A033428 -id:A033428 - cp's OEIS frontend

A092206 Positive integers that are not of the form n^2 or 3n^2.

2, 5, 6, 7, 8, 10, 11, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 82, 83, 84

Offset: 1

Eric W. Weisstein, Feb 24 2004

nonn

Values of n such that Q(sqrt(-n)) has 2 units.

A214295(a(n)) = 0. - Reinhard Zumkeller, Jul 12 2012

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Unit

Cf. A000290, A033428, A092205.

a092206 n = a092206_list !! (n-1)
a092206_list = filter ((== 0) . a214295) [1..]
-- Reinhard Zumkeller, Jul 12 2012

Select[ Range[100], Not[ IntegerQ[ Sqrt[#] ] || IntegerQ[ Sqrt[#/3] ] ]&] (* Jean-François Alcover, Oct 30 2012 *)

is(n)=!issquare(n/3^valuation(n,3)) \\ Charles R Greathouse IV, Oct 30 2012

A105020 Array read by antidiagonals: row n (n >= 0) contains the numbers m^2 - n^2, m >= n+1.

Original entry on oeis.org

1, 3, 4, 5, 8, 9, 7, 12, 15, 16, 9, 16, 21, 24, 25, 11, 20, 27, 32, 35, 36, 13, 24, 33, 40, 45, 48, 49, 15, 28, 39, 48, 55, 60, 63, 64, 17, 32, 45, 56, 65, 72, 77, 80, 81, 19, 36, 51, 64, 75, 84, 91, 96, 99, 100, 21, 40, 57, 72, 85, 96, 105, 112, 117, 120, 121

Offset: 0

Andrew S. Plewe and Franklin T. Adams-Watters, Jul 11 2005

A "Goldbach Conjecture" for this sequence: when there are n terms between consecutive odd integers (2n+1) and (2n+3) for n > 0, at least one will be the product of 2 primes (not necessarily distinct). Example: n=3 for consecutive odd integers a(7)=7 and a(11)=9 and of the 3 sequence entries a(8)=12, a(9)=15 and a(10)=16 between them, one is the product of 2 primes a(9)=15=3*5. - Michael Hiebl, Jul 15 2007
A024352 gives distinct values in the array, minus the first row (1, 4, 9, 16, etc.). a(n) gives all solutions to the equation x^2 + xy = n, with y mod 2 = 0, x > 0, y >= 0. - Andrew S. Plewe, Oct 19 2007
Alternatively, triangular sequence of coefficients of Dynkin diagram weights for the Cartan groups C_n: t(n,m) = m*(2*n - m). Row sums are A002412. - Roger L. Bagula, Aug 05 2008

			Array begins:
  1  4  9 16 25 36  49  64  81 100 ...
  3  8 15 24 35 48  63  80  99 120 ...
  5 12 21 32 45 60  77  96 117 140 ...
  7 16 27 40 55 72  91 112 135 160 ...
  9 20 33 48 65 84 105 128 153 180 ...
  ...
Triangle begins:
   1;
   3,  4;
   5,  8,  9;
   7, 12, 15, 16;
   9, 16, 21, 24, 25;
  11, 20, 27, 32, 35, 36;
  13, 24, 33, 40, 45, 48, 49;
  15, 28, 39, 48, 55, 60, 63, 64;
  17, 32, 45, 56, 65, 72, 77, 80, 81;
  19, 36, 51, 64, 75, 84, 91, 96, 99, 100;

R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0-486-44999-8, p. 139.

G. C. Greubel, Antidiagonals n = 0..50, flattened

Rows give A000290, A005563, A028347, A028560, A028566, A098603, A098847, A098848, A098849, A098850.
Columns give A005408, A008586, A016945, A008590, A017329, A008594, A008598, A008602, A008606, A000567.
Diagonals give A033428, A045944, A067725.
Cf. A000384, A002412, A024352, A105020, A128076.

