A132209 -id:A132209 - cp's OEIS frontend

A142463 a(n) = 2n^2 + 2n - 1.

-1, 3, 11, 23, 39, 59, 83, 111, 143, 179, 219, 263, 311, 363, 419, 479, 543, 611, 683, 759, 839, 923, 1011, 1103, 1199, 1299, 1403, 1511, 1623, 1739, 1859, 1983, 2111, 2243, 2379, 2519, 2663, 2811, 2963, 3119, 3279, 3443, 3611, 3783, 3959, 4139, 4323, 4511, 4703, 4899, 5099

Offset: 0

Roger L. Bagula, Sep 19 2008

Essentially the same as A132209.
From Vincenzo Librandi, Nov 25 2010: (Start)
Numbers k such that 2*k + 3 is a square.
First diagonal of A144562. (End)
The terms a(n) give the values for c of indefinite binary quadratic forms [a, b, c] = [2, 4n+2, a(n)] of discriminant D = 12, where a and c can be switched. The positive numbers represented by these forms are given in A084917. - Klaus Purath, Aug 31 2023

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Leo Tavares, Illustration: Hexagonic Diamonds.
Leo Tavares, Illustration: Hexagonic Rectangles.
Leo Tavares, Illustration: Hexagonic Crosses.
Leo Tavares, Illustration: Hexagonic Columns.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000096, A005408, A132209, A144562, A188653.

Cf. A005563, A016945, A001105, A000290, A004767, A046092, A002378, A028387.

Magma
```
[2*n^2+2*n-1: n in [0..100]];
```

A142463:= n-> 2*n^2 +2*n -1; seq(A142463(n), n=0..50); # G. C. Greubel, Mar 01 2021

Array[ -#*(2-#*2)-1&,5!,1] (* Vladimir Joseph Stephan Orlovsky, Dec 21 2008 *)
Table[2n^2+2n-1,{n,0,50}] (* Harvey P. Dale, Feb 29 2024 *)

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

[2*n^2 +2*n -1 for n in (0..50)] # G. C. Greubel, Mar 01 2021

a(n) = a(n-1) + 4*n.
From Paul Barry, Nov 03 2009: (Start)
G.f.: (1 - 6*x + x^2)/(1-x)^3.
a(n) = 4*C(n+1,2) - 1. (End)
a(n) = -A188653(2*n+1). - Reinhard Zumkeller, Apr 13 2011
a(n) = 3*( Sum_{k=1..n} k^5 )/( Sum_{k=1..n} k^3 ), n > 0. - Gary Detlefs, Oct 18 2011
a(n) = (A005408(n)^2 - 3)/2. - Zhandos Mambetaliyev, Feb 11 2017
E.g.f.: (-1 + 4*x + 2*x^2)*exp(x). - G. C. Greubel, Mar 01 2021
From Leo Tavares, Nov 22 2021: (Start)
a(n) = 2*A005563(n) - A005408(n). See Hexagonic Diamonds illustration.
a(n) = A016945(n-1) + A001105(n-1). See Hexagonic Rectangles illustration.
a(n) = A004767(n-1) + A046092(n-1). See Hexagonic Crosses illustration.
a(n) = A002378(n) + A028387(n-1). See Hexagonic Columns illustration. (End)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Wesley Ivan Hurt, Dec 03 2021
Sum_{n>=0} 1/a(n) = tan(sqrt(3)*Pi/2)*Pi/(2*sqrt(3)). - Amiram Eldar, Sep 16 2022

Edited by the Associate Editors of the OEIS, Sep 02 2009

A185787 Sum of first k numbers in column k of the natural number array A000027; by antidiagonals.

Original entry on oeis.org

1, 7, 25, 62, 125, 221, 357, 540, 777, 1075, 1441, 1882, 2405, 3017, 3725, 4536, 5457, 6495, 7657, 8950, 10381, 11957, 13685, 15572, 17625, 19851, 22257, 24850, 27637, 30625, 33821, 37232, 40865, 44727, 48825, 53166, 57757, 62605, 67717, 73100, 78761, 84707, 90945, 97482, 104325, 111481, 118957, 126760, 134897, 143375

