A026741 -id:A026741 - cp's OEIS frontend

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

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

A109626 Consider the array T(n,m) where the n-th row is the sequence of integer coefficients of A(x), where 1<=a(n)<=n, such that A(x)^(1/n) consists entirely of integer coefficients and where m is the (m+1)-th coefficient. This is the antidiagonal read from lower left to upper right.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 4, 3, 2, 1, 1, 5, 2, 1, 2, 1, 1, 6, 5, 4, 3, 2, 1, 1, 7, 3, 5, 3, 3, 1, 1, 1, 8, 7, 2, 5, 4, 3, 2, 1, 1, 9, 4, 7, 3, 1, 4, 3, 2, 1, 1, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 1, 11, 5, 3, 2, 7, 6, 5, 1, 3, 1, 1, 1, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 1, 13, 6, 11, 10, 9, 4, 1, 3, 5

Offset: 1

Paul D. Hanna and Robert G. Wilson v, Aug 01 2005

			Table begins:
\k...0...1...2...3...4...5...6...7...8...9..10..11..12..13
n\
 1|  1   1   1   1   1   1   1   1   1   1   1   1   1   1
 2|  1   2   1   2   2   2   1   2   2   2   1   2   1   2
 3|  1   3   3   1   3   3   3   3   3   3   3   3   1   3
 4|  1   4   2   4   3   4   4   4   1   4   4   4   3   4
 5|  1   5   5   5   5   1   5   5   5   5   4   5   5   5
 6|  1   6   3   2   3   6   6   6   3   4   6   6   6   6
 7|  1   7   7   7   7   7   7   1   7   7   7   7   7   7
 8|  1   8   4   8   2   8   4   8   7   8   8   8   4   8
 9|  1   9   9   3   9   9   3   9   9   1   9   9   6   9
10|  1  10   5  10  10   2   5  10  10  10   3  10   5  10
11|  1  11  11  11  11  11  11  11  11  11  11   1  11  11
12|  1  12   6   4   9  12   4  12  12   8   6  12   6  12
13|  1  13  13  13  13  13  13  13  13  13  13  13  13   1
14|  1  14   7  14   7  14  14   2   7  14  14  14  14  14
15|  1  15  15   5  15   3  10  15  15  10  15  15   5  15
16|  1  16   8  16   4  16   8  16  10  16   8  16  12  16

G. C. Greubel, Antidiagonals n = 1..100, flattened

Rows: A000012, A083952, A083953, A083954, A083945, A083946, A083947, A083948, A083949, A083950, A084066, A084067.
Columns: A000012, A111627, A026741, A051176, A111607, A060791, A111608, A106608, A111609, A111610, A111611, A106612, A106614, A106618, A106620.
Diagonals: A000027 (main), A111614 (first upper), A111627 (2nd), A111615 (3rd), A111618 (first lower), A111623 (2nd).
Other diagonals: A005408 (T(2*n-1, n)), A111626, A111627, A111628, A111629, A111630.
Cf. A111603, A111604.

f[n_]:= f[n]= Block[{a}, a[0] = 1; a[l_]:= a[l]= Block[{k = 1, s = Sum[ a[i]*x^i, {i,0,l-1}]}, While[ IntegerQ[Last[CoefficientList[Series[(s + k*x^l)^(1/n), {x, 0, l}], x]]] != True, k++ ]; k]; Table[a[j], {j,0,32}]];
T[n_, m_]:= f[n][[m]];
Flatten[Table[T[i,n-i], {n,15}, {i,n-1,1,-1}]]

A109626_row(n, len=40)={my(A=1, m); vector(len, k, if(k>m=1, while(denominator(polcoeff(sqrtn(O(x^k)+A+=x^(k-1), n), k-1))>1, m++); m, 1))} \\ M. F. Hasler, Jan 27 2025

When m is prime, column m is T(n,m) = n/gcd(m, n) = numerator of n/(n+m). - M. F. Hasler, Jan 27 2025

A195161 Multiples of 8 and odd numbers interleaved.

