A049450 -id:A049450 - cp's OEIS frontend

A194767 Denominator of the fourth increasing diagonal of the autosequence of second kind from (-1)^n / (n+1).

2, 2, 12, 20, 10, 42, 56, 24, 90, 110, 44, 156, 182, 70, 240, 272, 102, 342, 380, 140, 462, 506, 184, 600, 650, 234, 756, 812, 290, 930, 992, 352, 1122, 1190, 420, 1332, 1406, 494, 1560, 1640, 574, 1806, 1892, 660, 2070, 2162, 752, 2352, 2450, 850, 2652, 2756, 954, 2970, 3080, 1064, 3306, 3422, 1180, 3660

Offset: 0

Paul Curtz, Sep 02 2011

The autosequence of first kind from (-1)^n/(n+1) is A189733.
For the second kind (the second increasing diagonal is (-1)^n/(n+1), half of the main one):
      2,     1,     0,  -1/2,  -1/3,   1/6,   1/2,  5/12,
     -1,    -1,  -1/2,   1/6,   1/2,   1/3, -1/12, -7/20,
      0,   1/2,   2/3,   1/3,  -1/6, -5/12, -4/15,  1/12,
    1/2,   1/6,  -1/3,  -1/2,  -1/4,  3/20,  7/20, 13/60,
   -1/3,  -1/2,  -1/6,   1/4,   2/5,   1/5, -2/15, -3/10,
   -1/6,   1/3,  5/12,  3/20,  -1/5,  -1/3,  -1/6,  5/42,
    1/2,  1/12, -4/15, -7/20, -2/15,   1/6,   2/7,   1/7,
  -5/12, -7/20, -1/12, 13/60,  3/10,  5/42,  -1/7,  -1/4.
Main diagonal: (period 2:repeat 2, -1)/A026741(n+1).
Second (increasing) diagonal: (-1)^n / (n+1).
Third (increasing) diagonal: (-1)^(n+1)*A026741(n) / A045896(n).
Fourth (increasing) diagonal: (-1)^(n+1)*A146535(n)/ a(n).

OEIS Wiki, Autosequence.
Index entries for linear recurrences with constant coefficients, signature (0,0,3,0,0,-3,0,0,1).

Cf. A001504 = 2*A060544, A049450 = 2*A000326, A045945 = 6*A005449.

Cf. A026741, A045896, A051176, A146535, A189733, A306368.

c = Table[1/9 (7 n + 7 n^2 + 2 n Cos[2 n *Pi/3] + 2 n^2 Cos[2 n *Pi/3] + 2 Sqrt[3] n Sin[2 n *Pi/3] + 2 Sqrt[3] n^2 Sin[2 n *Pi/3]), {n, 1, 50}] (* Roger Bagula, Mar 25 2012 *)
a[n_] := (n+1) * Numerator[(n+2)/3]; Array[a, 60, 0] (* Amiram Eldar, Sep 17 2023 *)
LinearRecurrence[{0,0,3,0,0,-3,0,0,1},{2,2,12,20,10,42,56,24,90},60] (* Harvey P. Dale, May 15 2025 *)

a(3*n) = (3*n+1)*(3*n+2), a(3*n+1) = (n+1)*(3*n+2), a(3*n+2) = 3*(n+1)*(3*n+4).
G.f.: 2*(1+x+6*x^2+7*x^3+2*x^4+3*x^5+x^6)/(1-x^3)^3. - Jean-François Alcover, Nov 11 2016
a(n+2) = 2 * A306368(n) for n >= 0. - Joerg Arndt, Aug 25 2023
a(n) = (n+1) * A051176(n+2) for n >= 0. - Paul Curtz, Sep 13 2023
Sum_{n>=0} 1/a(n) = 1 + log(3) - Pi/(3*sqrt(3)). - Amiram Eldar, Sep 17 2023

A250328 Denominator of the harmonic mean of the first n pentagonal numbers.

Original entry on oeis.org

1, 3, 77, 877, 6271, 36049, 36423, 422137, 49691099, 1448086909, 11631128477, 2334008785, 44471893747, 1827784004699, 832564679309, 39202882860913, 196334425398149, 3473612060358899, 3478128507653999, 205449856947685261, 303604578504856471

Offset: 1

Colin Barker, Nov 18 2014

a(n+1) is, for n >= 0, also the numerator of the partial sums of the reciprocals of twice the pentagonal numbers {A049450(k+1)}A294513(n)%20(assuming%20that%20A250327(n+1)/(n+1)%20=%20A294513(n)/2).%20-%20_Wolfdieter%20Lang">{k>=0} with the denominators given in A294513(n) (assuming that A250327(n+1)/(n+1) = A294513(n)/2). - _Wolfdieter Lang, Nov 02 2017

			a(3) = 77 because the pentagonal numbers A000326(n), for n = 1,2,3 are 1, 5, 12 and 3/(1/1+1/5+1/12) = 180/77.

