A055999 -id:A055999 - cp's OEIS frontend

A134227 Row sums of triangle A134226.

1, 4, 9, 15, 22, 30, 39, 49, 60, 72, 85, 99, 114, 130, 147, 165, 184, 204, 225, 247, 270, 294, 319, 345, 372, 400, 429, 459, 490, 522, 555, 589, 624, 660, 697, 735, 774, 814, 855, 897, 940, 984, 1029, 1075, 1122, 1170, 1219, 1269, 1320, 1372, 1425, 1479, 1534, 1590, 1647, 1705, 1764, 1824, 1885, 1947

Offset: 1

Gary W. Adamson, Oct 14 2007

nonn

Essentially the same as A055999. - R. J. Mathar, Mar 28 2012

			a(4) = 15 = sum of row 4 terms of triangle A134226: (1 + 2 + 8 + 4).
a(4) = 15 = (1, 3, 3, 1) dot (1, 3, 2, -1) = (1 + 9 + 6 - 1).

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

Cf. A134226.

[1] cat [(n-1)*(n+6)/2: n in [2..70]]; // G. C. Greubel, Feb 17 2021

Table[(n-1)*(n+6)/2 + Boole[n==1], {n, 70}] (* G. C. Greubel, Feb 17 2021 *)
LinearRecurrence[{3,-3,1},{1,4,9,15},70] (* Harvey P. Dale, Aug 13 2024 *)

[1]+[(n-1)*(n+6)/2 for n in (2..70)] # G. C. Greubel, Feb 17 2021

Binomial transform of (1, 3, 2, -1, 1, -1, 1, -1, 1, ...).
From G. C. Greubel, Feb 17 2021: (Start)
a(n) = (n-1)*(n+6)/2 + [n=1].
G.f.: x*(1 +x -x^3)/(1-x)^3.
E.g.f.: 3 + x + (-6 +6*x +x^2)*exp(x)/2. (End)

Terms a(37) onward added by G. C. Greubel, Feb 17 2021

A185732 Accumulation array of the polygonal number array (A086270), by antidiagonals.

Original entry on oeis.org

1, 4, 2, 10, 9, 3, 20, 24, 15, 4, 35, 50, 42, 22, 5, 56, 90, 90, 64, 30, 6, 84, 147, 165, 140, 90, 39, 7, 120, 224, 273, 260, 200, 120, 49, 8, 165, 324, 420, 434, 375, 270, 154, 60, 9, 220, 450, 612, 672, 630, 510, 350, 192, 72, 10, 286, 605, 855, 984, 980, 861, 665, 440, 234, 85, 11, 364, 792, 1155, 1380, 1440, 1344, 1127, 840, 540, 280, 99, 12, 455, 1014, 1518, 1870, 2025, 1980, 1764, 1428, 1035, 650, 330, 114, 13, 560, 1274, 1950, 2464, 2750, 2790, 2604, 2240

Offset: 1

Clark Kimberling, Feb 01 2011

This is the (first) accumulation array of A086270; the second is A185733. See A144112 for the definition of accumulation array.

			Northwest corner:
1....4....10...20...35
2....9....24...50...90
3....15...42...90...165
4....22...64...140..260
5....30...90...200..375

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Rows 1 to 5: A000292, A006002, A059270, A177814, 5*A002411.
Columns 1 to 4: A000027, A055999, A067728, 10*A000096.
Cf. A144112, A086270, A185732.

f[n_,k_]:=k+n*k(k-1)/2;
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]]  (* Array A086270 *)
Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten  (* A086270 *)
s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}]; (* acc. arr. of {f(n,k)} *)
Factor[s[n,k]]  (* formula for A185732 *)
TableForm[Table[s[n,k],{n,1,10},{k,1,15}]] (* acc. arr. A185732 *)
Table[s[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten (* A185732 *)

T(n,k) = k*(k+1)*n*(n+1)*(k*n-n+k+5)/12.

A174183 a(n) is the period k such that binomial(m, n) (mod 10) = binomial(m + k, n) (mod 10).