Original entry on oeis.org

0, 1, 8, 3, 16, 5, 24, 7, 32, 9, 40, 11, 48, 13, 56, 15, 64, 17, 72, 19, 80, 21, 88, 23, 96, 25, 104, 27, 112, 29, 120, 31, 128, 33, 136, 35, 144, 37, 152, 39, 160, 41, 168, 43, 176, 45, 184, 47, 192, 49, 200, 51, 208, 53, 216, 55, 224, 57, 232, 59

Offset: 0

Omar E. Pol, Sep 10 2011

A008590 and A005408 interleaved. This is 8*n if n is even, n if n is odd, if n>=0.
Partial sums give the generalized 12-gonal (or dodecagonal) numbers A195162.
The moment generating function of p(x, m=2, n=1, mu=2) = 4*x*E(x, 2, 1), see A163931 and A274181, is given by M(a) = (- 4*log(1-a) - 4 * polylog(2, a))/a^2. The series expansion of M(a) leads to the sequence given above. - Johannes W. Meijer, Jul 03 2016
a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 12-gonal numbers. - Omar E. Pol, Jul 27 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Column 8 of A195151.
Sequences whose partial sums give the generalized n-gonal numbers, if n>=5: A026741, A001477, zero together with A080512, A022998, A195140, zero together with A165998, A195159, this sequence, A195312.
Cf. A144433.

&cat[[8*n, 2*n+1]: n in [0..30]]; // Vincenzo Librandi, Sep 27 2011

a := proc(n): (6*(-1)^n+10)*n/4 end: seq(a(n), n=0..59); # Johannes W. Meijer, Jul 03 2016

With[{nn=30},Riffle[8*Range[0,nn],2*Range[0,nn]+1]] (* or *) LinearRecurrence[{0,2,0,-1},{0,1,8,3},60] (* Harvey P. Dale, Nov 24 2013 *)

concat(0, Vec(x*(1+8*x+x^2)/((1-x)^2*(1+x)^2) + O(x^99))) \\ Altug Alkan, Jul 04 2016

a(2n) = 8n, a(2n+1) = 2n+1. [corrected by Omar E. Pol, Jul 26 2018]
a(n) = (6*(-1)^n+10)*n/4. - Vincenzo Librandi, Sep 27 2011
a(n) = 2*a(n-2)-a(n-4). G.f.: x*(1+8*x+x^2)/((1-x)^2*(1+x)^2). - Colin Barker, Aug 11 2012
From Ilya Gutkovskiy, Jul 03 2016: (Start)
a(m*2^k) = m*2^(k+2), k>0.
E.g.f.: x*(4*sinh(x) + cosh(x)).
Dirichlet g.f.: 2^(-s)*(2^s + 6)*zeta(s-1). (End)
Multiplicative with a(2^e) = 4*2^e, a(p^e) = p^e for odd prime p. - Andrew Howroyd, Jul 23 2018
a(n) = A144433(n-1) for n > 1. - Georg Fischer, Oct 14 2018

A106612 a(n) = numerator(n/(n+11)).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 2, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 3, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 4, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 5, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 6, 67, 68, 69, 70, 71, 72, 73, 74, 75

Offset: 0

N. J. A. Sloane, May 15 2005

In general, the numerators of n/(n+p) for prime p and n >= 0, form a sequence with the g.f.: x/(1-x)^2 - (p-1)*x^p/(1-x^p)^2. - Paul D. Hanna, Jul 27 2005

a(n) <> n iff n = 11 * k, in this case, a(n) = k. - Bernard Schott, Feb 19 2019

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,-1).

Cf. A109052, A137564 (differs, e.g., for n=100).

Cf. Sequences given by the formula numerator(n/(n + k)): A026741 (k = 2), A051176 (k = 3), A060819 (k = 4), A060791 (k = 5), A060789 (k = 6), A106608 thru A106611 (k = 7 thru 10), A051724 (k = 12), A106614 thru A106621 (k = 13 thru 20).

