A033991 -id:A033991 - 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

A033952 Write 1,2,... in a clockwise spiral on a square lattice, writing each digit at a separate lattice point, starting with 1 at the origin and 2 at x=0, y=-1; sequence gives the numbers on the positive x-axis.

Original entry on oeis.org

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

Offset: 1

Olivier Gorin (gorin(AT)roazhon.inra.fr)

Same as the South spoke of the Champernowne spiral (A244677).

			The spiral begins
.
  3---1---4---1---5
  |               |
  1   5---6---7   1
  |   |       |   |
  2   4   1   8   6
  |   |   |   |   |
  1   3---2   9   1
  |           |   |
  1---1---0---1   7
.

Andrey Zabolotskiy, Table of n, a(n) for n = 1..10000

Sequences from spirals: A001107, A002939, A007742, A033951, A033952, A033953, A033954, A033989, A033990, A033991, A002943, A033996, A033988.

almostNatural[n_, b_] := Block[{m = 0, d = n, i = 1, l, p}, While[m <= d, l = m; m = (b - 1) i*b^(i - 1) + l; i++]; i--; p = Mod[d - l, i]; q = Floor[(d - l)/i] + b^(i - 1); If[p != 0, IntegerDigits[q, b][[p]], Mod[q - 1, b]]];
f[n_] := 4n^2 - 5n + 2; Array[ almostNatural[ f@#, 10] &, 105] (* Robert G. Wilson v, Aug 08 2014 *)

More terms from Andrew J. Gacek (andrew(AT)dgi.net)

Edited by Charles R Greathouse IV, Nov 01 2009

A005476 a(n) = n(5n - 1)/2.

Original entry on oeis.org

0, 2, 9, 21, 38, 60, 87, 119, 156, 198, 245, 297, 354, 416, 483, 555, 632, 714, 801, 893, 990, 1092, 1199, 1311, 1428, 1550, 1677, 1809, 1946, 2088, 2235, 2387, 2544, 2706, 2873, 3045, 3222, 3404, 3591

Offset: 0

N. J. A. Sloane

a(n) is half the number of ways to divide an n X n square into 3 rectangles whose side-lengths are integers. See Matthew Scroggs link. - George Witty, Feb 06 2024

G. C. Greubel, Table of n, a(n) for n = 0..1000
Matthew Scroggs, Advent Calendar 2023 Solutions.
Leo Tavares, Illustration: Triangulated Diamonds
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A005475, A033994, A294833.
Cf. numbers of the form n*(n*k-k+4)/2 listed in A226488.
Cf. similar sequences listed in A022288.
Cf. A016754, A133694, A000290.

List([0..40], n-> Binomial(5*n,2)/5); # G. C. Greubel, Jul 30 2019

[Binomial(5*n,2)/5: n in [0..40]]; // G. C. Greubel, Jul 30 2019

[seq(binomial(5*n,2)/5,n=0..40)]; # Zerinvary Lajos, Jan 02 2007

Table[n(5n-1)/2, {n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Oct 25 2008 *)

a(n)=n*(5*n-1)/2 \\ Charles R Greathouse IV, Sep 24 2015

[binomial(5*n,2)/5 for n in (0..40)] # G. C. Greubel, Jul 30 2019

a(n) = C(5*n,2)/5 for n>=0. - Zerinvary Lajos, Jan 02 2007
a(n) = A033991(n) - A000326(n). - Zerinvary Lajos, Jun 11 2007
a(n) = a(n-1) + 5*n - 3 for n>0, a(0)=0. - Vincenzo Librandi, Nov 18 2010
a(n) = A000217(n) + A000384(n) = A000290(n) + A000326(n). - Omar E. Pol, Jan 11 2013
a(n) = A130520(5*n+1). - Philippe Deléham, Mar 26 2013
a(n) = A033994(n) - A033994(n-1). - J. M. Bergot, Jun 12 2013
From Bruno Berselli, Oct 17 2016: (Start)
G.f.: x*(2 + 3*x)/(1 - x)^3.
a(n) = A000217(3*n-1) - A000217(2*n-1). (End)
E.g.f.: x*(4 + 5*x)*exp(x)/2. - G. C. Greubel, Jul 30 2019
Sum_{n>=1} 1/a(n) = 2 * A294833. - Amiram Eldar, Nov 16 2020
From Leo Tavares, Nov 20 2021: (Start)
a(n) = A016754(n) - A133694(n+1). See Triangulated Diamonds illustration.
a(n) = A000290(n) + A000217(n) + 2*A000217(n-1)
a(n) = 2*A000217(n) + 3*A000217(n-1). (End)