Offset: 1

Clark Kimberling, Feb 03 2011

This is one of many interesting sequences and arrays that stem from the natural number array A000027, of which a northwest corner is as follows:
  1....2.....4.....7...11...16...22...29...
  3....5.....8....12...17...23...30...38...
  6....9....13....18...24...31...39...48...
  10...14...19....25...32...40...49...59...
  15...20...26....33...41...50...60...71...
  21...27...34....42...51...61...72...84...
  28...35...43....52...62...73...85...98...
Blocking out all terms below the main diagonal leaves columns whose sums comprise A185787.  Deleting the main diagonal and then summing give A185787.  Analogous treatments to the left of the main diagonal give A100182 and A101165.  Further sequences obtained directly from this array are easily obtained using the following formula for the array: T(n,k)=n+(n+k-2)(n+k-1)/2.
Examples:
row 1:  A000124
row 2:  A022856
row 3:  A016028
row 4:  A145018
row 5:  A077169
col 1:  A000217
col 2:  A000096
col 3:  A034856
col 4:  A055998
col 5:  A046691
col 6:  A052905
col 7:  A055999
diag. (1,5,...) ...... A001844
diag. (2,8,...) ...... A001105
diag. (4,12,...)...... A046092
diag. (7,17,...)...... A056220
diag. (11,23,...) .... A132209
diag. (16,30,...) .... A054000
diag. (22,38,...) .... A090288
diag. (3,9,...) ...... A058331
diag. (6,14,...) ..... A051890
diag. (10,20,...) .... A005893
diag. (15,27,...) .... A097080
diag. (21,35,...) .... A093328
antidiagonal sums:  (1,5,15,34,...)=A006003=partial sums of A002817.
Let S(n,k) denote the n-th partial sum of column k.  Then
S(n,k)=n*(n^2+3k*n+3*k^2-6*k+5)/6.
S(n,1)=n(n+1)(n+2)/6
S(n,2)=n(n+1)(n+5)/6
S(n,3)=n(n+2)(n+7)/6
S(n,4)=n(n^2+12n+29)/6
S(n,5)=n(n+5)(n+10)/6
S(n,6)=n(n+7)(n+11)/6
S(n,7)=n(n+10)(n+11)/6
Weight array of T: A144112
Accumulation array of T: A185506
Second rectangular sum array of T: A185507
Third rectangular sum array of T: A185508
Fourth rectangular sum array of T: A185509

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

Cf. A000027, A185788, A100182, A101165, A079824.

[n*(7*n^2-6*n+5)/6: n in [1..50]]; // Vincenzo Librandi, Jul 04 2012

f[n_,k_]:=n+(n+k-2)(n+k-1)/2;
s[k_]:=Sum[f[n,k],{n,1,k}];
Factor[s[k]]
Table[s[k],{k,1,70}]  (* A185787 *)
CoefficientList[Series[(3*x^2+3*x+1)/(1-x)^4,{x,0,50}],x] (* Vincenzo Librandi, Jul 04 2012 *)

a(n)=n*(7*n^2-6*n+5)/6.

G.f.: x*(3*x^2+3*x+1)/(1-x)^4. - Vincenzo Librandi, Jul 04 2012

Edited by Clark Kimberling, Feb 25 2023

A132355 Numbers of the form 9h^2 + 2h, for h an integer.

Original entry on oeis.org

0, 7, 11, 32, 40, 75, 87, 136, 152, 215, 235, 312, 336, 427, 455, 560, 592, 711, 747, 880, 920, 1067, 1111, 1272, 1320, 1495, 1547, 1736, 1792, 1995, 2055, 2272, 2336, 2567, 2635, 2880, 2952, 3211, 3287, 3560, 3640, 3927, 4011, 4312, 4400, 4715, 4807

Offset: 1

Mohamed Bouhamida, Nov 08 2007

X values of solutions to the equation 9*X^3 + X^2 = Y^2.
The set of all m such that 9*m + 1 is a perfect square. - Gary Detlefs, Feb 22 2010
The concatenation of any term with 11..11 (1 repeated an even number of times, see A099814) belongs to the list. Example: 87 is a term, so also 8711, 871111, 87111111, 871111111111, ... are terms of this sequence. - Bruno Berselli, May 15 2017

Jason Kimberley, Table of n, a(n) for n = 1..2108
S. Cooper and M. D. Hirschhorn, Results of Hurwitz type for three squares. Discrete Math., Vol. 274, No. 1-3 (2004), pp. 9-24. See A(q).
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