Original entry on oeis.org

1, 10, 20, 60, 240, 1200, 7200, 50400, 403200, 3628800, 36288000, 399168000, 4790016000, 62270208000, 871782912000, 13076743680000, 209227898880000, 3556874280960000, 64023737057280000, 1216451004088320000

Offset: 0

Michel Lagneau, Mar 11 2010

a(n) is the period (mod 10) of the numbers in each column n of Pascal's triangle.

			x(0)= 0.C(1,0)C(2,0)C(3,0) ... = 0.11111111111... and p(0)=1 ;
x(1)= 0.C(1,1)C(2,1)C(3,1) ... = 0.12345678901234... and p(1) = 10 ;
x(2)= 0.C(2,2)C(3,2)C(4,2) ... = 0.13605186556815063100 13605186556815063100... and p(2)=20.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.

Harvey P. Dale, Table of n, a(n) for n = 0..449
Michel Lagneau, Proof
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081, 2014

Cf. A000142, A002415, A007318, A002024, A000096, A000124, A002378, A000292, A000330, A055998, A055999, A056000, A056115, A056119, A056121, A056126, A051942, A101859, A001477.

Join[{1},Array[10#!&,20]] (* Harvey P. Dale, Feb 18 2018 *)

from math import factorial
def A174183(n): return 10*factorial(n) if n else 1 # Chai Wah Wu, Aug 07 2025

a(0)=1, and a(n) = 10 * n! for n >= 1.

Additional comments, and errors in examples corrected by Michel Lagneau, May 07 2010

A193002 Triangle T(n,k)=0 (k odd), T(0,0)=-3, T(n,0)=1 (n > 0) and T(n,k) = T(n-1,k) - T(n-2,k-2).

Original entry on oeis.org

-3, 1, 0, 1, 0, 3, 1, 0, 2, 0, 1, 0, 1, 0, -3, 1, 0, 0, 0, -5, 0, 1, 0, -1, 0, -6, 0, 3, 1, 0, -2, 0, -6, 0, 8, 0, 1, 0, -3, 0, -5, 0, 14, 0, -3, 1, 0, -4, 0, -3, 0, 20, 0, -11, 0, 1, 0, -5, 0, 0, 0, 25, 0, -25, 0, 3, 1, 0, -6

Offset: 0

Paul Curtz, Jul 14 2011

Consider an array with recurrence BB(m,n) = BB(m,n-1) + BB(m-1,n), m >= 0:
  3, -1, -1, -1,  -1,  -1,  -1,  -1,  -1,  -1,   -1,
  3,  2,  1,  0,  -1,  -2,  -3,  -4,  -5,  -6,   -7,
  3,  5,  6,  6,   5,   3,   0,  -4,  -9, -15,  -22,
  3,  8, 14, 20,  25,  28,  28,  24,  15,   0,  -22,
  3, 11, 25, 45,  70,  98, 126, 150, 165, 165,  143,
  3, 14, 39, 84, 154, 252, 378, 528, 693, 858, 1001,
with BB(m,n) = (3m-n)*binomial(n+m-1,n)/m if m > 0. So the BB are polynomials of degree m in n:
BB(1,n) = -(n-3)/1,
BB(2,n) = -(n-6)*(n+1)/2,    (see A055999)
BB(3,n) = -(n-9)*(n+1)*(n+2)/6,
BB(4,n) = -(n-12)*(n+1)*(n+2)*(n+3)/24,
BB(5,n) = -(n-15)*(n+1)*(n+2)*(n+3)*(n+4)/120.
Columns in the array are A010701, A016789, A095794, A005564, A059302.
T(n,k) is a zero-padded, column-shifted, sign-modified transpose of this array.

			Triangle begins
  -3;
   1,   0;
   1,   0,   3;
   1,   0,   2,   0;
   1,   0,   1,   0,  -3;
   1,   0,   0,   0,  -5,   0;
   1,   0,  -1,   0,  -6,   0,   3;
   1,   0,  -2,   0,  -6,   0,   8,   0;
   1,   0,  -3,   0,  -5,   0,  14,   0,  -3;
   1,   0,  -4,   0,  -3,   0,  20,   0, -11,   0;

Cf. A174559.