A033988 Write 0,1,2,... in a clockwise spiral on a square lattice, writing each digit at a separate lattice point, starting with 0 at the origin and 1 at x=0, y=-1; sequence gives the numbers on the positive y-axis.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

In other words, write 0 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 ... in a clockwise spiral, starting with the 0 and taking the first step south; the sequence is then picked out from the resulting spiral by starting at the origin and moving north.

			  1---3---1---4---1
  |               |
  2   4---5---6   5
  |   |       |   |
  1   3   0   7   1
  |   |   |   |   |
  1   2---1   8   6
  |           |   |
  1---0---1---9   1
.
We begin with the 0 and wrap the numbers 1 2 3 4 ... around it.
Then the sequence is obtained by reading vertically upwards, starting from the initial 0.

Andrew Woods, Table of n, a(n) for n = 0..1000

Sequences based on the same spiral: A033953, A033989, A033990. Spiral without zero: A033952.
Other sequences from spirals: A001107, A002939, A007742, A033951, A033954, A033991, A002943, A033996.
Cf. A033307.

nmax = 105; A033307 = Flatten[IntegerDigits /@ Range[0, nmax^2 + 10 nmax]]; a[n_] := If[n==0, 0, A033307[[4n^2 + n + 1]]]; Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Apr 24 2017, after Andrew Woods *)

a(n) = A033307(4*n^2 + n - 1) for n > 0. - Andrew Woods, May 18 2012

More terms from Andrew Gacek (andrew(AT)dgi.net)

Edited by Jon E. Schoenfield, Aug 12 2018

A033989 Write 0,1,2,... in a clockwise spiral on a square lattice, writing each digit at a separate lattice point, starting with 0 at the origin and 1 at x=0, y=-1; sequence gives the numbers on the negative x-axis.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

			  2---3---2---4---2---5---2
  |                       |
  2   1---3---1---4---1   6
  |   |               |   |
  2   2   4---5---6   5   2
  |   |   |       |   |   |
  1   1   3   0   7   1   7
  |   |   |   |   |   |   |
  2   1   2---1   8   6   2
  |   |           |   |   |
  0   1---0---1---9   1   8
  |                   |   |
  2---9---1---8---1---7   2
We begin with the 0 and wrap the numbers 1 2 3 4 ... around it. Then the sequence is obtained by reading leftwards, starting from the initial 0. - _Andrew Woods_, May 20 2012

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Sequences based on the same spiral: A033953, A033988, A033990. Spiral without zero: A033952.

Other sequences from spirals: A001107, A002939, A007742, A033951, A033954, A033991, A002943, A033996, A033988.

a033989 0 = 0
a033989 n = a033307 $ a185950 n  -- Reinhard Zumkeller, Aug 14 2013

nmax = 1000; A033307 = Flatten[IntegerDigits /@ Range[0, nmax^2+10*nmax]]; a[n_] := If[n==0, 0, A033307[[4n^2-n+1]]]; Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Apr 23 2017, after Andrew Woods *)

a(n) = A033307(4*n^2-n-1) for n > 0. - Andrew Woods, May 20 2012

More terms from Andrew J. Gacek (andrew(AT)dgi.net)

Edited by Charles R Greathouse IV, Nov 01 2009

A115258 Isolated primes in Ulam's lattice (1, 2, ... in spiral).

Original entry on oeis.org

83, 101, 127, 137, 163, 199, 233, 311, 373, 443, 463, 491, 541, 587, 613, 631, 641, 659, 673, 683, 691, 733, 757, 797, 859, 881, 911, 919, 953, 971, 991, 1013, 1051, 1061, 1103, 1109, 1117, 1193, 1201, 1213, 1249, 1307, 1319, 1409, 1433, 1459, 1483, 1487

Offset: 1

Giorgio Balzarotti and Paolo P. Lava, Feb 17 2006

nonn

Isolated prime numbers have no adjacent primes in a lattice generated by writing consecutive integers starting from 1 in a spiral distribution. If n0 is the number of isolated primes and p the number of primes less than N, the ratio n0/p approaches 1 as N increases. If n1, n2, n3, n4 denote the number of primes with respectively 1, 2, 3, 4 adjacent primes in the lattice, the ratios n1/n0, n2/n1, n3/n2, n4/n3 approach 0 as N increases. The limits stand for any 2D lattice of integers generated by a priori criteria (i.e., not knowing distributions of primes) as Ulam's lattice.

			83 is an isolated prime as the adjacent numbers in lattice 50, 51, 81, 82, 84, 123, 124, 125 are not primes.
From _Michael De Vlieger_, Dec 22 2015: (Start)
Spiral including n <= 17^2 showing only primes, with the isolated primes in parentheses (redrawn by _Jon E. Schoenfield_, Aug 06 2017):
  257 .  .  .  .  . 251 .  .  .  .  .  .  .  .  . 241
   . 197 .  .  . 193 . 191 .  .  .  .  .  .  .  .  .
   .  .  .  .  .  .  .  . 139 .(137).  .  .  .  . 239
   .(199).(101).  .  . 97  .  .  .  .  .  .  . 181 .
   .  .  .  .  .  .  .  . 61  . 59  .  .  . 131 .  .
   .  .  . 103 . 37  .  .  .  .  . 31  . 89  . 179 .
  263 . 149 . 67  . 17  .  .  . 13  .  .  .  .  .  .
   .  .  .  .  .  .  .  5  .  3  . 29  .  .  .  .  .
   .  . 151 .  .  . 19  .  .  2 11  . 53  .(127).(233)
   .  .  . 107 . 41  .  7  .  .  .  .  .  .  .  .  .
   .  .  .  . 71  .  .  . 23  .  .  .  .  .  .  .  .
   .  .  . 109 . 43  .  .  . 47  .  .  .(83) . 173 .
  269 .  .  . 73  .  .  .  .  . 79  .  .  .  .  . 229
   .  .  .  .  . 113 .  .  .  .  .  .  .  .  .  .  .
  271 . 157 .  .  .  .  .(163).  .  . 167 .  .  . 227
   . 211 .  .  .  .  .  .  .  .  .  .  . 223 .  .  .
   .  .  .  . 277 .  .  . 281 . 283 .  .  .  .  .  .
(End)

G. Balzarotti and P. P. Lava, Le sequenze di numeri interi, Hoepli, 2008, p. 22.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Spiral.

Cf. A001107, A002939, A007742, A033951-A033954, A033989, A033990, A033991, A002943, A033996, A033988, A014848.

Cf. A113688 (isolated semiprimes in the semiprime spiral), A156859.

# A is Ulam's lattice
if (isprime(A[x,y])and(not(isprime(A[x+1,y]) or isprime(A[x-1,y])or isprime(A[x,y+1])or isprime(A[x,y-1])or isprime(A[x-1,y-1])or isprime(A[x+1,y+1])or isprime(A[x+1,y-1])or isprime(A[x-1,y+1])))) then print (A[x,y]) ; fi;