[(k+1)*(2*n-k+1): k in [0..n], n in [0..15]]; // G. C. Greubel, Mar 15 2023

t[n_, m_]:= (n^2 - m^2); Flatten[Table[t[i, j], {i,12}, {j,i-1,0,-1}]]
(* to view table *) Table[t[i, j], {j,0,6}, {i,j+1,10}]//TableForm (* Robert G. Wilson v, Jul 11 2005 *)
Table[(k+1)*(2*n-k+1), {n,0,15}, {k,0,n}]//Flatten (* Roger L. Bagula, Aug 05 2008 *)

def A105020(n,k): return (k+1)*(2*n-k+1)
flatten([[A105020(n,k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Mar 15 2023

a(n) = r^2 - (r^2 + r - m)^2/4, where r = round(sqrt(m)) and m = 2*n+2. - Wesley Ivan Hurt, Sep 04 2021
a(n) = A128076(n+1) * A105020(n+1). - Wesley Ivan Hurt, Jan 07 2022
From G. C. Greubel, Mar 15 2023: (Start)
Sum_{k=0..n} T(n, k) = A002412(n+1).
Sum_{k=0..n} (-1)^k*T(n, k) = (1/2)*((1+(-1)^n)*A000384((n+2)/2) - (1- (-1)^n)*A000384((n+1)/2)). (End)

More terms from Robert G. Wilson v, Jul 11 2005

A200741 Table of distinct numbers of the form vw + wu + u*v with 1 <= u <= v <= w <= n.

Original entry on oeis.org

3, 3, 5, 8, 12, 3, 5, 7, 8, 11, 12, 15, 16, 21, 27, 3, 5, 7, 8, 9, 11, 12, 14, 15, 16, 19, 20, 21, 24, 26, 27, 32, 33, 40, 48, 3, 5, 7, 8, 9, 11, 12, 14, 15, 16, 17, 19, 20, 21, 23, 24, 26, 27, 29, 31, 32, 33, 35, 38, 39, 40, 45, 47, 48, 55, 56, 65, 75, 3, 5

Offset: 1

Reinhard Zumkeller, Nov 21 2011

A100440(n) = number of terms in row n;
T(1,1) = 3; right edge: T(n,A100440(n)) = A033428(n);
T(n,k) = T(n+1,k) for k <= A200742(n);
distinct terms per row of table in A200737.

Reinhard Zumkeller, Rows n=1..25 of triangle, flattened

import Data.List (nub)
a200741 n k = a200741_tabl !! (n-1) !! (k-1)
a200741_row = nub . a200737_row
a200741_tabl = map a200741_row [1..]

row[n_] := Table[v*w + w*u + u*v, {u, 1, n}, {v, u, n}, {w, v, n}] // Flatten // Union; Table[row[n], {n, 1, 6}] // Flatten (* Jean-François Alcover, Dec 04 2013 *)

A219056 a(n) = 3*n^4.

Original entry on oeis.org

0, 3, 48, 243, 768, 1875, 3888, 7203, 12288, 19683, 30000, 43923, 62208, 85683, 115248, 151875, 196608, 250563, 314928, 390963, 480000, 583443, 702768, 839523, 995328, 1171875, 1370928, 1594323, 1843968, 2121843, 2430000, 2770563, 3145728, 3557763, 4009008

Offset: 0

Reinhard Zumkeller, Nov 11 2012

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A008585, A000583, A000290, A033428, A219069.

Haskell
```
a219056 = (* 3) . (^ 4)
```

LinearRecurrence[{5,-10,10,-5,1},{0, 3, 48, 243, 768},100] (* or *) Table[3*n^4, {n,0,50}] (* G. Greubel, Jun 22 2016 *)

makelist(3*n^4,n,0,30); /* Martin Ettl, Nov 12 2012 */

a(n) = 3*n^4; \\ Michel Marcus, Jan 26 2022

a(n) = A219069(n,n) for n > 0;
a(n) = A008585(A000583(n)) = A000290(n)*A033428(n).
From Chai Wah Wu, Jun 22 2016: (Start)
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 4.
G.f.: 3*x*(1 + x)*(1 + 10*x + x^2)/(1 - x)^5. (End)
E.g.f.: 3*x*(1 + 7*x + 6*x^2 + x^3)*exp(x). - G. C. Greubel, Jun 22 2016

A242669 a(n) = n*floor(n/3).