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A000326, A250327 (numerators).

With[{s = Array[PolygonalNumber[5, #] &, 21]}, Denominator@ Array[HarmonicMean@ Take[s, #] &, Length@ s]] (* Michael De Vlieger, Nov 02 2017 *)

harmonicmean(v) = #v / sum(k=1, #v, 1/v[k])
s=vector(30); for(k=1, #s, s[k]=denominator(harmonicmean(vector(k, i, (3*i^2-i)/2)))); s

A064045 Square array read by antidiagonals of number of length 2k walks on an n-dimensional hypercubic lattice starting and finishing at the origin and staying in the nonnegative part.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 5, 10, 3, 1, 0, 14, 70, 24, 4, 1, 0, 42, 588, 285, 44, 5, 1, 0, 132, 5544, 4242, 740, 70, 6, 1, 0, 429, 56628, 73206, 16016, 1525, 102, 7, 1, 0, 1430, 613470, 1403028, 410928, 43470, 2730, 140, 8, 1, 0, 4862, 6952660, 29082339, 11925672, 1491210, 96684, 4445, 184, 9, 1

Offset: 0

Henry Bottomley, Aug 23 2001

			Rows start:
1, 0,  0,   0,    0,     0,       0, ...
1, 1,  2,   5,   14,    42,     132, ...
1, 2, 10,  70,  588,  5544,   56628, ...
1, 3, 24, 285, 4242, 73206, 1403028, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6

Rows include A000007, A000108, A005568, A064037. Columns include A000012, A001477, A049450, A064046. Cf. A064044.

a:= proc(n, k) option remember; `if`(n=0, `if`(k=0, 1, 0),
       add(binomial(2*k, 2*j)*binomial(2*j, j)/
       (j+1)*a(n-1, k-j), j=0..k))
    end:
seq(seq(a(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, May 06 2014

a[n_, k_] := a[n, k] = If[n == 0, If[k == 0, 1, 0], Sum[Binomial[2*k, 2*j]* Binomial[2*j, j]/(j+1)*a[n-1, k-j], {j, 0, k}]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Feb 26 2015, after Alois P. Heinz *)

a(n,k) = Sum_{j=0..k} C(2k,2j) c(j) a(n-1,k-j) where c(j) = C(2j,j)/(j+1) = A000108(j) with a(0,0) = 1 and a(0,k) = 0 for k>0.

A294514 Decimal expansion of (3/2)log(3) - Pi/(2sqrt(3)).

Original entry on oeis.org

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

Offset: 0

Wolfdieter Lang, Nov 02 2017

This is the limit of the series V(3,2) := Sum_{k>=0} 1/((k + 1)*(3*k + 1)) = Sum_{k>=0} 1/A049450(k+1) = (1/2)*Sum_{k>=0} (3/(3*k + 1) - 1/(k+1)) with partial sums given in A250328(n+1)/A294513(n).

			0.7410187508850556117958287265627106908292027126877538898170990327...

Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, pp. 189 - 193, with v_2(3) = (1/3)*V(3,2).

Cf. A049450, A250328/A294513.

RealDigits[N[(3/2)*Log[3] - Pi/(2*Sqrt[3]), 157]][[1]] (* Georg Fischer, Apr 04 2020 *)

(3/2)*log(3) - Pi/(2*sqrt(3)) \\ Michel Marcus, Nov 02 2017

Equals V(3,2) = Sum_{k>=0} 1/((k + 1)*(3*k + 1)).

Equals Sum_{k>=2} zeta(k)/3^(k-1). - Amiram Eldar, May 31 2021

a(100) ff. corrected by Georg Fischer, Apr 04 2020

Data truncated by Sean A. Irvine, Apr 10 2020

A306242 Number of ways to write n as x(3x+1) + y(3y-1) + z(3z+2) + w(3w-2), where x,y,z,w are nonnegative integers.

Original entry on oeis.org

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

Offset: 0

Zhi-Wei Sun, Jan 31 2019

nonn

Conjecture: a(n) > 0 for all n >= 0, and a(n) = 1 only for n = 0, 1, 2, 3, 4, 9, 13. Moreover, any nonnegative integer n can be written as x*(3x+1) + y*(3y-1) + z*(3z+2) + w*(3w-2), where x,y,z,w are nonnegative integers with x or y even.
The conjecture has been verified for n up to 10^6.
By Theorem 1.3 of the linked 2017 paper of the author, each nonnegative integer can be written as x*(3x+1) + y*(3y-1) + z*(3z+2) + 0*(3*0-2) with x,y,z integers.
We also have some other similar conjectures. For example, we conjecture that every n = 0,1,2,... can be written as x*(5x+1)/2 + y*(5y-1)/2 + z*(5z+3)/2 + w*(5w-3)/2 with x,y,z,w nonnegative integers.

			a(1) = 1 with 1 = 0*(3*0+1) + 0*(3*0-1) + 0*(3*0+2) + 1*(3*1-2).
a(3) = 1 with 3 = 0*(3*0+1) + 1*(3*1-1) + 0*(3*0+2) + 1*(3*1-2).
a(4) = 1 with 4 = 1*(3*1+1) + 0*(3*0-1) + 0*(3*0+2) + 0*(3*0-2).
a(9) = 1 with 9 = 1*(3*1+1) + 0*(3*0-1) + 1*(3*1+2) + 0*(3*0-2).
a(13) = 1 with 13 = 0*(3*0+1) + 0*(3*0-1) + 1*(3*1+2) + 2*(3*2-2).

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, A result similar to Lagrange's theorem, J. Number Theory 162(2016), 190-211.
Zhi-Wei Sun, On x(ax+1)+y(by+1)+z(cz+1) and x(ax+b)+y(ay+c)+z(az+d), J. Number Theory 171(2017), 275-283.

Cf. A000567, A045944, A049450, A049451, A255350.

OctQ[n_]:=OctQ[n]=IntegerQ[Sqrt[3n+1]]&&(n==0||Mod[Sqrt[3n+1]+1,3]==0);
tab={};Do[r=0;Do[If[OctQ[n-x(3x+2)-y(3y+1)-z(3z-1)],r=r+1],{x,0,(Sqrt[3n+1]-1)/3},{y,0,(Sqrt[12(n-x(3x+2))+1]-1)/6},{z,0,(Sqrt[12(n-x(3x+2)-y(3y+1))+1]+1)/6}];tab=Append[tab,r],{n,0,80}];Print[tab]

A309805 Maximum number of nonattacking kings placeable on a hexagonal board with edge-length n in Glinski's hexagonal chess.

Original entry on oeis.org

1, 2, 7, 10, 19, 24, 37, 44, 61, 70, 91, 102, 127, 140, 169, 184, 217, 234, 271, 290, 331, 352, 397, 420, 469, 494, 547, 574, 631, 660, 721, 752, 817, 850, 919, 954, 1027, 1064, 1141, 1180, 1261, 1302, 1387, 1430, 1519, 1564, 1657, 1704, 1801, 1850, 1951, 2002

Offset: 1

Sangeet Paul, Aug 17 2019

			a(1) = 1
.
  o
.
a(2) = 2
.
   . .
  o . o
   . .
.
a(3) = 7
.
    o . o
   . . . .
  o . o . o
   . . . .
    o . o
.
a(4) = 10
.
     . . . .
    o . o . o
   . . . . . .
  o . o . o . o
   . . . . . .
    o . o . o
     . . . .
.

Chess variants, Glinski's Hexagonal Chess
Wikipedia, Hexagonal chess - Gliński's hexagonal chess
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A003215, A049450.

Partial sums of A133090.

nn:=51; CoefficientList[Series[- (1 + x + 3*x^2 + x^3)/((- 1 + x)^3*(1 + x)^2),{x, 0, nn}], x] (* Georg Fischer, May 10 2020 *)

a(n) = n^2 - (n\2) - (n\2)^2; \\ Andrew Howroyd, Aug 17 2019

def A309805(n): return n**2-(m:=n>>1)*(m+1) # Chai Wah Wu, Apr 04 2024

a(n) = n^2 - floor(n/2) - floor(n/2)^2.
From Stefano Spezia, Aug 18 2019 (Start)
G.f.: - (1 + x + 3*x^2 + x^3)/((- 1 + x)^3*(1 + x)^2).
E.g.f.: (1/8)*exp(-x)*(-1 + 2*x + exp(2*x)*(1 + 4*x + 6*x^2)).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 5.
a(n) = (1/16)*(3 + (-1)^(1+2*n) - 4*n + 12*n^2 - 2*(-1)^n*(1 + 2*n)).
a(2*n-1) = A003215(n).
a(2*n) = A049450(n).
(End)

A064046 Number of length 6 walks on an n-dimensional hypercubic lattice starting and finishing at the origin and staying in the nonnegative part.

Original entry on oeis.org

0, 5, 70, 285, 740, 1525, 2730, 4445, 6760, 9765, 13550, 18205, 23820, 30485, 38290, 47325, 57680, 69445, 82710, 97565, 114100, 132405, 152570, 174685, 198840, 225125, 253630, 284445, 317660, 353365, 391650, 432605, 476320, 522885, 572390

Offset: 0

Henry Bottomley, Aug 23 2001

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Numbers of walks of length 0, 1, 2, 3, 4 and 5 are A000012, A000004, A001477, A000004, A049450 and A000004.

[5*n*(3*n^2-3*n+1): n in [0..40]]; // Vincenzo Librandi, Jun 16 2011

LinearRecurrence[{4,-6,4,-1},{0,5,70,285},40] (* Harvey P. Dale, Dec 02 2012 *)

a(n) = 5*n*(3*n^2 - 3*n + 1).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) = A064045(n, 3).
a(n) = a(n-1) + 15*A049450(n-1) + 30*A001477(n-1) + 5*A000012(n-1).
G.f.: 5*x*(7*x^2 + 10*x + 1)/(x-1)^4. [Colin Barker, Jul 21 2012]

A153780 10 times pentagonal numbers: a(n) = 5n(3*n-1).

Original entry on oeis.org

0, 10, 50, 120, 220, 350, 510, 700, 920, 1170, 1450, 1760, 2100, 2470, 2870, 3300, 3760, 4250, 4770, 5320, 5900, 6510, 7150, 7820, 8520, 9250, 10010, 10800, 11620, 12470, 13350, 14260, 15200, 16170, 17170, 18200, 19260, 20350

Offset: 0

Omar E. Pol, Jan 01 2009

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

Cf. A000326, A049450, A152734, A152996, A153449.

Table[5*n*(3*n - 1), {n,0,25}] (* or *) LinearRecurrence[{3,-3,1},{0,10,50},25] (* G. C. Greubel, Aug 28 2016 *)

a(n) = 5*n*(3*n-1); \\ Michel Marcus, Aug 28 2016

a(n) = 15*n^2 - 5*n = 10*A000326(n) = 5*A049450(n) = 2*A152734(n).
a(n) = 30*n + a(n-1) - 20 for n>0, a(0) = 0. - Vincenzo Librandi, Aug 03 2010
G.f.: 10*x*(1+2*x)/(1-x)^3. - Colin Barker, Feb 14 2012
From G. C. Greubel, Aug 28 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
E.g.f.: 5*x*(2 + 3*x)*exp(x). (End)

A238377 Row sums of triangle in A204028.

Original entry on oeis.org

1, 2, 6, 10, 17, 24, 34, 44, 57, 70, 86, 102, 121, 140, 162, 184, 209, 234, 262, 290, 321, 352, 386, 420, 457, 494, 534, 574, 617, 660, 706, 752, 801, 850, 902, 954, 1009, 1064, 1122, 1180

Offset: 0

Philippe Deléham, Feb 25 2014

			Triangle in A204028 begins:
  1;............................sum = 1
  1, 1;.........................sum = 2
  1, 4, 1;......................sum = 6
  1, 4, 4, 1;...................sum = 10
  1, 4, 7, 4, 1;................sum = 17
  1, 4, 7, 7, 4, 1;.............sum = 24
  1, 4, 7, 10, 7, 4, 1;.........sum = 34
  1, 4, 7, 10, 10, 7, 4, 1;.....sum = 44

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A056109, A049450, A204028.

G.f.: (1+2*x^2)/((1+x)*(1-x)^3).
a(2*n) = A056109(n).
a(2*n+1) = A049450(n+1) = 2*A000326(n+1).
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4), a(0) = 1, a(1) = 2, a(2) = 6, a(3) = 10.

Corrected by R. J. Mathar, Aug 08 2015

A272104 Sum of the even numbers among the larger parts of the partitions of n into two parts.

Original entry on oeis.org

0, 0, 0, 2, 2, 4, 4, 10, 10, 14, 14, 24, 24, 30, 30, 44, 44, 52, 52, 70, 70, 80, 80, 102, 102, 114, 114, 140, 140, 154, 154, 184, 184, 200, 200, 234, 234, 252, 252, 290, 290, 310, 310, 352, 352, 374, 374, 420, 420, 444, 444, 494, 494, 520, 520, 574, 574, 602

Offset: 0

Wesley Ivan Hurt, Apr 20 2016

Essentially, repeated values of A152749.

Sum of the lengths of the distinct rectangles with even length and integer width such that L + W = n, W <= L. For example, a(10) = 14; the rectangles are 2 X 8 and 4 X 6, so 8 + 6 = 14. - Wesley Ivan Hurt, Nov 04 2017

			a(5) = 4; the partitions of 5 into 2 parts are (4,1),(3,2) and the sum of the larger even parts is 4.
a(6) = 4; the partitions of 6 into 2 parts are (5,1),(4,2),(3,3) and the sum of the larger even parts is also 4.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (1,0,0,2,-2,0,0,-1,1).

Cf. A000326, A001318, A026741, A049450, A049452, A109043, A126964, A152749, A268351, A272212.

[(1+3*(2*n-3-(-1)^n)/2+3*(2*n-3-(-1)^n)^2/8+(2*n-1-(-1)^n)*(-1)^((2*n+1-(-1)^n) div 4)/2)/8 : n in [0..50]];

A272104:=n->(1+3*(2*n-3-(-1)^n)/2+3*(2*n-3-(-1)^n)^2/8+(2*n-1-(-1)^n)*(-1)^((2*n+1-(-1)^n)/4)/2)/8: seq(A272104(n), n=0..100);

Table[(1 + 3(2n-3-(-1)^n)/2 + 3(2n-3-(-1)^n)^2/8 + (2n-1-(-1)^n) * (-1)^((2n+1-(-1)^n)/4)/2) / 8, {n, 0, 50}]
Table[Total@ Map[First, IntegerPartitions[n, {2}] /. {k_, } /; OddQ@ k -> Nothing], {n, 0, 57}] (* _Michael De Vlieger, Apr 20 2016, Version 10.2 *)

concat(vector(3), Vec(2*x^3*(1-x+x^2)*(1+x+x^2)/((1-x)^3*(1+x)^2*(1+x^2)^2) + O(x^50))) \\ Colin Barker, Apr 20 2016

a(n) = (1 + 3*(2n-3-(-1)^n)/2 + 3*(2n-3-(-1)^n)^2/8 + (2n-1-(-1)^n) * (-1)^((2n+1-(-1)^n)/4)/2) / 8.
a(n) = Sum_{i=ceiling(n/2)..n-1} i * (i+1 mod 2).
a(n) = Sum_{i=1..floor(n/2)} (n-i) * (n-i+1 mod 2).
a(2n+1) = a(2n+2) = A152749(n) = 2*A001318(n).
G.f.: 2*x^3*(1-x+x^2)*(1+x+x^2) / ((1-x)^3*(1+x)^2*(1+x^2)^2). - Colin Barker, Apr 20 2016
From Wesley Ivan Hurt, Apr 22 2016, Apr 23 2016: (Start)
a(2n+2)-a(2n) = A109043(n) = 2*A026741(n).
a(4n) = A049450(n) = 2*A000326(n),
a(8n) = A126964(n) = 2*A049452(n),
a(12n) = 2*A268351(n).
a(n+1) = A001318(n) - A272212(n+1). (End)
E.g.f.: ((2 + 3*x*(1 + x))*cosh(x) - 2*(cos(x) + x*cos(x) + x*sin(x)) + (-1 + 3*(-1 + x)*x)*sinh(x))/16. - Ilya Gutkovskiy, Apr 29 2016

cp's OEIS Frontend

A194767 Denominator of the fourth increasing diagonal of the autosequence of second kind from (-1)^n / (n+1).

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A250328 Denominator of the harmonic mean of the first n pentagonal numbers.

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A064045 Square array read by antidiagonals of number of length 2k walks on an n-dimensional hypercubic lattice starting and finishing at the origin and staying in the nonnegative part.

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A294514 Decimal expansion of (3/2)*log(3) - Pi/(2*sqrt(3)).

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A306242 Number of ways to write n as x*(3x+1) + y*(3y-1) + z*(3z+2) + w*(3w-2), where x,y,z,w are nonnegative integers.

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A309805 Maximum number of nonattacking kings placeable on a hexagonal board with edge-length n in Glinski's hexagonal chess.

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A064046 Number of length 6 walks on an n-dimensional hypercubic lattice starting and finishing at the origin and staying in the nonnegative part.

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Magma

Mathematica

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A153780 10 times pentagonal numbers: a(n) = 5*n*(3*n-1).

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A294514 Decimal expansion of (3/2)log(3) - Pi/(2sqrt(3)).

A306242 Number of ways to write n as x(3x+1) + y(3y-1) + z(3z+2) + w(3w-2), where x,y,z,w are nonnegative integers.

A153780 10 times pentagonal numbers: a(n) = 5n(3*n-1).