spiral[n_] := Block[{o = 2 n - 1, t, w}, t = Table[0, {o}, {o}]; t = ReplacePart[t, {n, n} -> 1]; Do[w = Partition[Range[(2 (# - 1) - 1)^2 + 1, (2 # - 1)^2], 2 (# - 1)] &@ k; Do[t = ReplacePart[t, {(n + k) - (j + 1), n + (k - 1)} -> #[[1, j]]]; t = ReplacePart[t, {n - (k - 1), (n + k) - (j + 1)} -> #[[2, j]]]; t = ReplacePart[t, {(n - k) + (j + 1), n - (k - 1)} -> #[[3, j]]]; t = ReplacePart[t, {n + (k - 1), (n - k) + (j + 1)} -> #[[4, j]]], {j, 2 (k - 1)}] &@ w, {k, 2, n}]; t]; f[w_] := Block[{d = Dimensions@ w, t, g}, t = Reap[Do[Sow@ Take[#[[k]], {2, First@ d - 1}], {k, 2, Last@ d - 1}]][[-1, 1]] &@ w; g[n_] := If[n != 0, Total@ Join[Take[w[[Last@ # - 1]], {First@ # - 1, First@ # + 1}], {First@ #, Last@ #} &@ Take[w[[Last@ #]], {First@ # - 1, First@ # + 1}], Take[w[[Last@ # + 1]], {First@ # - 1, First@# + 1}]] &@(Reverse@ First@ Position[t, n] + {1, 1}) == 0, False]; Select[Union@ Flatten@ t, g@ # &]]; f[spiral@ 21 /. n_ /; CompositeQ@ n -> 0] (* Michael De Vlieger, Dec 22 2015, Version 10 *)

A118729 Rectangular array where row r contains the 8 numbers 4r^2 - 3r, 4r^2 - 2r, ..., 4r^2 + 4r.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 18, 20, 22, 24, 27, 30, 33, 36, 39, 42, 45, 48, 52, 56, 60, 64, 68, 72, 76, 80, 85, 90, 95, 100, 105, 110, 115, 120, 126, 132, 138, 144, 150, 156, 162, 168

Offset: 0

Stuart M. Ellerstein (ellerstein(AT)aol.com), May 21 2006

The numbers in row r span the interval ]8*A000217(r-1), 8*A000217(r)].
The first difference between the entries in row r is r.
Partial sums of floor(n/8). - Philippe Deléham, Mar 26 2013
Apart from the initial zeros, the same as A008726. - Philippe Deléham, Mar 28 2013
a(n+7) is the number of key presses required to type a word of n letters, all different, on a keypad with 8 keys where 1 press of a key is some letter, 2 presses is some other letter, etc., and under an optimal mapping of letters to keys and presses (answering LeetCode problem 3014). - Christopher J. Thomas, Feb 16 2024

			The array starts, with row r=0, as
  r=0:   0  0  0  0  0  0  0  0;
  r=1:   1  2  3  4  5  6  7  8;
  r=2:  10 12 14 16 18 20 22 24;
  r=3:  27 30 33 36 39 42 45 48;

G. C. Greubel, Table of n, a(n) for n = 0..1000
LeetCode, 3014. Minimum Number of Pushes to Type Word I.
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,1,-2,1).

Cf. similar sequences: A000217, A002620, A130518, A130519, A130520, A174709, A174738, A218470, A131242.

Flatten[Table[4r^2+r(Range[-3,4]),{r,0,6}]] (* or *) LinearRecurrence[ {2,-1,0,0,0,0,0,1,-2,1},{0,0,0,0,0,0,0,0,1,2},60] (* Harvey P. Dale, Nov 26 2015 *)

From Philippe Deléham, Mar 26 2013: (Start)
a(8k)   = A001107(k).
a(8k+1) = A002939(k).
a(8k+2) = A033991(k).
a(8k+3) = A016742(k).
a(8k+4) = A007742(k).
a(8k+5) = A002943(k).
a(8k+6) = A033954(k).
a(8k+7) = A033996(k). (End)
G.f.: x^8/((1-x)^2*(1-x^8)). - Philippe Deléham, Mar 28 2013
a(n) = floor(n/8)*(n-3-4*floor(n/8)). - Ridouane Oudra, Jun 04 2019
a(n+7) = (1/2)*(n+(n mod 8))*(floor(n/8)+1). - Christopher J. Thomas, Feb 13 2024

Redefined as a rectangular tabf array and description simplified by R. J. Mathar, Oct 20 2010

A226492 a(n) = n(11n-5)/2.