List([0..80],n->NumeratorRat(n/(n+11))); # Muniru A Asiru, Feb 19 2019

[Numerator(n/(n+11)): n in [0..100]]; // Vincenzo Librandi, Apr 18 2011

seq(numer(n/(n+11)),n=0..80); # Muniru A Asiru, Feb 19 2019

f[n_]:=Numerator[n/(n+11)];Array[f,100,0] (* Vladimir Joseph Stephan Orlovsky, Feb 17 2011 *)
LinearRecurrence[{0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,-1},{0,1,2,3,4,5,6,7,8,9,10,1,12,13,14,15,16,17,18,19,20,21},80] (* Harvey P. Dale, Jul 05 2021 *)

vector(100, n, n--; numerator(n/(n+11))) \\ G. C. Greubel, Feb 19 2019

[lcm(n,11)/11 for n in range(0, 54)] # Zerinvary Lajos, Jun 09 2009

G.f.: x/(1-x)^2 - 10*x^11/(1-x^11)^2. - Paul D. Hanna, Jul 27 2005
a(n) = lcm(n,11)/11.
From R. J. Mathar, Apr 18 2011: (Start)
a(n) = A109052(n)/11.
Dirichlet g.f.: zeta(s-1)*(1-10/11^s). (End)
a(n) = 2*a(n-11) - a(n-22). - G. C. Greubel, Feb 19 2019
From Amiram Eldar, Nov 25 2022: (Start)
Multiplicative with a(11^e) = 11^(e-1), and a(p^e) = p^e if p != 11.
Sum_{k=1..n} a(k) ~ (111/242) * n^2. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = 21*log(2)/11. - Amiram Eldar, Sep 08 2023

A115361 Inverse of matrix (1,x)-(x,x^2) (expressed in Riordan array notation).

Original entry on oeis.org

1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Paul Barry, Jan 21 2006

Row sums are the 'ruler function' A001511. Columns are stretched Fredholm-Rueppel sequences (A036987). Inverse is A115359.
Eigensequence of triangle A115361 = A018819 starting with offset 1: (1, 2, 2, 4, 4, 6, 6, 10, 10, 14, 14, 20, 20, ...). - Gary W. Adamson, Nov 21 2009
From Gary W. Adamson, Nov 27 2009: (Start)
A115361 * [1, 2, 3, ...] = A129527 = (1, 3, 3, 7, 5, 9, 7, 15, ...).
(A115361)^(-1) * [1, 2, 3, ...] = A115359 * [1, 2, 3, ...] = A026741 starting /Q (1, 1, 3, 2, 5, 3, 7, 4, 9, ...). (End)
This is the lower-left triangular part of the inverse of the infinite matrix A_{ij} = [i=j] - [i=2j], its upper-right part (above / right to the diagonal) being zero. The n-th row has 1 in column n/2^i, i = 0, 1, ... as long as this is an integer. - M. F. Hasler, May 13 2018
The rows are the reversed binary expansions of A127804. - Tilman Piesk, Jun 10 2025

			Triangle begins:
  1;
  1,1;
  0,0,1;
  1,1,0,1;
  0,0,0,0,1;
  0,0,1,0,0,1;
  0,0,0,0,0,0,1;
  1,1,0,1,0,0,0,1;
  0,0,0,0,0,0,0,0,1;
  0,0,0,0,1,0,0,0,0,1;
  0,0,0,0,0,0,0,0,0,0,1;

Tilman Piesk, Illustration of first 32 rows.

Cf. A016741, A018819, A129527. - Gary W. Adamson, Nov 21 2009

Cf. A036987, A209229, A127804.

A115361 := proc(n,k)
    for j from 0 do
        if k+(2*j-1)*(k+1) > n then
            return 0 ;
        elif k+(2^j-1)*(k+1) = n then
            return 1 ;
        end if;
    end do;
end proc: # R. J. Mathar, Jul 14 2012

(*recurrence*)
Clear[t]
t[1, 1] = 1;
t[n_, k_] :=
t[n, k] =
  If[k == 1, Sum[t[n, k + i], {i, 1, 2 - 1}],
   If[Mod[n, k] == 0, t[n/k, 1], 0], 0]
Flatten[Table[Table[t[n, k], {k, 1, n}], {n, 14}]] (* Mats Granvik, Jun 26 2014 *)

tabl(nn) = {T = matrix(nn, nn, n, k, n--; k--; if ((n==k), 1, if (n==2*k+1, -1, 0))); Ti = T^(-1); for (n=1, nn, for (k=1, n, print1(Ti[n, k], ", ");); print(););} \\ Michel Marcus, Mar 28 2015

