A047212 -id:A047212 - cp's OEIS frontend

A191740 Dispersion of A047220, (numbers >1 and congruent to 0 or 1 or 3 mod 5), by antidiagonals.

1, 3, 2, 6, 5, 4, 11, 10, 8, 7, 20, 18, 15, 13, 9, 35, 31, 26, 23, 16, 12, 60, 53, 45, 40, 28, 21, 14, 101, 90, 76, 68, 48, 36, 25, 17, 170, 151, 128, 115, 81, 61, 43, 30, 19, 285, 253, 215, 193, 136, 103, 73, 51, 33, 22, 476, 423, 360, 323, 228, 173, 123

Offset: 1

Clark Kimberling, Jun 14 2011

For a background discussion of dispersions and their fractal sequences, see A191426.  For dispersions of congruence sequences mod 3, mod 4, or mod 5, see A191655, A191663, A191667, A191702.
...
Suppose that {2,3,4,5,6} is partitioned as {x1, x2} and {x3,x4,x5}.  Let S be the increasing sequence of numbers >1 and congruent to x1 or x2 mod 5, and let T be the increasing sequence of numbers >1 and congruent to x3 or x4 or x5 mod 5.  There are 10 sequences in S, each matched by a (nearly) complementary sequence in T.  Each of the 20 sequences generates a dispersion, as listed here:
...
A191722=dispersion of A008851 (0, 1 mod 5 and >1)
A191723=dispersion of A047215 (0, 2 mod 5 and >1)
A191724=dispersion of A047218 (0, 3 mod 5 and >1)
A191725=dispersion of A047208 (0, 4 mod 5 and >1)
A191726=dispersion of A047216 (1, 2 mod 5 and >1)
A191727=dispersion of A047219 (1, 3 mod 5 and >1)
A191728=dispersion of A047209 (1, 4 mod 5 and >1)
A191729=dispersion of A047221 (2, 3 mod 5 and >1)
A191730=dispersion of A047211 (2, 4 mod 5 and >1)
A191731=dispersion of A047204 (3, 4 mod 5 and >1)
...
A191732=dispersion of A047202 (2,3,4 mod 5 and >1)
A191733=dispersion of A047206 (1,3,4 mod 5 and >1)
A191734=dispersion of A032793 (1,2,4 mod 5 and >1)
A191735=dispersion of A047223 (1,2,3 mod 5 and >1)
A191736=dispersion of A047205 (0,3,4 mod 5 and >1)
A191737=dispersion of A047212 (0,2,4 mod 5 and >1)
A191738=dispersion of A047222 (0,2,3 mod 5 and >1)
A191739=dispersion of A008854 (0,1,4 mod 5 and >1)
A191740=dispersion of A047220 (0,1,3 mod 5 and >1)
A191741=dispersion of A047217 (0,1,2 mod 5 and >1)
...
For further information about these 20 dispersions, see A191722.
...
Regarding the dispersions A191722-A191741, there are general formulas for sequences of the type "(a or b mod m)" and "(a or b or c mod m)" used in the relevant Mathematica programs.

			Northwest corner:
1....3....6....11...20
2....5....10...18...31
4....8....15...26...45
7....13...23...40...68
9....16...28...48...81
12...21...36...61...103

Ivan Neretin, Table of n, a(n) for n = 1..5050 (first 100 antidiagonals, flattened)

Cf. A047211, A047220, A191722, A191730, A191426.

(* Program generates the dispersion array t of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
a=3; b=5; c2=6; m[n_]:=If[Mod[n,3]==0,1,0];
f[n_]:=a*m[n+2]+b*m[n+1]+c2*m[n]+5*Floor[(n-1)/3]
Table[f[n], {n, 1, 30}]  (* A047220 *)
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}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
  (* A191740 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191740  *)

A191741 Dispersion of A047217, (numbers >1 and congruent to 0 or 1 or 2 mod 5), by antidiagonals.