BB := proc(m,n) if m=0 then if n= 0 then 3 ; else -1; end if; else (3*m-n)*binomial(n+m-1,n)/m ; end if; end proc:
A193002 := proc(n,k) if type(k,'odd') then 0; else (-1)^(1+k/2)*BB(k/2,n-k) ; end if; end proc:
seq(seq(A193002(n,k),k=0..n),n=0..15) ; # R. J. Mathar, Aug 30 2011

Sum_{k=0..n} T(n,k) = A130806(n+5). (row sums)
Sum_{k=0..n} (-1)^(k/2)*T(n,k) = -A000032(n-2). (alternating row sums)
T(n,k) = (-1)^(1+k/2)*BB(k/2,n-k). - R. J. Mathar, Aug 30 2011
T(n,2k) = (-1)^(1+k)*(5-n/k)*binomial(n-k-1,k-1), k > 0. - R. J. Mathar, Aug 30 2011

A214859 Triangle read by rows, T(n,k) = n^2 - k(k+1)/2 if k(k+1)/2 <= n^2.

Original entry on oeis.org

0, 1, 0, 4, 3, 1, 9, 8, 6, 3, 16, 15, 13, 10, 6, 1, 25, 24, 22, 19, 15, 10, 4, 36, 35, 33, 30, 26, 21, 15, 8, 0, 49, 48, 46, 43, 39, 34, 28, 21, 13, 4, 64, 63, 61, 58, 54, 49, 43, 36, 28, 19, 9, 81, 80, 78, 75, 71, 66, 60, 53, 45, 36, 26, 15, 3, 100, 99, 97

Offset: 0

Philippe Deléham, Mar 09 2013

Row lengths are in A214857.

			Triangle begins:
    0;
    1,   0;
    4,   3,   1;
    9,   8,   6,   3;
   16,  15,  13,  10,   6,   1;
   25,  24,  22,  19,  15,  10,   4;
   36,  35,  33,  30,  26,  21,  15,  8,  0;
   49,  48,  46,  43,  39,  34,  28, 21, 13,  4;
   64,  63,  61,  58,  54,  49,  43, 36, 28, 19,  9;
   81,  80,  78,  75,  71,  66,  60, 53, 45, 36, 26, 15,  3;
  100,  99,  97,  94,  90,  85,  79, 72, 64, 55, 45, 34, 22,  9;
  121, 120, 118, 115, 111, 106, 100, 93, 85, 76, 66, 55, 43, 30, 16, 1;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Cf. Diagonals: A000217, A034856, A055999,

Columns: A000290, A005563, A028872, A028878.

Table[s = {}; k = 0; While[tri = k*(k + 1)/2; tri <= n^2, AppendTo[s, n^2 - tri]; k++]; s, {n, 0, 10}] (* T. D. Noe, Mar 11 2013 *)

T(2*n,n) = A022264(n).

T(n,n) = n*(n-1)/2 = A000217(n-1).

A279895 a(n) = n(5n + 11)/2.

Original entry on oeis.org

0, 8, 21, 39, 62, 90, 123, 161, 204, 252, 305, 363, 426, 494, 567, 645, 728, 816, 909, 1007, 1110, 1218, 1331, 1449, 1572, 1700, 1833, 1971, 2114, 2262, 2415, 2573, 2736, 2904, 3077, 3255, 3438, 3626, 3819, 4017, 4220, 4428, 4641, 4859, 5082, 5310, 5543, 5781, 6024, 6272, 6525

Offset: 0

Bruno Berselli, Dec 22 2016

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

Cf. A000566, A008589, A017281.
Second bisection of A165720.
The first differences are in A016885.
Cf. similar sequences provided by P(s,m)+s*m, where P(s,m) = ((s-2)*m^2-(s-4)*m)/2 is the m-th s-gonal number: A008585 (s=2), A055999 (s=3), A028347 (s=4), A140091 (s=5), A033537 (s=6), this sequence (s=7), A067725 (s=8).