A115361_row(n,v=vector(n))={until(bittest(n,0)||!n\=2,v[n]=1);v} \\ Yields the n-th row (of length n). - M. F. Hasler, May 13 2018

T(n,k)={if(n%k, 0, my(e=valuation(n/k,2)); n/k==1<Andrew Howroyd, Aug 03 2018

# translation of Maple code by R. J. Mathar
def a115361(n, k):
    j = 0
    while True:
        if k + (2*j - 1) * (k + 1) > n:
            return 0
        elif k + (2**j - 1) * (k + 1) == n:
            return 1
        else:
            j += 1  #  Tilman Piesk, Jun 10 2025

Number triangle whose k-th column has g.f. x^k*sum{j>=0} x^((2^j-1)*(k+1)).

T(n,k) = A209229((n+1)/(k+1)) for k+1 divides n+1, T(n,k) = 0 otherwise. - Andrew Howroyd, Aug 05 2018

A195019 Multiples of 3 and of 4 interleaved: a(2n-1) = 3n, a(2n) = 4n.

Original entry on oeis.org

3, 4, 6, 8, 9, 12, 12, 16, 15, 20, 18, 24, 21, 28, 24, 32, 27, 36, 30, 40, 33, 44, 36, 48, 39, 52, 42, 56, 45, 60, 48, 64, 51, 68, 54, 72, 57, 76, 60, 80, 63, 84, 66, 88, 69, 92, 72, 96, 75, 100, 78, 104, 81, 108, 84, 112, 87, 116, 90, 120, 93, 124, 96, 128

Offset: 1

Omar E. Pol, Sep 07 2011, Sep 12 2011

First differences of A195020.

a(n) is also the length of the n-th edge of a square spiral in which the first two edges are the legs of the primitive Pythagorean triple [3, 4, 5]. The spiral contains infinitely many Pythagorean triples in which the hypotenuses are the positives A008587. Zero together with partial sums give A195020; the vertices of the spiral.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Ron Knott, Pythagorean triangles and triples
Eric Weisstein's World of Mathematics, Pythagorean Triple
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A008585, A008586, A026741, A195020-A195029, A195031, A195033, A195035.

[((n-3)*(-1)^n+7*n+3)/4: n in [1..60]]; // Vincenzo Librandi, Sep 12 2011

Table[((n-3)*(-1)^n + 7*n + 3)/4, {n,1,50}] (* G. C. Greubel, Aug 19 2017 *)

a(n)=(n+1)\2*(4-n%2)  \\ M. F. Hasler, Sep 08 2011

pair(3*n, 4*n).
a(2*n-1) = 3*n, a(2*n) = 4*n. - M. F. Hasler, Sep 08 2011
G.f.: x*(3+4*x) / ( (x-1)^2*(1+x)^2 ). - R. J. Mathar, Sep 09 2011
From Bruno Berselli, Sep 12 2011: (Start)
a(n) = ((n-3)*(-1)^n + 7*n + 3)/4.
a(n) + a(n+1) = A047355(n+2). (End)
E.g.f.: (1/4)*((3 + 7*x)*exp(x) - (3 + x)*exp(-x)). - G. C. Greubel, Aug 19 2017

A195310 Triangle read by rows with T(n,k) = n - A001318(k), n >= 1, k >= 1, if (n - A001318(k)) >= 0.

Original entry on oeis.org

0, 1, 0, 2, 1, 3, 2, 4, 3, 0, 5, 4, 1, 6, 5, 2, 0, 7, 6, 3, 1, 8, 7, 4, 2, 9, 8, 5, 3, 10, 9, 6, 4, 11, 10, 7, 5, 0, 12, 11, 8, 6, 1, 13, 12, 9, 7, 2, 14, 13, 10, 8, 3, 0, 15, 14, 11, 9, 4, 1, 16, 15, 12, 10, 5, 2, 17, 16, 13, 11, 6, 3, 18, 17, 14, 12, 7, 4

Offset: 1

Omar E. Pol, Sep 21 2011

Also triangle read by rows in which column k lists the nonnegative integers A001477 starting at the row A001318(k). This sequence is related to Euler's Pentagonal Number Theorem. A000041(a(n)) gives the absolute value of A175003(n). To get the number of partitions of n see the example.