A205808 is the characteristic function.
Cf. A000217, A001082, A002378, A005563, A028347, A036666, A046092, A054000, A056220, A062717, A087475, A132209, A010701, A056020.
Numbers of the form 9*n^2+k*n, for integer n: A016766 (k=0), this sequence (k=2), A185039 (k=4), A057780 (k=6), A218864 (k=8). - Jason Kimberley, Nov 09 2012
For similar sequences of numbers m such that 9*m+k is a square, see list in A266956.

a:=func; [0] cat [a(n*m): m in [-1,1], n in [1..25]]; // Jason Kimberley, Nov 08 2012

readlib(issqr); for n from 0 to 3560 do if(issqr(9*n+1)) then print(n) fi od; # Gary Detlefs, Feb 22 2010
seq(n^2+n+5*ceil(n/2)^2,n=0..39); # Gary Detlefs, Feb 23 2010

f[n_]:=IntegerQ[Sqrt[1+9*n]]; Select[Range[0,8! ],f[ # ]&] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
Sort[Table[9n^2+2n,{n,-30,30}]] (* Harvey P. Dale, Dec 06 2013 *)

a(n)=n^2-n+5*(n\2)^2 \\ Charles R Greathouse IV, Sep 28 2015

a(2*k) = k*(9*k-2), a(2*k+1) = k*(9*k+2).
a(n) = n^2 - n + 5*floor(n/2)^2. - Gary Detlefs, Feb 23 2010
From R. J. Mathar, Mar 17 2010: (Start)
a(n) = +a(n-1) +2*a(n-2) -2*a(n-3) -a(n-4) +a(n-5).
G.f.: x^2*(7 + 4*x + 7*x^2)/((1 + x)^2*(1 - x)^3). (End)
a(n) = (2*n - 1 + (-1)^n)*(9*(2*n - 1) + (-1)^n)/16. - Luce ETIENNE, Sep 13 2014
Sum_{n>=2} 1/a(n) = 9/4 - cot(2*Pi/9)*Pi/2. - Amiram Eldar, Mar 15 2022

Simpler definition and minor edits from N. J. A. Sloane, Feb 03 2012

Since this is a list, offset changed to 1 and formulas translated by Jason Kimberley, Nov 18 2012

A132356 a(2k) = k(10k+2), a(2k+1) = 10k^2 + 18k + 8, with k >= 0.

Original entry on oeis.org

0, 8, 12, 36, 44, 84, 96, 152, 168, 240, 260, 348, 372, 476, 504, 624, 656, 792, 828, 980, 1020, 1188, 1232, 1416, 1464, 1664, 1716, 1932, 1988, 2220, 2280, 2528, 2592, 2856, 2924, 3204, 3276, 3572, 3648, 3960, 4040, 4368, 4452, 4796, 4884, 5244, 5336, 5712

Offset: 0

Mohamed Bouhamida, Nov 08 2007

X values of solutions to the equation 10*X^3 + X^2 = Y^2.
Polygonal number connection: 2*H_n + 6S_n, where H_n is the n-th hexagonal number and S_n is the n-th square number. This is the base formula that is expanded upon to achieve the full series. See contributing formula below. - William A. Tedeschi, Sep 12 2010
Equivalently, numbers of the form 2*h*(5*h+1), where h = 0, -1, 1, -2, 2, -3, 3, -4, 4, ... . - Bruno Berselli, Feb 02 2017

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

Cf. A054000, A056220, A087475, A028347, A046092, A132209.
Cf. numbers m such that k*m+1 is a square: A005563 (k=1), A046092 (k=2), A001082 (k=3), A002378 (k=4), A036666 (k=5), A062717 (k=6), A132354 (k=7), A000217 (k=8), A132355 (k=9), A219257 (k=11), A152749 (k=12), A219389 (k=13), A219390 (k=14), A204221 (k=15), A074378 (k=16), A219394 (k=17), A219395 (k=18), A219396 (k=19), A219190 (k=20), A219391 (k=21), A219392 (k=22), A219393 (k=23), A001318 (k=24), A219259 (k=25), A217441 (k=26), A219258 (k=27), A219191 (k=28).
Cf. A220082 (numbers k such that 10*k-1 is a square).

CoefficientList[Series[4*x*(2*x^2 + x + 2)/((1 - x)^3*(1 + x)^2), {x, 0, 50}], x] (* G. C. Greubel, Jun 12 2017 *)
LinearRecurrence[{1,2,-2,-1,1},{0,8,12,36,44},50] (* Harvey P. Dale, Dec 15 2023 *)

my(x='x+O('x^50)); concat([0], Vec(4*x*(2*x^2+x+2)/((1-x)^3*(1+x)^2))) \\ G. C. Greubel, Jun 12 2017

a(n) = n^2 + n + 6*((n+1)\2)^2 \\ Charles R Greathouse IV, Sep 11 2022

G.f.: 4*x*(2*x^2+x+2)/((1-x)^3*(1+x)^2). - R. J. Mathar, Apr 07 2008
a(n) = 10*x^2 - 2*x, where x = floor(n/2)*(-1)^n for n >= 1. - William A. Tedeschi, Sep 12 2010
a(n) = ((2*n+1-(-1)^n)*(10*(2*n+1)-2*(-1)^n))/16. - Luce ETIENNE, Sep 13 2014
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 4. - Chai Wah Wu, May 24 2016
Sum_{n>=1} 1/a(n) = 5/2 - sqrt(1+2/sqrt(5))*Pi/2. - Amiram Eldar, Mar 15 2022
a(n) = n^2 + n + 6*ceiling(n/2)^2. - Ridouane Oudra, Aug 06 2022

More terms from Max Alekseyev, Nov 13 2009

A171176 Triangle read by rows in which row n lists 3n-1 together with the first 2n-1 positive integers, in reverse order.

Original entry on oeis.org

2, 1, 5, 3, 2, 1, 8, 5, 4, 3, 2, 1, 11, 7, 6, 5, 4, 3, 2, 1, 14, 9, 8, 7, 6, 5, 4, 3, 2, 1, 17, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 20, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 23, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 26, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1

Offset: 1

Omar E. Pol, Feb 23 2010

a(n) is also the length of the n-th L-toothpick added to the structure of A171166.

			Triangle begins:
   2,  1;
   5,  3,  2,  1;
   8,  5,  4,  3,  2,  1;
  11,  7,  6,  5,  4,  3,  2,  1;
  14,  9,  8,  7,  6,  5,  4,  3,  2, 1;
  17, 11, 10,  9,  8,  7,  6,  5,  4, 3, 2, 1;
  20, 13, 12, 11, 10,  9,  8,  7,  6, 5, 4, 3, 2, 1;
  23, 15, 14, 13, 12, 11, 10,  9,  8, 7, 6, 5, 4, 3, 2, 1;
  26, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1;

Harvey P. Dale, Table of n, a(n) for n = 1..1000
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A000027, A016789, A172310, A171164, A161165, A161166, A171172, A171173, A171174, A171175, A171177, A171178.

Cf. A132209 (row sums). - R. J. Mathar, Mar 01 2010

Table[{3n-1,Reverse[Range[2n-1]]},{n,10}]//Flatten (* Harvey P. Dale, Jun 26 2022 *)

A141530 a(n) = 4n^3 - 6n^2 + 1.

Original entry on oeis.org

1, -1, 9, 55, 161, 351, 649, 1079, 1665, 2431, 3401, 4599, 6049, 7775, 9801, 12151, 14849, 17919, 21385, 25271, 29601, 34399, 39689, 45495, 51841, 58751, 66249, 74359, 83105, 92511, 102601, 113399, 124929, 137215, 150281, 164151, 178849, 194399, 210825, 228151

Offset: 0

Paul Curtz, Aug 12 2008

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A046092, A141047, A141417.

See Librandi's comment in A078371.

[4*n^3 -6*n^2 +1: n in [0..50]]; // G. C. Greubel, Mar 29 2021

A141530:= n-> 4*n^3 -6*n^2 +1; seq(A141530(n), n=0..50); # G. C. Greubel, Mar 29 2021

Array[4*#^3-6*#^2+1&,50,0] (* Vladimir Joseph Stephan Orlovsky, Nov 03 2009 *)
LinearRecurrence[{4,-6,4,-1},{1,-1,9,55},50] (* Harvey P. Dale, Nov 30 2011 *)

a(n)=4*n^3-6*n^2+1 \\ Charles R Greathouse IV, Oct 07 2015

def A141530(n): return (m:=(n<<1)-1)*(n*(m-1)-1) # Chai Wah Wu, Mar 11 2024

[4*n^3 -6*n^2 +1 for n in (0..50)] # G. C. Greubel, Mar 29 2021

a(n) = (2*n-1)*(2*n^2 - 2*n - 1) = A060747(n)*A132209(n-1), n > 1. - R. J. Mathar, Feb 22 2009
G.f.: (1 - 5*x + 19*x^2 + 9*x^3)/(1-x)^4. - Jaume Oliver Lafont, Aug 30 2009
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) with a(0)=1, a(1)=-1, a(2)=9, a(3)=55. - Harvey P. Dale, Nov 30 2011
E.g.f.: (1 - 2*x + 6*x^2 + 4*x^3)*exp(x). - G. C. Greubel, Mar 29 2021

Corrected, completed and edited, following an observation from Vincenzo Librandi, by M. F. Hasler, Feb 12 2009

Further edited by N. J. A. Sloane, Feb 13 2009

A330613 Triangle read by rows: T(n, k) = 1 + k - 2n - 2kn + 2n^2, with 0 <= k < n.

Original entry on oeis.org

1, 5, 2, 13, 8, 3, 25, 18, 11, 4, 41, 32, 23, 14, 5, 61, 50, 39, 28, 17, 6, 85, 72, 59, 46, 33, 20, 7, 113, 98, 83, 68, 53, 38, 23, 8, 145, 128, 111, 94, 77, 60, 43, 26, 9, 181, 162, 143, 124, 105, 86, 67, 48, 29, 10, 221, 200, 179, 158, 137, 116, 95, 74, 53, 32, 11

Offset: 1

Stefano Spezia, Dec 20 2019

T(n, k) is the k-th super- and subdiagonal sum of the matrix M(n) whose permanent is A330287(n).

			n\k|   0   1   2   3   4   5
---+------------------------
1  |   1
2  |   5   2
3  |  13   8   3
4  |  25  18  11   4
5  |  41  32  23  14   5
6  |  61  50  39  28  17   6
...
For n = 3 the matrix M is
      1, 2, 3
      2, 4, 6
      3, 6, 8
and therefore T(3, 0) = 1 + 4 + 8 = 13, T(3, 1) = 2 + 6 = 8 and T(3, 2) = 3.

Stefano Spezia, First 150 rows of the triangle, flattened

Cf. A005408, A330287.

Cf. A000027: diagonal; A001105: 2nd column; A001844: 1st column; A016789: 1st subdiagonal; A016885: 2nd subdiagonal; A017029: 3rd subdiagonal; A017221: 4th subdiagonal; A017461: 5th subdiagonal; A081436: row sums; A132209: 3rd column; A164284: 7th subdiagonal; A269044: 6th subdiagonal.

Flatten[Table[1+k-2n-2k*n+2n^2,{n,1,11},{k,0,n-1}]] (* or *)
r[n_] := Table[SeriesCoefficient[(1-x*(2-5x+2(1+x)y))/((1-x)^3*(1-y)^2), {x, 0, i}, {y, 0, j}], {i, n, n}, {j, 0, n-1}]; Flatten[Array[r, 11]] (* or *)
r[n_] := Table[SeriesCoefficient[Exp[x+y]*(1+2x(x-y)+y), {x, 0, i}, {y, 0, j}]*i!*j!, {i, n, n}, {j, 0, n-1}]; Flatten[Array[r, 11]]

O.g.f.: (1 - x*(2 - 5*x + 2*(1 + x)*y))/((1 - x)^3*(1 - y)^2).
E.g.f.: exp(x+y)*(1 + 2*x*(x - y) + y).
T(n, k) = A001844(n-1) - k*A005408(n-1), with 0 <= k < n. [Typo corrected by Stefano Spezia, Feb 14 2020]

A294774 a(n) = 2n^2 + 2n + 5.

Original entry on oeis.org

5, 9, 17, 29, 45, 65, 89, 117, 149, 185, 225, 269, 317, 369, 425, 485, 549, 617, 689, 765, 845, 929, 1017, 1109, 1205, 1305, 1409, 1517, 1629, 1745, 1865, 1989, 2117, 2249, 2385, 2525, 2669, 2817, 2969, 3125, 3285, 3449, 3617, 3789, 3965, 4145, 4329, 4517, 4709, 4905

Offset: 0

Bruno Berselli, Nov 08 2017

This is the case k = 9 of 2*n^2 + (1-(-1)^k)*n + (2*k-(-1)^k+1)/4 (similar sequences are listed in Crossrefs section). Note that:

2*( 2*n^2 + (1-(-1)^k)*n + (2*k-(-1)^k+1)/4 ) - k = ( 2*n + (1-(-1)^k)/2 )^2. From this follows an alternative definition for the sequence: Numbers h such that 2*h - 9 is a square. Therefore, if a(n) is a square then its base is a term of A075841.

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

1st diagonal of A154631, 3rd diagonal of A055096, 4th diagonal of A070216.
Second column of Mathar's array in A016813 (Comments section).
Subsequence of A001481, A001983, A004766, A020668, A046711 and A057653 (because a(n) = (n+2)^2 + (n-1)^2); A097268 (because it is also a(n) = (n^2+n+3)^2 - (n^2+n+2)^2); A047270; A243182 (for y=1).
Similar sequences (see the first comment): A161532 (k=-14), A181510 (k=-13), A152811 (k=-12), A222182 (k=-11), A271625 (k=-10), A139570 (k=-9), (-1)*A147973 (k=-8), A059993 (k=-7), A268581 (k=-6), A090288 (k=-5), A054000 (k=-4), A142463 or A132209 (k=-3), A056220 (k=-2), A046092 (k=-1), A001105 (k=0), A001844 (k=1), A058331 (k=2), A051890 (k=3), A271624 (k=4), A097080 (k=5), A093328 (k=6), A271649 (k=7), A255843 (k=8), this sequence (k=9).

seq(2*n^2 + 2*n + 5, n=0..100); # Robert Israel, Nov 10 2017

Table[2n^2+2n+5,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{5,9,17},50] (* Harvey P. Dale, Sep 18 2023 *)

Vec((5 - 6*x + 5*x^2) / (1 - x)^3 + O(x^50)) \\ Colin Barker, Nov 13 2017

O.g.f.: (5 - 6*x + 5*x^2)/(1 - x)^3.
E.g.f.: (5 + 4*x + 2*x^2)*exp(x).
a(n) = a(-1-n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = 5*A000217(n+1) - 6*A000217(n) + 5*A000217(n-1).
n*a(n) - Sum_{j=0..n-1} a(j) = A002492(n) for n>0.
a(n) = Integral_{x=0..2n+4} |3-x| dx. - Pedro Caceres, Dec 29 2020

A143199 Triangle read by rows: T(n, k) = (n+1)A000096(k-1) + n if k <= floor(n/2), otherwise T(n, k) = (n+1)A000096(n-k-1) + n.

Original entry on oeis.org

-1, -1, -1, -1, 2, -1, -1, 3, 3, -1, -1, 4, 14, 4, -1, -1, 5, 17, 17, 5, -1, -1, 6, 20, 41, 20, 6, -1, -1, 7, 23, 47, 47, 23, 7, -1, -1, 8, 26, 53, 89, 53, 26, 8, -1, -1, 9, 29, 59, 99, 99, 59, 29, 9, -1, -1, 10, 32, 65, 109, 164, 109, 65, 32, 10, -1

Offset: 0

Roger L. Bagula and Gary W. Adamson, Oct 20 2008

			Triangle begins as:
  -1;
  -1, -1;
  -1,  2, -1;
  -1,  3,  3, -1;
  -1,  4, 14,  4,  -1;
  -1,  5, 17, 17,   5,  -1;
  -1,  6, 20, 41,  20,   6,  -1;
  -1,  7, 23, 47,  47,  23,   7, -1;
  -1,  8, 26, 53,  89,  53,  26,  8, -1;
  -1,  9, 29, 59,  99,  99,  59, 29,  9, -1;
  -1, 10, 32, 65, 109, 164, 109, 65, 32, 10, -1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000096, A132209, A142463.

function T(n,k) // A143199
  if k le Floor(n/2) then return n + (n+1)*(k-1)*(k+2)/2;
  else return T(n,n-k);
  end if;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 10 2024

seq(print(seq((n + 1) * (if m <= n/2 then (m - 1) * (m + 2)\
 / 2 else (n - m + 2) * (n - (m + 1)) / 2 fi) + n, m=0..n)), n=0..10); # Georg Fischer, Oct 28 2023