Magma
```
[n*(5*n+11)/2: n in [0..60]];
```

Table[n (5 n + 11)/2, {n, 0, 60}]
LinearRecurrence[{3,-3,1},{0,8,21},60] (* Harvey P. Dale, Nov 14 2022 *)

PARI
```
vector(60, n, n--; n*(5*n+11)/2)
```
Python
```
[n*(5*n+11)/2 for n in range(60)]
```
Sage
```
[n*(5*n+11)/2 for n in range(60)]
```

O.g.f.: x*(8 - 3*x)/(1 - x)^3.
E.g.f.: x*(16 + 5*x)*exp(x)/2.
a(n+h) - a(n-h) = h*A017281(n+1), with h>=0. A particular case:
a(n) - a(-n) = 11*n = A008593(n).
a(n+h) + a(n-h) = 2*a(n) + A033429(h), with h>=0. A particular case:
a(n) + a(-n) = A033429(n).
a(n) - a(n-2) = A017281(n) for n>1. Also:
40*a(n) + 121 = A017281(n+1)^2.
a(n) = A000566(n) + 7*n, also a(n) = A000566(n) + A008589(n). - Michel Marcus, Dec 22 2016

A095668 Sixth column (m=5) of (1,4)-Pascal triangle A095666.

Original entry on oeis.org

4, 21, 66, 161, 336, 630, 1092, 1782, 2772, 4147, 6006, 8463, 11648, 15708, 20808, 27132, 34884, 44289, 55594, 69069, 85008, 103730, 125580, 150930, 180180, 213759, 252126, 295771, 345216, 401016, 463760, 534072, 612612, 700077, 797202, 904761

Offset: 0

Wolfdieter Lang, Jun 11 2004

If Y is a 4-subset of an n-set X, then, for n >= 8, a(n-8) is the number of 5-subsets of X having at most one element in common with Y. - Milan Janjic, Dec 08 2007

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A000389, A095666.

Cf. A000217, A055999.

[(n+20)*Binomial(n+4, 4)/5: n in [0..30]]; // G. C. Greubel, Nov 25 2017

A095668:=n->(n+20)*binomial(n+4, 4)/5: seq(A095668(n), n=0..80); # Wesley Ivan Hurt, Nov 25 2017

Table[(n + 20)*Binomial[n + 4, 4]/5, {n, 0, 50}] (* G. C. Greubel, Nov 25 2017 *)

for(n=0,30, print1((n+20)*binomial(n+4, 4)/5, ", ")) \\ G. C. Greubel, Nov 25 2017

G.f.: (4-3*x)/(1-x)^6.
a(n) = (n+20)*binomial(n+4, 4)/5.
a(n) = 4*b(n) - 3*b(n-1), with b(n) = binomial(n+5, 5) = A000389(n+5, 5).
E.g.f.: (480 + 2040*x + 1680*x^2 + 440*x^3 + 40*x^4 + x^5)*exp(x)/120. - G. C. Greubel, Nov 25 2017
a(n) = Sum_{i=0..n+1} A000217(i)*A055999(n+2-i). - Bruno Berselli, Mar 05 2018

A209274 Table T(n,k) = n*(n+2^k-1)/2, n, k > 0 read by antidiagonals.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 9, 10, 16, 17, 15, 14, 15, 32, 33, 27, 22, 20, 21, 64, 65, 51, 38, 30, 27, 28, 128, 129, 99, 70, 50, 39, 35, 36, 256, 257, 195, 134, 90, 63, 49, 44, 45, 512, 513, 387, 262, 170, 111, 77, 60, 54, 55, 1024, 1025, 771, 518, 330, 207, 133, 92, 72, 65, 66

Offset: 1

Boris Putievskiy, Jan 15 2013

Column number 1 A000217 n*(n+1)/2,
column number 2 A000096 n*(n+3)/2,
column number 3 A055999 n*(n+7)/2,
column number 4 A056121 n*(n+15)/2,
column number 5 A132758 n*(n+31)/2.
Row number 1 A000079 2^k,
row number 2 A000051 2^k + 1.

			The start of the sequence as table:
  1....2...4...8...16...32...64...
  3....5...9..17...33...65..129...
  6....9..15..27...51...99..195...
  10..14..22..38...70..134..262...
  15..20..30..50...90..170..330...
  21..27..39..63..111..207..399...
  28..35..49..77..133..245..469...
  . . .
The start of the sequence as triangle array read by rows:
  1;
  2,3;
  4,5,6;
  8,9,9,10;
  16,17,15,14,15;
  32,33,27,22,20,21;
  64,65,51,38,30,27,28;
  . . .
Row number r contains r numbers.