			Written as a triangle:
   0;
   1,  0;
   2,  1;
   3,  2;
   4,  3,  0;
   5,  4,  1;
   6,  5,  2,  0;
   7,  6,  3,  1;
   8,  7,  4,  2;
   9,  8,  5,  3;
  10,  9,  6,  4;
  11, 10,  7,  5,  0;
  12, 11,  8,  6,  1;
  13, 12,  9,  7,  2;
  14, 13, 10,  8,  3,  0;
.
For n = 15, consider row 15 which lists the numbers 14, 13, 10, 8, 3, 0. From Euler's Pentagonal Number Theorem we have that the number of partitions of 15 is p(15) = p(14) + p(13) - p(10) - p(8) + p(3) + p(0) = 135 + 101 - 42 - 22 + 3 + 1 = 176.

L. Euler, De mirabilibus proprietatibus numerorum pentagonalium
L. Euler, On the remarkable properties of the pentagonal numbers, arXiv:math/0505373 [math.HO], 2005.
Eric Weisstein's World of Mathematics, Pentagonal Number Theorem
Wikipedia, Pentagonal number theorem

Row sums give A195311.

Cf. A000041, A001318, A010815, A026741, A057077, A175003.

rows = 20;
a1318[n_] := If[EvenQ[n], n(3n/2+1)/4, (n+1)(3n+1)/8];
T[n_, k_] := n - a1318[k];
Table[DeleteCases[Table[T[n, k], {k, 1, n}], ?Negative], {n, 1, rows}] // Flatten (* _Jean-François Alcover, Sep 22 2018 *)

A175003(n,k) = A057077(k-1)*A000041(T(n,k)), n >= 1, k >= 1.

Name essentially suggested by Franklin T. Adams-Watters (see history), Sep 21 2011

A060791 a(n) = n / gcd(n,5).

Original entry on oeis.org

1, 2, 3, 4, 1, 6, 7, 8, 9, 2, 11, 12, 13, 14, 3, 16, 17, 18, 19, 4, 21, 22, 23, 24, 5, 26, 27, 28, 29, 6, 31, 32, 33, 34, 7, 36, 37, 38, 39, 8, 41, 42, 43, 44, 9, 46, 47, 48, 49, 10, 51, 52, 53, 54, 11, 56, 57, 58, 59, 12, 61, 62, 63, 64, 13, 66, 67, 68, 69

Offset: 1

Len Smiley, Apr 26 2001

As well as being a multiplicative sequence, a(n) is also strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for n, m >= 1. In particular, a(n) is a divisibility sequence: if n divides m then a(n) divides a(m). Peter Bala, Feb 20 2019

Harry J. Smith, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,2,0,0,0,0,-1).

Cf. Sequences given by the formula n/gcd(n,k) = numerator(n/(n + k)): A026741 (k = 2), A051176 (k = 3), A060819 (k = 4), A060789 (k = 6), A106608 thru A106612 (k = 7 thru 11), A051724 (k = 12), A106614 thru A106621 (k = 13 thru 20).

Cf. A109046.

List([1..80],n->n/Gcd(n,5)); # Muniru A Asiru, Feb 20 2019

[n/GCD(n, 5): n in [1..100]]; // G. C. Greubel, Feb 20 2019

seq(n/gcd(n,5),n=1..80); # Muniru A Asiru, Feb 20 2019

f[n_]:=Numerator[n/(n+5)]; Array[f,100] (* Vladimir Joseph Stephan Orlovsky, Feb 16 2011*)