Original entry on oeis.org

1, 2, 3, 5, 6, 4, 10, 11, 7, 8, 17, 20, 12, 15, 9, 30, 35, 21, 26, 16, 13, 51, 60, 36, 45, 27, 22, 14, 86, 101, 61, 76, 46, 37, 25, 18, 145, 170, 102, 127, 77, 62, 42, 31, 19, 242, 285, 171, 212, 130, 105, 71, 52, 32, 23, 405, 476, 286, 355, 217, 176, 120

Offset: 1

Clark Kimberling, Jun 14 2011

For a background discussion of dispersions and their fractal sequences, see A191426.  For dispersions of congruence sequences mod 3, mod 4, or mod 5, see A191655, A191663, A191667, A191702.
...
Suppose that {2,3,4,5,6} is partitioned as {x1, x2} and {x3,x4,x5}.  Let S be the increasing sequence of numbers >1 and congruent to x1 or x2 mod 5, and let T be the increasing sequence of numbers >1 and congruent to x3 or x4 or x5 mod 5.  There are 10 sequences in S, each matched by a (nearly) complementary sequence in T.  Each of the 20 sequences generates a dispersion, as listed here:
...
A191722=dispersion of A008851 (0, 1 mod 5 and >1)
A191723=dispersion of A047215 (0, 2 mod 5 and >1)
A191724=dispersion of A047218 (0, 3 mod 5 and >1)
A191725=dispersion of A047208 (0, 4 mod 5 and >1)
A191726=dispersion of A047216 (1, 2 mod 5 and >1)
A191727=dispersion of A047219 (1, 3 mod 5 and >1)
A191728=dispersion of A047209 (1, 4 mod 5 and >1)
A191729=dispersion of A047221 (2, 3 mod 5 and >1)
A191730=dispersion of A047211 (2, 4 mod 5 and >1)
A191731=dispersion of A047204 (3, 4 mod 5 and >1)
...
A191732=dispersion of A047202 (2,3,4 mod 5 and >1)
A191733=dispersion of A047206 (1,3,4 mod 5 and >1)
A191734=dispersion of A032793 (1,2,4 mod 5 and >1)
A191735=dispersion of A047223 (1,2,3 mod 5 and >1)
A191736=dispersion of A047205 (0,3,4 mod 5 and >1)
A191737=dispersion of A047212 (0,2,4 mod 5 and >1)
A191738=dispersion of A047222 (0,2,3 mod 5 and >1)
A191739=dispersion of A008854 (0,1,4 mod 5 and >1)
A191740=dispersion of A047220 (0,1,3 mod 5 and >1)
A191741=dispersion of A047217 (0,1,2 mod 5 and >1)
...
For further information about these 20 dispersions, see A191722.
...
Regarding the dispersions A191722-A191741, there are general formulas for sequences of the type "(a or b mod m)" and "(a or b or c mod m)" used in the relevant Mathematica programs.

			Northwest corner:
1....2....5....10...17
3....6....11...20...35
4....7....12...21...36
8....15...26...45...76
9....16...27...46...77
13...22...37...62...105

Ivan Neretin, Table of n, a(n) for n = 1..5050 (first 100 antidiagonals, flattened)

Cf. A047204, A047217, A191722, A191731, A191426.

(* Program generates the dispersion array t of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
a=2; b=5; c2=6; m[n_]:=If[Mod[n,3]==0,1,0];
f[n_]:=a*m[n+2]+b*m[n+1]+c2*m[n]+5*Floor[(n-1)/3]
Table[f[n], {n, 1, 30}]  (* A047217 *)
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}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191741 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191741  *)

A143977 Rectangular array R by antidiagonals: label each unit square in the first quadrant lattice by its northeast vertex (x,y) and mark squares having |x-y| == 0 (mod 3); then R(m,n) is the number of marked squares in the rectangle [0,m] X [0,n].

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 06 2008

Rows numbered 3,6,9,12,15,... are, except for initial terms, multiples of (1,2,3,4,5,6,7,...) = A000027.