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.

Cf. A000217, A000096, A055999, A056121, A132758, A000079, A000051.

b[n_] := n - d[n]*(d[n] + 1)/2; c[n_] := (d[n]^2 + 3*d[n] + 4)/2 - n; d[n_] := Floor[(-1 + Sqrt[8*n - 7])/2]; a[n_] := b[n]*(b[n] + 2^c[n] - 1)/2; Table[a[n], {n, 1, 50}] (* G. C. Greubel, Jan 04 2018 *)

a(n, k) = n*(n+2^k-1)/2
array(rows, cols) = for(x=1, rows, for(y=1, cols, print1(a(x, y), ", ")); print(""))
/* Print initial 7 rows and 8 columns of table as follows */
array(7, 8) \\ Felix Fröhlich, Jan 05 2018

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
result = i*(i+2**j-1)/2

a(n) = A002260(n)*(A002260(n)+2^A004736(n)-1)/2.
a(n) = i*(i+2^j-1)/2,
where i = n-t*(t+1)/2, j = (t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2).

A358269 a(n) is the position m of the last prime term in the sequence {b(m)} defined by b(1) = n, if b(m) is prime then b(m+1) = b(m) - m, else b(m+1) = b(m) + m.

Original entry on oeis.org

3, 1004, 3, 1004, 3, 1004, 30, 349, 30, 5, 19, 5, 30, 1004, 30, 8, 11, 8, 30, 5, 86, 17, 67, 17, 15, 9, 19, 9, 15, 9, 19, 484, 19, 13, 30, 9, 19, 9, 19, 13, 374, 13, 19, 13, 11, 484, 86, 484, 19, 13, 67, 16, 19, 16, 19, 484, 374, 484, 19, 484, 374, 24, 19, 13

Offset: 0

Samuel Harkness, Nov 06 2022

nonn

A sequence {b(m)} is guaranteed to have no more primes when the m-th term "k" with value "s" is the sum of at least 3 consecutive positive integers where the sum is "s" and the last consecutive positive integer in the sum is k-1. Any number which is the sum of at least three consecutive positive integers is guaranteed to be composite. By the definition of the sequence, the next term k + 1 = s + k, and this term will be the sum of at least three consecutive positive integers with the last consecutive positive integer being k. This guarantees that this term is also guaranteed to be composite, and by induction, all future terms in {b(m)} will be composite.
In a sequence {b(m)}, if the m-th term k with value s satisfies c = (sqrt(-8*s + 4*k^2 - 4*k + 1) + 1)/2 for a positive integer c with s being nonprime and k > 3 then the value of all terms >= k will be composite.
It is unknown whether all initial conditions "n" guarantee a final prime. All terms up to n = 1000 have a final prime.
Treat negative numbers in the sequence {b(m)} as nonprime. The only n whose {b(m)} contain negative terms b(m) are 1, 3, 6, and 7.

			For n = 9: b(1) = 9. Nonprime, b(2) = 9 + 1 = 10. Nonprime, b(3) = 10 + 2 = 12. Nonprime, b(4) = 12 + 3 = 15. Nonprime, b(5) = 15 + 4 = 19. Prime, b(6) = 19 - 5 = 14. Note 14 = 2 + 3 + 4 + 5 and is nonprime, so b(7) = 2 + 3 + 4 + 5 + 6 and nonprime. All b(m) after this will be nonprime by the same pattern, thus the final prime for b(1) = 9 occurs at b(5), and a(9) = 5.

Samuel Harkness, Table of n, a(n) for n = 0..1111

Cf. A074171, A358166

Examples of sequences of the sum of consecutive positive integers, where the sum of at least three is guaranteed to be composite: A055999, A212427.

T = {}; For[f = 0, f <= 63, f++, a = 0; t = f; q = 0; While[a == 0, q++; If[t < 0, t += q, If[PrimeQ[t], t -= q; If[t >= 0, If[q != 2 && q != 1 && ! PrimeQ[t], s = t; k = q + 1; z = (Sqrt[-8 s + 4 k^2 - 4 k + 1] + 1)/2; If[Element[z, Reals] && z > 0 && Mod[z, 1] == 0, AppendTo[T, q]; Break[]]]], t += q]]]]; