 663, 350, 369, 777, 817, 429, 450, 943, 987, 516, 539, 1125, 1173, 611, 636, 1323, 1375, 714, 741, 1537, 1593

Offset: 0

Paul Curtz, Dec 20 2010

Sequence of differences denominator(n) - numerator(n) = 1,2,2,1... = A014695(n).

Denominator: A160050(n+2).

			The reduced fractions are 0, 3/5, 7/9, 6/7, 9/10, 25/27, 33/35, 21/22, 26/27, 63/65, 75/77, 44/45, ..

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Wikipedia, Quasi-polynomial.
Index entries for linear recurrences with constant coefficients, signature (3,-6,10,-12,12,-10,6,-3,1).

Cf. A001793, A014695, A028557, A060819, A160050, A176027.

[Floor(n*(n+5)*((-1)^((2*n-(-1)^n-3)/4)+3)/8) : n in [0..50]]; // Vincenzo Librandi, Oct 08 2011

A178242 := proc(n) n*(5+n)/(n+1)/(n+4) ;  numer(%) ;end proc:
seq(A178242(n),n=0..80) ; # R. J. Mathar, Dec 20 2010

f[n_] := n/GCD[n, 4]; Array[f[#] f[# + 5] &, 50, 0]
Table[Numerator[n*(5+n)/((n+1)*(n+4))], {n,0,50}] (* G. C. Greubel, Sep 21 2018 *)

vector(50, n, n--; numerator(n*(5+n)/((n+1)*(n+4)))) \\ G. C. Greubel, Sep 21 2018

a(n) = numerator(A176027(n)/A001793(n+1)).
a(n) = A060819(n)*A060819(n+5).
a(n) = +3*a(n-1) -6*a(n-2) +10*a(n-3) -12*a(n-4) +12*a(n-5) -10*a(n-6) +6*a(n-7) -3*a(n-8) +a(n-9).
a(n) = 3*a(n-4) -3*a(n-8) +a(n-12).
G.f.:  x*(-3+2*x-3*x^2-3*x^3+x^7) / ( (x-1)^3*(x^2+1)^3 ).
a(n) = n*(n+5)*((-1)^((2*n-(-1)^n-3)/4)+3)/8 = n*(n+5)*(3-i^(n*(n+1)))/8, where i=sqrt(-1); also a(n) = a(n-4)*A028557(n)/A028557(n-4) for n>4. - Bruno Berselli, Dec 30 2010
From Peter Bala, Aug 07 2022: (Start)
a(n) = numerator of n*(n+5)/4.
a(n) is quasi-polynomial in n: a(4*n) = n*(4*n+5) = A343560(n+1); a(4*n+1) = (2*n+3)*(4*n+1); a(4*n+2) = (2*n+1)*(4*n+7); a(4*n+3) = (n+2)*(4*n+3) = A180863(n+2). (End)
Sum_{n>=1} 1/a(n) = 112/75 - Pi/10. - Amiram Eldar, Aug 16 2022

A139131 Squarefree kernel of n*(n+1)/2.

Original entry on oeis.org

1, 3, 6, 10, 15, 21, 14, 6, 15, 55, 66, 78, 91, 105, 30, 34, 51, 57, 190, 210, 231, 253, 138, 30, 65, 39, 42, 406, 435, 465, 62, 66, 561, 595, 210, 222, 703, 741, 390, 410, 861, 903, 946, 330, 345, 1081, 282, 42, 35, 255, 1326, 1378, 159, 165, 770, 798, 1653, 1711

Offset: 1

Reinhard Zumkeller, Apr 10 2008

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A000217, A007947, A014601, A014695, A078636.

Times@@(Transpose[FactorInteger[#]][[1]])&/@Accumulate[Range[60]] (* Harvey P. Dale, Jun 06 2013 *)

a(n) = vecprod(factor(n*(n+1)/2)[, 1]); \\ Amiram Eldar, May 12 2025

a(n) = A007947(A000217(n)).
a(A014601(n)) = A078636(A014601(n)).
a(n) = A078636(n) / A014695(n).

A330707 a(n) = ( 3*n^2 + n - 1 + (-1)^floor(n/2) )/4.

Original entry on oeis.org

0, 1, 3, 7, 13, 20, 28, 38, 50, 63, 77, 93, 111, 130, 150, 172, 196, 221, 247, 275, 305, 336, 368, 402, 438, 475, 513, 553, 595, 638, 682, 728, 776, 825, 875, 927, 981, 1036, 1092, 1150, 1210, 1271, 1333, 1397, 1463

Offset: 0

Paul Curtz, Dec 27 2019

Essentially four odds followed by four evens.
Last digit is neither 4 nor 9.
Essentially twice or twin sequences in the hexagonal spiral from A002265.
                  21  21  21  22  22  22  22
                21  14  14  14  14  15  15  23
              20  13   8   8   8   9   9  15  23
            20  13   8   4   4   4   4   9  15  23
          20  13   7   3   1   1   1   5   9  16  23
        20  13   7   3   1   0   0   2   5  10  16  24
          19  12   7   3   0   0   2   5  10  16  24
            19  12   7   3   2   2   5  10  16  24
              19  12   6   6   6   6  10  17  24
                19  12  11  11  11  11  17  25
                  18  18  18  18  17  17  25
.
There are 12 twin sequences. 6 of them (A001859, A006578, A077043, A231559, A024219, A281026) are in the OEIS. a(n) is the seventh.
  0, 1, 3, 7, 13, 20, 28, 38, 50, ...
  1, 2, 4, 6,  7,  8, 10, 12, 13, ...
  1, 2, 2, 1,  1,  2,  2,  1,  1, ... period 4. See A014695.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-4,4,-3,1).

Cf. A001859, A002265, A006578, A014695, A016921, A024219, A033579, A077043, A231559, A281026.

[(3*n^2+n-1+ (-1)^Floor(n/2))/4: n in [0..60]]; // G. C. Greubel, Dec 30 2019

seq((3*n^2+n-1+sqrt(2)*sin((2*n+1)*Pi/4))/4, n = 0..60); # G. C. Greubel, Dec 30 2019

LinearRecurrence[{3,-4,4,-3,1}, {0,1,3,7,13}, 60] (* Amiram Eldar, Dec 27 2019 *)

concat(0, Vec(x*(1 + 2*x^2) / ((1 - x)^3*(1 + x^2)) + O(x^60))) \\ Colin Barker, Dec 27 2019

[(3*n^2+n-1+(-1)^floor(n/2))/4 for n in (0..60)] # G. C. Greubel, Dec 30 2019

a(n) = A231559(-n).
a(1+2*n) + a(2+2*n) = A033579(n+1).
a(40+n) - a(n) = 1210, 1270, 1330, 1390, 1450, ... . See 10*A016921(n).
From Colin Barker, Dec 27 2019: (Start)
G.f.: x*(1 + 2*x^2) / ((1 - x)^3*(1 + x^2)).
a(n) = 3*a(n-1) - 4*a(n-2) + 4*a(n-3) - 3*a(n-4) + a(n-5) for n>4.
(End)
E.g.f.: (cos(x) + sin(x) + (-1 + 4*x + 3*x^2)*exp(x))/4. - Stefano Spezia, Dec 27 2019
a(n) = ( 3*n^2 + n - 1 + sqrt(2)*sin((2*n+1)*Pi/4) )/4 = ( 3*n^2 + n - 1 + (-1)^floor(n/2) )/4. - G. C. Greubel, Dec 30 2019

A078636 a(n) = rad(n*(n+1)).

Original entry on oeis.org

2, 6, 6, 10, 30, 42, 14, 6, 30, 110, 66, 78, 182, 210, 30, 34, 102, 114, 190, 210, 462, 506, 138, 30, 130, 78, 42, 406, 870, 930, 62, 66, 1122, 1190, 210, 222, 1406, 1482, 390, 410, 1722, 1806, 946, 330, 690, 2162, 282, 42, 70, 510, 1326, 1378, 318, 330, 770, 798

Offset: 1

Jon Perry, Dec 12 2002

			a(3) = 6 as rad(3*4) = rad(12) = rad(2*2*3) = 2*3 = 6.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A002378, A007947, A008683, A014601, A139131, A307868.

A078636 := proc(n)
    A007947(n)*A007947(n+1) ;
end proc:
seq( A078636(n),n=1..10) ; # R. J. Mathar, Mar 15 2023

rad[n_] := Times @@ FactorInteger[n][[All, 1]];
a[n_] := rad[n(n+1)];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Mar 27 2024 *)

rad(n)=local(p,i); p=factor(n)[,1]; prod(i=1,length(p),p[i])
for (k=1,100,print1(rad(k*(k+1))", "))

From Reinhard Zumkeller, Aug 05 2003: (Start)
a(n) = rad(n*(n+1)) = rad(n)*rad(n+1).
mu(a(n)) = mu(rad(n*(n+1))) = mu(rad(n))*mu(rad(n+1)), where rad=A007947 and mu=A008683. (End)
From Reinhard Zumkeller, Apr 10 2008: (Start)
a(A014601(n)) = A139131(A014601(n)).
a(n) = A139131(n) * A014695(n). (End)
From Amiram Eldar, Apr 04 2025: (Start)
a(n) = A007947(A002378(n)).
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = Product_{p prime} (1 - 2/(p*(p+1))) = 0.4716806... (A307868). (End)

A086314 Total number of edges in the distinct simple graphs on n nodes.

Original entry on oeis.org

0, 1, 6, 33, 170, 1170, 10962, 172844, 4944024, 270116280, 28022441260, 5448008695536, 1969579223350128, 1321964082404214704, 1649890513414726210320, 3840060942271653473695680, 16723638762440239422492944768, 136749695973639295091912681599872

Offset: 1

Eric W. Weisstein, Jul 15 2003

nonn

a(n) = A000088(n)*n(n-1)/4.

a(n) = A000088(n)*A064038(n)/A014695(n).

Eric Weisstein's World of Mathematics, Simple Graph
Eric Weisstein's World of Mathematics, Graph Edge

Cf. A000088, A014695, A064038.

<< Combinatorica`; Table[D[GraphPolynomial[n, x], x] /. x -> 1, {n, 18}]  (* Geoffrey Critzer, Sep 29 2012 *)
<< Combinatorica`; Table[Binomial[n, 2] NumberOfGraphs[n]/2, {n, 18}] (* Eric W. Weisstein, May 17 2017 *)

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A141479 a(n) = A000111(n) + A014695(n).

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A257942 a(n) = (n+1)*(n+2)/A014695(n+1), where A014695 is repeat (1, 2, 2, 1).

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A064038 Numerator of average number of swaps needed to bubble sort a string of n distinct letters.

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A130658 Period 4: repeat [1, 1, 2, 2].

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A176126 Numerator of -A127276(n)/A001788(n).

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A178242 Numerator of n*(5+n)/((n+1)*(n+4)).

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A139131 Squarefree kernel of n*(n+1)/2.

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A178242 Numerator of n(5+n)/((n+1)(n+4)).