Original entry on oeis.org

0, 0, 0, 3, 4, 5, 12, 14, 16, 27, 30, 33, 48, 52, 56, 75, 80, 85, 108, 114, 120, 147, 154, 161, 192, 200, 208, 243, 252, 261, 300, 310, 320, 363, 374, 385, 432, 444, 456, 507, 520, 533, 588, 602, 616, 675, 690, 705, 768, 784, 800, 867, 884, 901, 972, 990

Offset: 0

Bruno Berselli, Jul 01 2014

For n = 0, 1, 2, 4, 8, 49, 98, 676, 1352, 9409, 18818, 131044, 262088, 1825201, 3650402, ... a(n) is a square.

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

Cf. A000290 (n^2), A010762 (floor(n/2)*floor(n/3)), A093353 (n*floor(n/2)), A213033 (n*floor(n/2)*floor(n/3)), A233035 (n*floor(n/4)).
Cf. A033428, A045944, A049451.
Cf. A002264 (floor(n/3)).

Magma
```
[n*Floor(n/3): n in [0..60]];
```
Mathematica
```
Table[n Floor[n/3], {n, 0, 60}]
```

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

Sage
```
[n*floor(n/3) for n in (0..60)];
```

G.f.: x^3*(3 + x + x^2 + x^3)/((1 - x)^3*(1 + x + x^2)^2).
a(3m) = A033428(m), a(3m+1) = A049451(m), a(3m+2) = A045944(m).
Sum_{n>=3} (-1)^(n+1)/a(n) = 9/4 + Pi^2/36 - Pi/(2*sqrt(3)) - 2*log(2). - Amiram Eldar, Mar 30 2023

A244634 a(n) = 27*n^2.

Original entry on oeis.org

0, 27, 108, 243, 432, 675, 972, 1323, 1728, 2187, 2700, 3267, 3888, 4563, 5292, 6075, 6912, 7803, 8748, 9747, 10800, 11907, 13068, 14283, 15552, 16875, 18252, 19683, 21168, 22707, 24300, 25947, 27648, 29403, 31212, 33075, 34992, 36963, 38988, 41067, 43200, 45387

Offset: 0

Vincenzo Librandi, Jul 03 2014

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

Cf. similar sequences listed in A244630.

Cf. A000290, A016766, A033428, A305548.

Magma
```
[27*n^2: n in [0..40]];
```
Mathematica
```
Table[27 n^2, {n, 0, 40}]
```

a(n)=27*n^2 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: 27*x*(1 + x)/(1 - x)^3. [corrected by Bruno Berselli, Jul 03 2014]
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.
a(n) = 27*A000290(n) = 9*A033428(n) = 3*A016766(n). - Omar E. Pol, Jul 03 2014
From Elmo R. Oliveira, Dec 01 2024: (Start)
E.g.f.: 27*x*(1 + x)*exp(x).
a(n) = n*A305548(n). (End)

A249327 Rectangular array T(n,k) = f(n)*k^2, where f = A005117 (squarefree numbers); n, k >= 1; read by antidiagonals.

Original entry on oeis.org

1, 4, 2, 9, 8, 3, 16, 18, 12, 5, 25, 32, 27, 20, 6, 36, 50, 48, 45, 24, 7, 49, 72, 75, 80, 54, 28, 10, 64, 98, 108, 125, 96, 63, 40, 11, 81, 128, 147, 180, 150, 112, 90, 44, 13, 100, 162, 192, 245, 216, 175, 160, 99, 52, 14, 121, 200, 243, 320, 294, 252, 250

Offset: 1

Clark Kimberling, Oct 26 2014

Every positive integer occurs exactly once.

			Northwest corner:
1   4    9    16   25    36    49
2   8    18   32   50    72    98
3   12   27   48   75    108   147
5   20   45   80   125   180   245
6   24   54   96   150   216   294

Cf. A005117, A000037 (is partitioned by the rows of A249327, excluding the first).

z = 20; f = Select[Range[10000], SquareFreeQ[#] &];
u[n_, k_] := f[[n]]*k^2; t = Table[u[n, k], {n, 1, 20}, {k, 1, 20}];
TableForm[t] (* A249327 array *)
Table[u[k, n - k + 1], {n, 1, 15}, {k, 1, n}] // Flatten (* A249327 sequence *)

T(1,k) = A000290(k), T(2,k) = A001105(k), T(3,k) = A033428(k), T(4,k) = A033429(k), T(5,.) through T(10,.) are A033581, A033582, A033583, A033584, A152742 and A144555 without initial 0. - M. F. Hasler, Oct 31 2014

A308422 a(n) = n^2 if n odd, 3*n^2/4 if n even.

Original entry on oeis.org

0, 1, 3, 9, 12, 25, 27, 49, 48, 81, 75, 121, 108, 169, 147, 225, 192, 289, 243, 361, 300, 441, 363, 529, 432, 625, 507, 729, 588, 841, 675, 961, 768, 1089, 867, 1225, 972, 1369, 1083, 1521, 1200, 1681, 1323, 1849, 1452, 2025, 1587, 2209, 1728, 2401, 1875, 2601, 2028, 2809, 2187, 3025

Offset: 0

Ilya Gutkovskiy, May 26 2019

Moebius transform of A076577.

Amiram Eldar, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Cf. A000290, A007434, A016754, A026741, A033428, A076577, A308418, A309337.

a[n_] := If[OddQ[n], n^2, 3 n^2/4]; Table[a[n], {n, 0, 55}]
nmax = 55; CoefficientList[Series[x (1 + 3 x + 6 x^2 + 3 x^3 + x^4)/(1 - x^2)^3, {x, 0, nmax}], x]
LinearRecurrence[{0, 3, 0, -3, 0, 1}, {0, 1, 3, 9, 12, 25}, 56]
Table[(7 - (-1)^n) n^2/8, {n, 0, 55}]

G.f.: x*(1 + 3*x + 6*x^2 + 3*x^3 + x^4)/(1 - x^2)^3.
G.f.: Sum_{k>=1} J_2(k)*x^k/(1 - x^(2*k)), where J_2() is the Jordan function (A007434).
E.g.f.: x*((4 + 3*x)*cosh(x) + (3 + 4*x)*sinh(x))/4.
Dirichlet g.f.: zeta(s-2)*(1 - 1/2^s).
a(n) = (7 - (-1)^n)*n^2/8.
a(n) = Sum_{d|n, n/d odd} J_2(d).
a(2*k+1) = A016754(k), a(2*k) = A033428(k).
Sum_{n>=1} 1/a(n) = 13*Pi^2/72 = 1.7820119057522453061...
Sum_{n>=1} (-1)^(n+1)/a(n) = 5*Pi^2/72 = 0.68538919452009434853...
Multiplicative with a(2^e) = 3*2^(2*e-2), and a(p^e) = p^(2*e) for odd primes p. - Amiram Eldar, Oct 26 2020
For n >= 1, n*a(n) = A309337(n) = Sum_{d divides n} (-1)^(d+1) * J(3, n/d), where the Jordan totient function J_3(n) = A059376. - Peter Bala, Jan 21 2024

A064763 a(n) = 28*n^2.

Original entry on oeis.org

0, 28, 112, 252, 448, 700, 1008, 1372, 1792, 2268, 2800, 3388, 4032, 4732, 5488, 6300, 7168, 8092, 9072, 10108, 11200, 12348, 13552, 14812, 16128, 17500, 18928, 20412, 21952, 23548, 25200, 26908, 28672, 30492, 32368, 34300, 36288, 38332

Offset: 0

Roberto E. Martinez II, Oct 18 2001

Number of edges in a complete 8-partite graph of order 8n, K_n,n,n,n,n,n,n,n.