			Northwest corner:
  1  1  1  2  2  2  3
  1  2  2  3  4  4  5
  1  2  3  4  5  6  7
  2  3  4  6  7  8 10
  2  4  5  7  9 10 12

Stefano Spezia, First 140 antidiagonals of the array, flattened

Diagonals: A008810, A007984, A000212, A128422.
Rows and columns: A002264, A004523, A000027, A004772, A047212, et al.
Cf. A143974, A143976, A143979.

T[m_,n_]:=Ceiling[m n/3];Flatten[Table[T[m-n+1,n],{m,13},{n,m}]] (* Stefano Spezia, Oct 27 2022 *)

R(m,n) = ceiling(m*n/3). [Corrected by Stefano Spezia, Oct 27 2022]

A270109 a(n) = n^3 + (n+1)*(n+2).

Original entry on oeis.org

2, 7, 20, 47, 94, 167, 272, 415, 602, 839, 1132, 1487, 1910, 2407, 2984, 3647, 4402, 5255, 6212, 7279, 8462, 9767, 11200, 12767, 14474, 16327, 18332, 20495, 22822, 25319, 27992, 30847, 33890, 37127, 40564, 44207, 48062, 52135, 56432, 60959, 65722, 70727, 75980, 81487, 87254

Offset: 0

Bruno Berselli, Mar 11 2016, at the suggestion of Giuseppe Amoruso in BASE Cinque forum

For n>1, many consecutive terms of the sequence are generated by floor(sqrt(n^2 + 2)^3) + n^2 + 2.
It appears that this is a subsequence of A000037 (the nonsquares).
The primes in the sequence belong to A045326.
Inverse binomial transform is 2, 5, 8, 6, 0, 0, 0, ... (0 continued).

Bruno Berselli, Table of n, a(n) for n = 0..1000
MathsSmart, Number pattern and Puzzle - 7, 20, 47, 94, 167.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Subsequence of A001651, A047212.
Cf. A000037, A045326.
Cf. A027444: numbers of the form n^3+n*(n+1); A085490: numbers of the form n^3+(n-1)*n.
Cf. A008865: numbers of the form n+(n+1)*(n+2); A130883: numbers of the form n^2+(n+1)*(n+2).

Magma
```
[n^3+(n+1)*(n+2): n in [0..50]];
```

Table[n^3 + (n + 1) (n + 2), {n, 0, 50}]

Maxima
```
makelist(n^3+(n+1)*(n+2), n, 0, 50);
```
PARI
```
vector(50, n, n--; n^3+(n+1)*(n+2))
```
Sage
```
[n^3+(n+1)*(n+2) for n in (0..50)]
```

O.g.f.: (2 - x + 4*x^2 + x^3)/(1 - x)^4.
E.g.f.: (2 + x)*(1 + x)^2*exp(x).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), n>3.
a(n+h) - a(n) + a(n-h) = n^3 + n^2 + (6*h^2+3)*n + (2*h^2+2) for any h. This identity becomes a(n) = n^3 + n^2 + 3*n + 2 if h=0.
a(h*a(n) + n) = (h*a(n))^3 + (3*n+1)*(h*a(n))^2 + (3*n^2+2*n+3)*(h*a(n)) + a(n) for any h, therefore a(h*a(n) + n) is always a multiple of a(n).
a(n) + a(-n) = 2*A059100(n) = A255843(n).
a(n) - a(-n) = 4*A229183(n).

A238738 Expansion of (1 + 2x + 2x^2)/(1 - x - 2x^3 + 2x^4 + x^6 - x^7).

Original entry on oeis.org

1, 3, 5, 7, 11, 15, 18, 24, 30, 34, 42, 50, 55, 65, 75, 81, 93, 105, 112, 126, 140, 148, 164, 180, 189, 207, 225, 235, 255, 275, 286, 308, 330, 342, 366, 390, 403, 429, 455, 469, 497, 525, 540, 570, 600, 616, 648, 680, 697, 731, 765, 783, 819, 855, 874

Offset: 0

Bruno Berselli, Mar 04 2014

Subsequence of A008732: a(n) = A008732(A047212(n+1)).

See also Deléham's example in A008732: these numbers are in the first (A000566), third (A005475) and fifth (A028895) column.