T[n_, k_]:= If[k<=Floor[n/2], n +(n+1)*(k-1)*(k+2)/2, T[n,n-k]];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten

def A143199(n,k): return n +(n+1)*(k-1)*(k+2)//2 if (k<1+int(n//2)) else A143199(n,n-k)
flatten([[A143199(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jun 10 2024

T(n, m) = (n + 1)*(if m <= floor(n/2) then (m - 1)*(m + 2) / 2 else (n - m + 2)*(n - (m + 1)) / 2 fi) + n. - Georg Fischer, Oct 28 2023
From G. C. Greubel, Jun 10 2024: (Start)
T(n, k) = n + (n+1)*(k-1)*(k+2)/2 if 0 <= k <= floor(n/2), otherwise T(n, k) = T(n, n-k).
Sum_{k=0..n} T(n, k) = (1/48)*(n+1)*(-53 - 5*n + 3*(-1)^n*(n+1) + 2*(n + 1)^3). (End)

Definition clarified and offset corrected by Georg Fischer, Oct 28 2023

A214868 Triangle T read by rows: T(n,0) = T(n,n) = 1 for n>=0, for n>=2 and 1<=k<=n-1, T(n,k) = T(n-1,k-1) + T(n-1,k) if k = [n/2] or k = [(n+1)/2], else T(n,k) = T(n-1,k-1) + T(n-2,k-1) + T(n-1,k).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 6, 5, 1, 1, 7, 11, 11, 7, 1, 1, 9, 23, 22, 23, 9, 1, 1, 11, 39, 45, 45, 39, 11, 1, 1, 13, 59, 107, 90, 107, 59, 13, 1, 1, 15, 83, 205, 197, 197, 205, 83, 15, 1

Offset: 0

Philippe Deléham, Mar 10 2013

			Triangle begins
1
1, 1
1, 2, 1
1, 3, 3, 1
1, 5, 6, 5, 1
1, 7, 11, 11, 7, 1
1, 9, 23, 22, 23, 9, 1
1, 11, 39, 45, 45, 39, 11, 1
1, 13, 59, 107, 90, 107, 59, 13, 1
1, 15, 83, 205, 197, 197, 205, 83, 15, 1
1, 17, 111, 347, 509, 394, 509, 347, 111, 17, 1
1, 19, 143, 541, 1061, 903, 903, 1061, 541, 143, 19, 1
1, 21, 179, 795, 1949, 2473, 1806, 2473, 1949, 795, 179, 21, 1
...

Cf. A001003, A006318, A065096, A110110

Sum_{k, 0<=k<=n} T(n,k) = A110110(n), number of symmetric Schroeder paths of length 2n.
Sum_{k, 0<=k<=n-2} T(n+k,k) = A065096(n-1), n>=2.
T(2n,n) = A006318(n), large Schroeder numbers.
T(2n+1,n) = A001003(n+1), little Schroeder numbers.
T(n,0) = A000012(n).
T(n,1) = A004280(n).
T(n+2,2) = A142463(n) = A132209(n), n>0.

cp's OEIS Frontend

A142463 a(n) = 2*n^2 + 2*n - 1.

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A185787 Sum of first k numbers in column k of the natural number array A000027; by antidiagonals.

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A132355 Numbers of the form 9*h^2 + 2*h, for h an integer.

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A132356 a(2*k) = k*(10*k+2), a(2*k+1) = 10*k^2 + 18*k + 8, with k >= 0.

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A171176 Triangle read by rows in which row n lists 3n-1 together with the first 2n-1 positive integers, in reverse order.

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A141530 a(n) = 4*n^3 - 6*n^2 + 1.

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A142463 a(n) = 2n^2 + 2n - 1.

A132355 Numbers of the form 9h^2 + 2h, for h an integer.

A132356 a(2k) = k(10k+2), a(2k+1) = 10k^2 + 18k + 8, with k >= 0.

A141530 a(n) = 4n^3 - 6n^2 + 1.

A330613 Triangle read by rows: T(n, k) = 1 + k - 2n - 2kn + 2n^2, with 0 <= k < n.

A294774 a(n) = 2n^2 + 2n + 5.

A143199 Triangle read by rows: T(n, k) = (n+1)A000096(k-1) + n if k <= floor(n/2), otherwise T(n, k) = (n+1)A000096(n-k-1) + n.