a(n) = { n / gcd(n, 5) } \\ Harry J. Smith, Jul 12 2009

a(n) = n/(5-4*(n%5>0)) \\ Zak Seidov, Feb 17 2011

[lcm(n,5)/5 for n in range(1, 51)] # Zerinvary Lajos, Jun 07 2009

G.f.: x*(1 + 2*x + 3*x^2 + 4*x^3 + x^4 + 4*x^5 + 3*x^6 + 2*x^7 + x^8)/(1 - x^5)^2.
a(n) = n/5 if 5|n, otherwise a(n) = n.
From R. J. Mathar, Apr 18 2011: (Start)
a(n) = A109046(n)/5.
Dirichlet g.f.: zeta(s-1)*(1-4/5^s). (End)
G.f.: x*(x^4 + x^3 - x^2 + x + 1)*(x^4 + x^3 + 3*x^2 + x + 1)/((x - 1)^2*(x^4 + x^3 + x^2 + x + 1)^2). - R. J. Mathar, Oct 31 2015
From Peter Bala, Feb 20 2019: (Start)
a(n) = numerator(n/(n + 5)).
If gcd(n, m) = 1 then a(a(n)*a(m)) = a(a(n)) * a(a(m)), a(a(a(n))*a(a(m))) = a(a(a(n))) * a(a(a(m))) and so on.
G.f.: x/(1 - x)^2 - 4*x^5/(1 - x^5)^2. (End)
Sum_{k=1..n} a(k) ~ (21/50) * n^2. - Amiram Eldar, Nov 25 2022
Sum_{n>=1} (-1)^(n+1)/a(n) = 9*log(2)/5. - Amiram Eldar, Sep 08 2023

Extended (using terms from b-file) by Michel Marcus, Feb 08 2014

A195140 Multiples of 5 and odd numbers interleaved.

Original entry on oeis.org

0, 1, 5, 3, 10, 5, 15, 7, 20, 9, 25, 11, 30, 13, 35, 15, 40, 17, 45, 19, 50, 21, 55, 23, 60, 25, 65, 27, 70, 29, 75, 31, 80, 33, 85, 35, 90, 37, 95, 39, 100, 41, 105, 43, 110, 45, 115, 47, 120, 49, 125, 51, 130, 53, 135, 55, 140, 57, 145, 59, 150, 61, 155, 63

Offset: 0

Omar E. Pol, Sep 10 2011

This is 5*n/2 if n is even, n if n is odd.
Partial sums give the generalized enneagonal numbers A118277.
a(n) is also the length of the n-th line segment of a rectangular spiral on the infinite square grid. The vertices of the spiral are the generalized enneagonal numbers. - Omar E. Pol, Jul 27 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

A008587 and A005408 interleaved.
Column 5 of A195151.
Cf. Sequences whose partial sums give the generalized n-gonal numbers, if n>=5: A026741, A001477, zero together with A080512, A022998, this sequence, zero together with A165998, A195159, A195161, A195312.
Cf. A047336, A118277.

&cat[[5*n,2*n+1]: n in [0..31]]; // Bruno Berselli, Sep 27 2011

With[{nn=40},Riffle[5*Range[0,nn],Range[1,2nn+1,2]]] (* or *) LinearRecurrence[ {0,2,0,-1},{0,1,5,3},80] (* Harvey P. Dale, Dec 15 2014 *)

a(n)=(7+3*(-1)^n)*n/4 \\ Charles R Greathouse IV, Oct 07 2015

a(2n) = 5n, a(2n+1) = 2n+1.
G.f.: x*(1+5*x+x^2) / ((x-1)^2*(x+1)^2). - Alois P. Heinz, Sep 26 2011
From Bruno Berselli, Sep 27 2011: (Start)
a(n) = (7+3*(-1)^n)*n/4.
a(n) = -a(-n) = a(n-2)*n/(n-2) = 2*a(n-2)-a(n-4).
a(n) + a(n-1) = A047336(n). (End)
Multiplicative with a(2^e) = 5*2^(e-1), a(p^e) = p^e for odd prime p. - Andrew Howroyd, Jul 23 2018
Dirichlet g.f.: zeta(s-1) * (1 + 3/2^s). - Amiram Eldar, Oct 25 2023