			G.f.: 1 + 3*x + 5*x^2 + 7*x^3 + 11*x^4 + 15*x^5 + 18*x^6 + 24*x^7 + ...

Bruno Berselli, Table of n, a(n) for n = 0..1000
Bruno Berselli, Illustration of the initial terms.
Index entries for linear recurrences with constant coefficients, signature (1,0,2,-2,0,-1,1).

Cf. A000212 (see illustration above), A000217, A008732, A211538.

m:=60; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+2*x+2*x^2)/(1-x-2*x^3+2*x^4+x^6-x^7)));

CoefficientList[Series[(1 + 2 x + 2 x^2)/(1 - x - 2 x^3 + 2 x^4 + x^6 - x^7), {x, 0, 60}], x]

makelist(coeff(taylor((1+2*x+2*x^2)/(1-x-2*x^3+2*x^4+x^6-x^7), x, 0, n), x, n), n, 0, 60);

Vec((1+2*x+2*x^2)/(1-x-2*x^3+2*x^4+x^6-x^7)+O(x^60))

m = 60; L. = PowerSeriesRing(ZZ, m); f = (1+2*x+2*x^2)/(1-x-2*x^3+2*x^4+x^6-x^7); print(f.coefficients())

G.f.: (1 + 2*x + 2*x^2) / ((1 - x)^3*(1 + x + x^2)^2).
a(n) = a(n-1) + 2*a(n-3) - 2*a(n-4) - a(n-6) + a(n-7), with n>6.
a(3k)   = k*(5*k +  7)/2 + 1 (A000566);
a(3k+1) = k*(5*k + 11)/2 + 3 (A005475);
a(3k+2) = k*(5*k + 15)/2 + 5 (A028895).
a(n) = (floor(n/3)+1)*(4*n-7*floor(n/3)+2)/2. [Luce ETIENNE, Jun 14 2014]

A260307 a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-6) - a(n-7) - a(n-8) + a(n-9) with a(0) - a(8) as shown below.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 8, 7, 10, 9, 13, 10, 15, 12, 17, 14, 20, 15, 22, 17, 24, 19, 27, 20, 29, 22, 31, 24, 34, 25, 36, 27, 38, 29, 41, 30, 43, 32, 45, 34, 48, 35, 50, 37, 52, 39, 55, 40, 57, 42, 59, 44, 62, 45, 64, 47, 66, 49, 69, 50, 71, 52, 73, 54, 76, 55, 78

Offset: 0

Paul Curtz, Nov 22 2015

A260708 difference table rows have the same nine-step recurrence:
0, 1, 3,  6, 10, 16, 21, 29, 36, 46, 55, 65, 78,  93, ...
1, 2, 3,  4,  6,  5,  8,  7, 10,  9, 13, 10, 15,  12, ...     = a(n)
1, 1, 1,  2, -1,  3, -1,  3, -1,  4, -3,  5, -3,   5, ...     = b(n)
0, 0, 1, -3,  4, -4,  4, -4,  5, -7,  8, -8,  8,  -8, ... (see A042965(n)).
(b(2n) + b(2n+1) = A052901(n+2).)

G. C. Greubel, Table of n, a(n) for n = 0..1500
Index entries for linear recurrences with constant coefficients, signature (0,1,0,0,0,1,0,-1).

Cf. A004767, A010718, A042965, A047212, A047282, A052901, A152467, A260160 (eight-step recurrence), A260699 (nine-step recurrence), A260708.

