A087483 -id:A087483 - cp's OEIS frontend

A194067 Natural interspersion of A087483; a rectangular array, by antidiagonals.

1, 2, 3, 4, 5, 8, 6, 7, 11, 12, 9, 10, 15, 16, 21, 13, 14, 19, 20, 26, 27, 17, 18, 24, 25, 32, 33, 40, 22, 23, 30, 31, 38, 39, 47, 48, 28, 29, 36, 37, 45, 46, 55, 56, 65, 34, 35, 43, 44, 53, 54, 63, 64, 74, 75, 41, 42, 51, 52, 61, 62, 72, 73, 84, 85, 96, 49, 50, 59

Offset: 1

Clark Kimberling, Aug 14 2011

See A194029 for definitions of natural fractal sequence and natural interspersion. Every positive integer occurs exactly once (and every pair of rows intersperse), so that as a sequence, A194067 is a permutation of the positive integers; its inverse is A194068.

			Northwest corner:
1...2...4...6...9...13
3...5...7...10..14..18
8...11..15..19..24..30
12..16..20..25..31..37
21..26..32..38..45..53

Cf. A194029, A194066, A194068, A087483.

z = 70;
c[k_] := 1 + Floor[(1/3) k^2];
c = Table[c[k], {k, 1, z}]  (* A087483 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
f = Table[f[n], {n, 1, 300}]   (* A194066 *)
r[n_] := Flatten[Position[f, n]]
t[n_, k_] := r[n][[k]]
TableForm[Table[t[n, k], {n, 1, 7}, {k, 1, 7}]]
p = Flatten[Table[t[k, n - k + 1], {n, 1, 14}, {k, 1, n}]] (* A194067 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 90}]] (* A194068 *)

A194066 Natural fractal sequence of A087483.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Aug 14 2011

nonn

See A194029 for definitions of natural fractal sequence and natural interspersion.

Cf. A194029, A194067, A087483.

z = 70;
c[k_] := 1 + Floor[(1/3) k^2];
c = Table[c[k], {k, 1, z}]  (* A087483 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
f = Table[f[n], {n, 1, 300}]   (* A194066 *)
r[n_] := Flatten[Position[f, n]]
t[n_, k_] := r[n][[k]]
TableForm[Table[t[n, k], {n, 1, 7}, {k, 1, 7}]]
p = Flatten[Table[t[k, n - k + 1], {n, 1, 14}, {k, 1, n}]] (* A194067 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 90}]] (* A194068 *)

A087465 Rank array R of 3/2 read by antidiagonals; this array is the dispersion of the complement of the sequence given by r(n) = r(n-1) + 1 + floor(3n/2) for n>=1, with r(0) = 1; that is, A077043(n+1).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 6, 8, 10, 12, 9, 11, 14, 16, 19, 13, 15, 18, 21, 24, 27, 17, 20, 23, 26, 30, 33, 37, 22, 25, 29, 32, 36, 40, 44, 48, 28, 31, 35, 39, 43, 47, 52, 56, 61, 34, 38, 42, 46, 51, 55, 60, 65, 70, 75, 41, 45, 50, 54, 59, 64, 69, 74, 80, 85, 91, 49, 53, 58, 63, 68, 73

Offset: 0

Clark Kimberling, Sep 09 2003

The sequence is a permutation of the positive integers and the array is a transposable dispersion.
Let T(n,k) be the rectangular version of the array at A036561, with northwest corner as shown here:
   1     2     4     8    16    32
   3     6    12    24    48    96
   9    18    36    72   144   288
  27    54   108   216   432   864
Then R(n,k) is the rank of T(n,k) when all the numbers in {T(n,k)} are jointly ranked. - Clark Kimberling, Jan 25 2018

			Northwest corner of R:
   1    2    4    6    9   13   17   22
   3    5    8   11   15   20   25   31
   7   10   14   18   23   29   35   42
  12   16   21   26   32   39   46   54
  19   24   30   36   43   51   59   68
  27   33   40   47   55   64   73   83
  37   44   52   60   69   79   89  100
Let t=3/2; then R(i,j) = rank of (j,i) when all nonnegative integer pairs (a,b) are ranked by the relation << defined as follows: (a,b) << (c,d) if a + b*t < c + d*t, and also (a,b) << (c,d) if a + b*t = c + d*t and b < d.  Thus R(2,1) = 10 is the rank of (1,2) in the list (0,0) << (1,0) << (0,1) << (2,0) << (1,1) << (3,0) << (0,2) << (2,1) << (4,0) << (1,2).

Clark Kimberling, Antidiagonals n = 1..60, flattened
Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.

Cf. A087466, A087468, A087483, A007780 (row 1), A077043 (column 1).

r = 20; r1 = 12;(*r=# rows of T,r1=# rows to show*);
c = 20; c1 = 12;(*c=# cols of T,c1=# cols to show*);
s[0] = 1; s[n_] := s[n] = s[n - 1] + 1 + Floor[3 n/2]; u = Table[s[n], {n, 0, 100}]
v = Complement[Range[Max[u]], u]; f[n_] := v[[n]]; Table[f[n], {n, 1, 30}]
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1,   Length[Union[list]]]; rows = {NestList[f, 1, c]}; Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}]; w[i_, j_] := rows[[i, j]];
TableForm[Table[w[i, j], {i, 1, 10}, {j, 1, 10}]]   (* A087465 array *)
Flatten[Table[w[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A087465 sequence *)
TableForm[Table[w[i, 1] + w[1, j] + (i - 1)*(j - 1) - 1, {i, 1, 10}, {j, 1, 10}]] (* A087465 array, by formula *)

R(i,j) = R(i,0) + R(0,j) + i*j - 1, for i>=1, j>=1.

Updated by Clark Kimberling, Sep 23 2014

A281333 a(n) = 1 + floor(n/2) + floor(n^2/3).

Original entry on oeis.org

1, 1, 3, 5, 8, 11, 16, 20, 26, 32, 39, 46, 55, 63, 73, 83, 94, 105, 118, 130, 144, 158, 173, 188, 205, 221, 239, 257, 276, 295, 316, 336, 358, 380, 403, 426, 451, 475, 501, 527, 554, 581, 610, 638, 668, 698, 729, 760, 793, 825, 859, 893, 928, 963, 1000, 1036, 1074, 1112, 1151, 1190

Offset: 0

Bruno Berselli, Jan 20 2017

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

Subsequences: A033577, A244805 (numbers of the form 1 + k/2 + k^2/3), A212978 (second bisection).
Cf. A236771: n + floor(n/2) + floor(n^2/3).
Cf. A008619: 1 + floor(n/2); A087483: 1 + floor(n^2/3).
Cf. A000212, A004526.

[1 + n div 2 + n^2 div 3: n in [0..60]];

A281333:=n->1 + floor(n/2) + floor(n^2/3): seq(A281333(n), n=0..100); # Wesley Ivan Hurt, Feb 09 2017

Table[1 + Floor[n/2] + Floor[n^2/3], {n, 0, 60}]
LinearRecurrence[{1,1,0,-1,-1,1},{1,1,3,5,8,11},80] (* Harvey P. Dale, Sep 29 2024 *)

makelist(1+floor(n/2)+floor(n^2/3), n, 0, 60);

vector(60, n, n--; 1+floor(n/2)+floor(n^2/3))

[1+int(n/2)+int(n**2/3) for n in range(60)]

[1+floor(n/2)+floor(n^2/3) for n in range(60)]

G.f.: (1 + x^2 + x^3 + x^4)/((1 + x)*(1 + x + x^2)*(1 - x)^3).
a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6).
a(n) = 1 + floor(n/2 + n^2/3).
a(n) = (12*n^2 + 18*n + 4*(-1)^(2*n/3) + 4*(-1)^(-2*n/3) + 9*(-1)^n + 19)/36.
a(n) - n = a(-n).
a(6*k+r) = 12*k^2 + (4*r+3)*k + a(r), where 0 <= r <= 5. Particular cases:
a(6*k) = A244805(k+1), a(6*k+1) = A033577(k).
a(n+2) -  a(n) =   A004773(n+2).
a(n+3) -  a(n) =   A014601(n+2).
a(n+4) -  a(n) =   A047480(n+3).
a(n) - a(-n+3) = 2*A001651(n-1).
a(n) + a(-n+3) = 2*A097922(n-1).
a(n) = 1 + A004526(n) + A000212(n) = A008619(n) + A000212(n). - Omar E. Pol, Dec 23 2020

A192735 Left edge of the triangle in A033291.

Original entry on oeis.org

1, 2, 6, 16, 30, 54, 91, 136, 198, 280, 374, 492, 637, 798, 990, 1216, 1462, 1746, 2071, 2420, 2814, 3256, 3726, 4248, 4825, 5434, 6102, 6832, 7598, 8430, 9331, 10272, 11286, 12376, 13510, 14724, 16021, 17366, 18798, 20320, 21894, 23562, 25327, 27148, 29070

Offset: 1

Reinhard Zumkeller, Jul 08 2011

nonn

a(n) = A087483(n-2) * n.

Index entries for linear recurrences with constant coefficients, signature (2,-1,2,-4,2,-1,2,-1).

Cf. A192736.

Haskell
```
See A033291.
```

A192735 := proc(n)
    A087483(n-2)*n ;
end proc:
seq(A192735(n),n=1..10) ; # R. J. Mathar, Aug 10 2017

G.f.: x*(x^2+1)*(2*x^4+4*x^3+2*x^2+1) / ((x-1)^4*(x^2+x+1)^2). - Colin Barker, Mar 31 2013

A053799 Number of basis partitions of n+9 with Durfee square size 3.

Original entry on oeis.org

1, 2, 4, 8, 12, 18, 26, 34, 44, 56, 68, 82, 98, 114, 132, 152, 172, 194, 218, 242, 268, 296, 324, 354, 386, 418, 452, 488, 524, 562, 602, 642, 684, 728, 772, 818, 866, 914, 964, 1016, 1068, 1122, 1178, 1234, 1292, 1352, 1412, 1474, 1538, 1602, 1668, 1736

Offset: 0

James Sellers, Mar 27 2000

a(n) is the number of solutions in integers (x,y,z) of |x| + 2|y| + 3|z| = |n|. - Michael Somos, Jul 17 2018

			G.f. = 1 + 2*x + 4*x^2 + 8*x^3 + 12*x^4 + 18*x^5 + 26*x^6 + 34*x^7 + ... - _Michael Somos_, Jul 17 2018

Seiichi Manyama, Table of n, a(n) for n = 0..10000
M. D. Hirschhorn, Basis partitions and Rogers-Ramanujan partitions, Discrete Math. 205 (1999), 241-243.
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

LinearRecurrence[{2,-1,1,-2,1},{1,2,4,8,12,18},60] (* Harvey P. Dale, Aug 25 2015 *)
a[ n_] := 2 Quotient[ n^2, 3] + 2 - Boole[n == 0]; (* Michael Somos, Jul 17 2018 *)
a[ n_] := SeriesCoefficient[ (1 + x^2) (1 + x^3) / ((1 - x)^3 (1 + x + x^2)), {x, 0, Abs@n}]; (* Michael Somos, Jul 17 2018 *)
a[ n_] := Length @ FindInstance[ Abs[x] + 2 Abs[y] + 3 Abs[z] == Abs[n], {x, y, z}, Integers, 10^9]; (* Michael Somos, Jul 17 2018 *)

{a(n) = n^2 \ 3 * 2 + 2 - (n==0)}; /* Michael Somos, Jul 17 2018 */

For n>0, a(n) = 2*(1+floor(n^2/3)) = 2*A087483(n-1) = 2*(1+A000212(n)). - Max Alekseyev, Dec 05 2013
G.f.: (1+x)*(1+x^2)*(1+x^3)/((1-x)*(1-x^2)*(1-x^3)) = (1+x)*(1+x^2)*(1-x+x^2)/((1-x)^3*(1+x+x^2)).
a(n) = A000982(n)+A008749(n). - John Mason, Jan 08 2015
From Michael Somos, Jul 17 2018: (Start)
Euler transform of length 6 sequence [2, 1, 2, -1, 0, -1].
a(n+1) - 2*a(n) + a(n-1) = 1 + (-1)^n if |n|>1.
a(n) = a(-n) for all n in Z. (End)

cp's OEIS Frontend

A194067 Natural interspersion of A087483; a rectangular array, by antidiagonals.

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A194066 Natural fractal sequence of A087483.

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A087465 Rank array R of 3/2 read by antidiagonals; this array is the dispersion of the complement of the sequence given by r(n) = r(n-1) + 1 + floor(3n/2) for n>=1, with r(0) = 1; that is, A077043(n+1).

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A281333 a(n) = 1 + floor(n/2) + floor(n^2/3).

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A192735 Left edge of the triangle in A033291.

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A053799 Number of basis partitions of n+9 with Durfee square size 3.

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