Original entry on oeis.org

0, 3, 17, 42, 78, 125, 183, 252, 332, 423, 525, 638, 762, 897, 1043, 1200, 1368, 1547, 1737, 1938, 2150, 2373, 2607, 2852, 3108, 3375, 3653, 3942, 4242, 4553, 4875, 5208, 5552, 5907, 6273, 6650, 7038, 7437, 7847, 8268, 8700, 9143, 9597, 10062, 10538, 11025, 11523

Offset: 0

Bruno Berselli, Jun 11 2013

Sequences of numbers of the form n*(n*k - k + 6)/2:
. k from 0 to 10, respectively: A008585, A055998, A005563, A045943, A014105, A005475, A033428, A022264, A033991, A062741, A147874;
. k=11: a(n);
. k=12: A094159;
. k=13: 0, 3, 19, 48, 90, 145, 213, 294, 388, 495, 615, 748, 894, ...;
. k=14: 0, 3, 20, 51, 96, 155, 228, 315, 416, 531, 660, 803, 960, ...;
. k=15: A152773;
. k=16: A139272;
. k=17: 0, 3, 23, 60, 114, 185, 273, 378, 500, 639, 795, 968, ...;
. k=18: A152751;
. k=19: 0, 3, 25, 66, 126, 205, 303, 420, 556, 711, 885, 1078, ...;
. k=20: 0, 3, 26, 69, 132, 215, 318, 441, 584, 747, 930, 1133, ...;
. k=21: A152759;
. k=22: 0, 3, 28, 75, 144, 235, 348, 483, 640, 819, 1020, 1243, ...;
. k=23: 0, 3, 29, 78, 150, 245, 363, 504, 668, 855, 1065, 1298, ...;
. k=24: A152767;
. k=25: 0, 3, 31, 84, 162, 265, 393, 546, 724, 927, 1155, 1408, ...;
. k=26: 0, 3, 32, 87, 168, 275, 408, 567, 752, 963, 1200, 1463, ...;
. k=27: A153783;
. k=28: A195021;
. k=29: 0, 3, 35, 96, 186, 305, 453, 630, 836, 1071, 1335, 1628, ...;
. k=30: A153448;
. k=31: 0, 3, 37, 102, 198, 325, 483, 672, 892, 1143, 1425, 1738, ...;
. k=32: 0, 3, 38, 105, 204, 335, 498, 693, 920, 1179, 1470, 1793, ...;
. k=33: A153875.
Also:
a(n) - n         = A180223(n);
a(n) + n         = n*(11*n-3)/2 = 0, 4, 19, 45, 82, 130, 189, 259, ...;
a(n) - 2*n       = A051865(n);
a(n) + 2*n       = A022268(n);
a(n) - 3*n       = A152740(n-1);
a(n) + 3*n       = A022269(n);
a(n) - 4*n       = n*(11*n-13)/2 = 0, -1, 9, 30, 62, 105, 159, 224, ...;
a(n) + 4*n       = A254963(n);
a(n) - n*(n-1)/2 = A147874(n+1);
a(n) + n*(n-1)/2 = A094159(n) (case k=12);
a(n) - n*(n-1)   = A062741(n) (see above, this is the case k=9);
a(n) + n*(n-1)   = n*(13*n-7)/2 (case k=13);
a(n) - n*(n+1)/2 = A135706(n);
a(n) + n*(n+1)/2 = A033579(n);
a(n) - n*(n+1)   = A051682(n);
a(n) + n*(n+1)   = A186030(n);
a(n) - n^2       = A062708(n);
a(n) + n^2       = n*(13*n-5)/2 = 0, 4, 21, 51, 94, 150, 219, ..., etc.
Sum of reciprocals of a(n), for n > 0: 0.47118857003113149692081665034891...

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

Cf. sequences in Comments lines.
First differences are in A017425.
Cf. A033584, A180223.

Magma
```
[n*(11*n-5)/2: n in [0..50]];
```

I:=[0,3,17]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..46]]; // Vincenzo Librandi, Aug 18 2013

Table[n (11 n - 5)/2, {n, 0, 50}]
CoefficientList[Series[x (3 + 8 x) / (1 - x)^3, {x, 0, 45}], x] (* Vincenzo Librandi, Aug 18 2013 *)
LinearRecurrence[{3,-3,1},{0,3,17},50] (* Harvey P. Dale, Jan 14 2019 *)

a(n)=n*(11*n-5)/2 \\ Charles R Greathouse IV, Sep 24 2015

G.f.: x*(3+8*x)/(1-x)^3.
a(n) + a(-n) = A033584(n).
From Elmo R. Oliveira, Dec 27 2024: (Start)
E.g.f.: exp(x)*x*(6 + 11*x)/2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.
a(n) = n + A180223(n). (End)

A054925 a(n) = ceiling(n*(n-1)/4).

Original entry on oeis.org

0, 0, 1, 2, 3, 5, 8, 11, 14, 18, 23, 28, 33, 39, 46, 53, 60, 68, 77, 86, 95, 105, 116, 127, 138, 150, 163, 176, 189, 203, 218, 233, 248, 264, 281, 298, 315, 333, 352, 371, 390, 410, 431, 452, 473, 495, 518, 541, 564, 588, 613, 638, 663, 689, 716, 743, 770, 798

Offset: 0

N. J. A. Sloane, May 24 2000

Number of edges in "median" graph - gives positions of largest entries in rows of table in A054924.
Form the clockwise spiral starting 0,1,2,....; then A054925(n+1) interleaves 2 horizontal (A033951, A033991) and 2 vertical (A007742, A054552) branches. A bisection is A014848. - Paul Barry, Oct 08 2007
Consider the standard 4-dimensional Euclidean lattice. We take 1 step along the positive x-axis, 2 along the positive y-axis, 3 along the positive z-axis, 4 along the positive t-axis, and then back round to the x-axis. This sequence gives the floor of the Euclidean distance to the origin after n steps. - Jon Perry, Apr 16 2013
Jon Perry's JavaScript code is explained by A238604. - Michael Somos, Mar 01 2014
Ceiling of the area under the polygon connecting the lattice points (n, floor(n/2)) from 0..n. - Wesley Ivan Hurt, Jun 09 2014
Ceiling of one-half of each triangular number. - Harvey P. Dale, Oct 03 2016
For n > 2, also the edge cover number of the (n-1)-triangular honeycomb queen graph. - Eric W. Weisstein, Jul 14 2017
Conjecture: For n>11, there always exists a prime number p such that a(n)Raul Prisacariu, Sep 01 2024
For n = 1 up to at least n = 13, also the lower matching number of the triangular honeycomb bishop graph. - Eric W. Weisstein, Dec 13 2024
Conjecturally, apart from the first term, the sequence terms are the exponents in the expansion of Sum_{n >= 0} q^(3*n) * (Product_{k >= 2*n+1} 1 - q^k) = 1 - q - q^2 + q^3 + q^5 - q^8 - q^11 + + - - .... Cf. A039825. - Peter Bala, Apr 13 2025

			a(6) = 8; ceiling(6*(6-1)/4) = ceiling(30/4) = 8.
G.f. = x^2 + 2*x^3 + 3*x^4 + 5*x^5 + 8*x^6 + 11*x^7 + 14*x^8 + 18*x^9 + 23*x^10 + ...

Ivan Panchenko, Table of n, a(n) for n = 0..10000
Craig Knecht, Binary fill patterns in a triangle refilled with natural numbers.
Eric Weisstein's World of Mathematics, Edge Cover Number.
Eric Weisstein's World of Mathematics, Triangular Honeycomb Bishop Graph.
Eric Weisstein's World of Mathematics, Triangular Honeycomb Queen Graph.
Index entries for linear recurrences with constant coefficients, signature (3,-4,4,-3,1).

Cf. A007742, A033951, A033991, A039825, A054552, A054924, A054925 + A011848 = C(n, 2).

Cf. A213172, A238604.

p=new Array(0,0,0,0);
for (a=0;a<100;a++) {
p[a%4]+=a;
document.write(Math.floor(Math.sqrt(p[0]*p[0]+p[1]*p[1]+p[2]*p[2]+p[3]*p[3]))+", ");
} /* Jon Perry, Apr 16 2013 */

[ Ceiling(n*(n-1)/4) : n in [0..50] ]; // Wesley Ivan Hurt, Jun 09 2014

I:=[0,0,1,2,3]; [n le 5 select I[n] else 3*Self(n-1)-4*Self(n-2)+4*Self(n-3)-3*Self(n-4)+Self(n-5): n in [1..60]]; // Vincenzo Librandi, Jul 14 2015

seq(ceil(binomial(n,2)/2), n=0..57); # Zerinvary Lajos, Jan 12 2009

Table[Ceiling[(n^2 - n)/4], {n, 0, 20}] (* Wesley Ivan Hurt, Nov 01 2013 *)
LinearRecurrence[{3, -4, 4, -3, 1}, {0, 0, 1, 2, 3}, 60] (* Vincenzo Librandi, Jul 14 2015 *)
Join[{0}, Ceiling[#/2] &/ @ Accumulate[Range[0, 60]]] (* Harvey P. Dale, Oct 03 2016 *)
Ceiling[Binomial[Range[0, 20], 2]/2] (* Eric W. Weisstein, Dec 13 2024 *)
Table[Ceiling[Binomial[n, 2]/2], {n, 0, 20}] (* Eric W. Weisstein, Dec 13 2024 *)
Table[(1 + (n - 1) n - Cos[n Pi/2] - Sin[n Pi/2])/4, {n, 0, 20}] (* Eric W. Weisstein, Dec 13 2024 *)
CoefficientList[Series[x^2 (-1 + x - x^2)/((-1 + x)^3 (1 + x^2)), {x, 0, 20}], x] (* Eric W. Weisstein, Dec 13 2024 *)

{a(n) = ceil( n * (n-1)/4)}; /* Michael Somos, Feb 11 2004 */

[ceil(binomial(n,2)/2) for n in range(0,58)] # Zerinvary Lajos, Dec 01 2009

Euler transform of length 6 sequence [ 2, 0, 1, 1, 0, -1]. - Michael Somos, Sep 02 2006
From Michael Somos, Feb 11 2004: (Start)
G.f.: x^2 * (x^2 - x + 1) / ((1 - x)^3 * (1 + x^2)) = x^2 * (1 - x^6) / ((1 - x)^2 * (1 - x^3) * (1 - x^4)).
a(1-n) = a(n).
A011848(n) = a(-n). (End)
From Michael Somos, Mar 01 2014: (Start)
a(n + 4) = a(n) + 2*n + 3.
a(n+1) = floor( sqrt( A238604(n))). (End)
a(n) = A011848(n) + A133872(n+2). - Wesley Ivan Hurt, Jun 09 2014
Sum_{n>=2} 1/a(n) = 4 - Pi + 2*Pi*sinh(sqrt(7)*Pi/4)/(sqrt(7)*(1/sqrt(2)+cosh(sqrt(7)*Pi/4))). - Amiram Eldar, Dec 23 2024

A152743 6 times pentagonal numbers: a(n) = 3n(3*n-1).

Original entry on oeis.org

0, 6, 30, 72, 132, 210, 306, 420, 552, 702, 870, 1056, 1260, 1482, 1722, 1980, 2256, 2550, 2862, 3192, 3540, 3906, 4290, 4692, 5112, 5550, 6006, 6480, 6972, 7482, 8010, 8556, 9120, 9702, 10302, 10920, 11556, 12210, 12882, 13572, 14280, 15006, 15750, 16512, 17292

Offset: 0

Omar E. Pol, Dec 12 2008

a(n) is also the Wiener index of the windmill graph D(4,n). The windmill graph D(m,n) is the graph obtained by taking n copies of the complete graph K_m with a vertex in common (i.e. a bouquet of n pieces of K_m graphs). The Wiener index of a connected graph is the sum of distances between all unordered pairs of vertices in the graph. The Wiener index of D(m,n) is (1/2)n(m-1)[(m-1)(2n-1)+1]. For the Wiener indices of D(3,n), D(5,n), and D(6,n) see A033991, A028994, and A180577, respectively. - Emeric Deutsch, Sep 21 2010

a(n+1) gives the number of edges in a hexagon-like honeycomb built from A003215(n) congruent regular hexagons (see link). Example: a hexagon-like honeycomb consisting of 7 congruent regular hexagons has 1 core hexagon inside a perimeter of six hexagons. The perimeter consists of 18 external edges. There are 6 edges shared by the perimeter hexagons. The core hexagon has 6 edges. a(2) is the total number of edges, i.e. 18 + 6 + 6 = 30. - Ivan N. Ianakiev, Mar 10 2015

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Ivan N. Ianakiev, Hexagon-like honeycomb built from regular congruent hexagons.
Eric Weisstein's World of Mathematics, Windmill Graph.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000326, A152734, A152744, A049450, A062741, A033991, A028994, A180577.

[ 3*n*(3*n-1) : n in [0..50] ]; // Wesley Ivan Hurt, Jun 09 2014

A152743:=n->3*n*(3*n-1); seq(A152743(n), n=0..50); # Wesley Ivan Hurt, Jun 09 2014

Table[3n(3n-1),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{0,6,30},40] (* Harvey P. Dale, Jun 30 2011 *)
CoefficientList[Series[-6x (2x+1)/(x-1)^3,{x,0,40}],x] (* Robert G. Wilson v, Mar 10 2015 *)

a(n)=3*n*(3*n-1) \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 9n^2 - 3n = A000326(n)*6.
a(n) = A049450(n)*3 = A062741(n)*2. - Omar E. Pol, Dec 15 2008
a(n) = a(n-1) + 18*n - 12 (with a(0)=0). - Vincenzo Librandi, Nov 26 2010
G.f.: -((6*x*(2*x+1))/(x-1)^3). - Harvey P. Dale, Jun 30 2011
E.g.f.: 3*x*(2+3*x)*exp(x). - G. C. Greubel, Sep 01 2018
From Amiram Eldar, Feb 27 2022: (Start)
Sum_{n>=1} 1/a(n) = (9*log(3) - sqrt(3)*Pi)/18.
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi*sqrt(3) - 6*log(2))/9. (End)

Converted reference to link by Omar E. Pol, Oct 07 2010

cp's OEIS Frontend

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

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A033952 Write 1,2,... in a clockwise spiral on a square lattice, writing each digit at a separate lattice point, starting with 1 at the origin and 2 at x=0, y=-1; sequence gives the numbers on the positive x-axis.

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A005476 a(n) = n*(5*n - 1)/2.

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A033988 Write 0,1,2,... in a clockwise spiral on a square lattice, writing each digit at a separate lattice point, starting with 0 at the origin and 1 at x=0, y=-1; sequence gives the numbers on the positive y-axis.

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A033989 Write 0,1,2,... in a clockwise spiral on a square lattice, writing each digit at a separate lattice point, starting with 0 at the origin and 1 at x=0, y=-1; sequence gives the numbers on the negative x-axis.

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A115258 Isolated primes in Ulam's lattice (1, 2, ... in spiral).

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A118729 Rectangular array where row r contains the 8 numbers 4*r^2 - 3*r, 4*r^2 - 2*r, ..., 4*r^2 + 4*r.

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A005476 a(n) = n(5n - 1)/2.

A118729 Rectangular array where row r contains the 8 numbers 4r^2 - 3r, 4r^2 - 2r, ..., 4r^2 + 4r.

A226492 a(n) = n(11n-5)/2.

A152743 6 times pentagonal numbers: a(n) = 3n(3*n-1).