I:=[1,2,3,4,6,5,8,7];[n le 8 select I[n] else Self(n-2) + Self(n-6) - Self(n-8): n in [1..70]]; // Vincenzo Librandi, Dec 26 2015

RecurrenceTable[{a[n] == a[n-2] + a[n-6] - a[n-8], a[0]=1, a[1]=2, a[2]=3, a[3]=4, a[4]=6, a[5]=5, a[6]=8, a[7]=7}, a, {n,0,100}] (* G. C. Greubel, Nov 23 2015 *)

Vec((x^6+x^5+3*x^4+2*x^3+2*x^2+2*x+1)/((x-1)^2*(x+1)^2*(x^2-x+1)*(x^2+x+1)) + O(x^100)) \\ Colin Barker, Nov 22 2015

vector(100, n, n--; n + (-1)^n *((n+2)\6) + 1) \\ Altug Alkan, Nov 24 2015

a(2n) = A047282(n). a(2n+1) = A047212(n+1).
a(n) = A260708(n+1) - A260708(n).
a(n+6) = a(n) + period of length 2: repeat 7, 5.
a(2n) + a(2n+1) = 3 + 4*n.
a(n) = n + 1 + (-1)^n*A152467(n+2).
From Colin Barker, Nov 22 2015: (Start)
a(n) = a(n-2) + a(n-6) - a(n-8) for n>7.
G.f.: (x^6+x^5+3*x^4+2*x^3+2*x^2+2*x+1) / ((x-1)^2*(x+1)^2*(x^2-x+1)*(x^2+x+1)).
(End)

A093303 a(n) = a(n-1)*(2n-1) + 2n with a(0)=0.

Original entry on oeis.org

0, 2, 10, 56, 400, 3610, 39722, 516400, 7746016, 131682290, 2501963530, 52541234152, 1208448385520, 30211209638026, 815702660226730, 23655377146575200, 733316691543831232, 24199450820946430690, 846980778733125074186

Offset: 0

Emrehan Halici (emrehan(AT)halici.com.tr), Apr 24 2004

Obviously, a(n) is always an even number. a(2) and a(6) are even semiprimes. - Altug Alkan, Dec 07 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A005843.

Flatten[{0,Table[n!*Binomial[2*n-1,n]/2^(n-1)*Sum[2^k*k/(k!*Binomial[2*k-1,k]), {k,1,n}],{n,1,20}]}] (* Vaclav Kotesovec, Oct 28 2012 *)

a(n) = if(n==0, 0, n!*binomial(2*n-1,n)/2^(n-1) * sum(k=1, n, 2^k*k/(k!*binomial(2*k-1,k)))) \\ Altug Alkan, Dec 07 2015

a(n) = if(n==0, 0, a(n-1)*(2*n-1) + 2*n); \\ Altug Alkan, Dec 07 2015

a(n) = n!*C(2*n-1,n)/2^(n-1) * Sum_{k=1..n} 2^k*k/(k!*C(2*k-1,k)), for n>0. - Vaclav Kotesovec, Oct 28 2012
From Altug Alkan, Dec 07 2015: (Start)
a(A047212(k)) mod 10 = 0.
a(A016861(k)) mod 10 = 2.
a(A016885(k)) mod 10 = 6. (End)
a(n) ~ (sqrt(2) + 2*sqrt(Pi)*exp(1/2)*erf(1/sqrt(2))) * 2^n * n^n / exp(n). - Vaclav Kotesovec, Dec 18 2015

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 24 2004

cp's OEIS Frontend

A191740 Dispersion of A047220, (numbers >1 and congruent to 0 or 1 or 3 mod 5), by antidiagonals.

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A191741 Dispersion of A047217, (numbers >1 and congruent to 0 or 1 or 2 mod 5), by antidiagonals.

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A143977 Rectangular array R by antidiagonals: label each unit square in the first quadrant lattice by its northeast vertex (x,y) and mark squares having |x-y| == 0 (mod 3); then R(m,n) is the number of marked squares in the rectangle [0,m] X [0,n].

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A270109 a(n) = n^3 + (n+1)*(n+2).

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A238738 Expansion of (1 + 2*x + 2*x^2)/(1 - x - 2*x^3 + 2*x^4 + x^6 - x^7).

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A260307 a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-6) - a(n-7) - a(n-8) + a(n-9) with a(0) - a(8) as shown below.

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A093303 a(n) = a(n-1)*(2n-1) + 2n with a(0)=0.

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A238738 Expansion of (1 + 2x + 2x^2)/(1 - x - 2x^3 + 2x^4 + x^6 - x^7).