Sequence found by reading the line from 0, in the direction 0, 28, ..., in the square spiral whose vertices are the generalized 16-gonal numbers. - Omar E. Pol, Jul 03 2014

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

Similar sequences are listed in A244630.
Cf. A000217, A000290, A001105, A016742, A033428, A033581, A033582, A033583.
Cf. A144555, A135628.

[28*n^2: n in [0..40]]; // Vincenzo Librandi, Mar 30 2015

I:=[0,28,112]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..40]]; // Vincenzo Librandi, Mar 30 2015

CoefficientList[Series[28 x (1 + x) / (1 - x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Mar 30 2015 *)

a(n)=28*n^2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 56*n + a(n-1) - 28 (with a(0)=0). - Vincenzo Librandi, Aug 07 2010
a(n) = 28*A000290(n) = 14*A001105(n) = 7*A016742(n) = 4*A033582(n) = 2*A144555(n). - Omar E. Pol, Jul 03 2014
From Vincenzo Librandi, Mar 30 2015: (Start)
G.f.: 28*x*(1+x)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
a(n) = t(8*n) - 8*t(n), where t(i) = i*(i+k)/2 for any k. Special case (k=1): a(n) = A000217(8*n) - 8*A000217(n). - Bruno Berselli, Aug 31 2017
From Elmo R. Oliveira, Dec 01 2024: (Start)
E.g.f.: 28*x*(1 + x)*exp(x).
a(n) = n*A135628(n). (End)

A256320 Number of partitions of 4n into exactly 3 parts.

Original entry on oeis.org

0, 1, 5, 12, 21, 33, 48, 65, 85, 108, 133, 161, 192, 225, 261, 300, 341, 385, 432, 481, 533, 588, 645, 705, 768, 833, 901, 972, 1045, 1121, 1200, 1281, 1365, 1452, 1541, 1633, 1728, 1825, 1925, 2028, 2133, 2241, 2352, 2465, 2581, 2700, 2821, 2945, 3072, 3201

Offset: 0

Colin Barker, Mar 24 2015

			For n=2 the 5 partitions of 4*2 = 8 are [1,1,6], [1,2,5], [1,3,4], [2,2,4] and [2,3,3].

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

Cf. A033428 (6n), A256321 (5n), A256322 (7n).

Cf. A184637.

Length /@ (Total /@ IntegerPartitions[4 #, {3}] & /@ Range[0, 49]) (* Michael De Vlieger, Mar 24 2015 *)
CoefficientList[Series[-x (x + 1)^3/((x - 1)^3 (x^2 + x + 1)), {x, 0, 49}], x] (* or *)
Table[2 (6 n^2 + Cos[(2 Pi n)/3] - 1)/9, {n, 0, 49}] (* Michael De Vlieger, Jun 06 2016 *)

concat(0, vector(40, n, k=0; forpart(p=4*n, k++, , [3,3]); k))

concat(0, Vec(-x*(x+1)^3/((x-1)^3*(x^2+x+1)) + O(x^100)))

a(n) = A184637(n) for n > 2.
a(n) = 2*a(n-1)-a(n-2)+a(n-3)-2*a(n-4)+a(n-5) for n>4.
G.f.: -x*(x+1)^3 / ((x-1)^3*(x^2+x+1)).
a(n) = 2*(6*n^2+cos((2*Pi*n)/3)-1)/9. - Colin Barker, Jun 06 2016

cp's OEIS Frontend

A092206 Positive integers that are not of the form n^2 or 3n^2.

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A105020 Array read by antidiagonals: row n (n >= 0) contains the numbers m^2 - n^2, m >= n+1.

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A200741 Table of distinct numbers of the form v*w + w*u + u*v with 1 <= u <= v <= w <= n.

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A219056 a(n) = 3*n^4.

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Maxima

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A242669 a(n) = n*floor(n/3).

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A244634 a(n) = 27*n^2.

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A249327 Rectangular array T(n,k) = f(n)*k^2, where f = A005117 (squarefree numbers); n, k >= 1; read by antidiagonals.

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A200741 Table of distinct numbers of the form vw + wu + u*v with 1 <= u <= v <= w <= n.