Corrected and edited by Alois P. Heinz, Sep 25 2011

A195151 Square array read by antidiagonals upwards: T(n,k) = n((k-2)(-1)^n+k+2)/4, n >= 0, k >= 0.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 3, 1, 1, 0, 0, 3, 2, 1, 0, 5, 2, 3, 3, 1, 0, 0, 5, 4, 3, 4, 1, 0, 7, 3, 5, 6, 3, 5, 1, 0, 0, 7, 6, 5, 8, 3, 6, 1, 0, 9, 4, 7, 9, 5, 10, 3, 7, 1, 0, 0, 9, 8, 7, 12, 5, 12, 3, 8, 1, 0, 11, 5, 9, 12, 7, 15, 5, 14, 3, 9, 1, 0, 0, 11, 10, 9, 16, 7

Offset: 0

Omar E. Pol, Sep 14 2011

Also square array T(n,k) read by antidiagonals in which column k lists the multiples of k and the odd numbers interleaved, n>=0, k>=0. Also square array T(n,k) read by antidiagonals in which if n is even then row n lists the multiples of (n/2), otherwise if n is odd then row n lists a constant sequence: the all n's sequence. Partial sums of the numbers of column k give the column k of A195152. Note that if k >= 1 then partial sums of the numbers of the column k give the generalized m-gonal numbers, where m = k + 4.

All columns are multiplicative. - Andrew Howroyd, Jul 23 2018

			Array begins:
.  0,   0,   0,   0,   0,   0,   0,   0,   0,   0,...
.  1,   1,   1,   1,   1,   1,   1,   1,   1,   1,...
.  0,   1,   2,   3,   4,   5,   6,   7,   8,   9,...
.  3,   3,   3,   3,   3,   3,   3,   3,   3,   3,...
.  0,   2,   4,   6,   8,  10,  12,  14,  16,  18,...
.  5,   5,   5,   5,   5,   5,   5,   5,   5,   5,...
.  0,   3,   6,   9,  12,  15,  18,  21,  24,  27,...
.  7,   7,   7,   7,   7,   7,   7,   7,   7,   7,...
.  0,   4,   8,  12,  16,  20,  24,  28,  32,  36,...
.  9,   9,   9,   9,   9,   9,   9,   9,   9,   9,...
.  0,   5,  10,  15,  20,  25,  30,  35,  40,  45,...
...

Rows: A000004, A000012, A001477, A010701, A005843, A010716, A008585, A010727, A008586, A010734, A008587.

Columns k: A026741 (k=1), A001477 (k=2), zero together with A080512 (k=3), A022998 (k=4), A195140 (k=5), zero together with A165998 (k=6), A195159 (k=7), A195161 (k=8), A195312 k=(9), A195817 (k=10), A317311 (k=11), A317312 (k=12), A317313 (k=13), A317314 k=(14), A317315 (k=15), A317316 (k=16), A317317 (k=17), A317318 (k=18), A317319 k=(19), A317320 (k=20), A317321 (k=21), A317322 (k=22), A317323 (k=23), A317324 k=(24), A317325 (k=25), A317326 (k=26).

T(n,k) = n*((k-2)*(-1)^n+k+2)/4 \\ Andrew Howroyd, Jul 23 2018

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A109626 Consider the array T(n,m) where the n-th row is the sequence of integer coefficients of A(x), where 1<=a(n)<=n, such that A(x)^(1/n) consists entirely of integer coefficients and where m is the (m+1)-th coefficient. This is the antidiagonal read from lower left to upper right.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A195161 Multiples of 8 and odd numbers interleaved.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A106612 a(n) = numerator(n/(n+11)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A115361 Inverse of matrix (1,x)-(x,x^2) (expressed in Riordan array notation).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

PARI

Python

Formula

A195019 Multiples of 3 and of 4 interleaved: a(2*n-1) = 3*n, a(2*n) = 4*n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

A195019 Multiples of 3 and of 4 interleaved: a(2n-1) = 3n, a(2n) = 4n.

A195151 Square array read by antidiagonals upwards: T(n,k) = n((k-2)(-1)^n+k+2)/4, n >= 0, k >= 0.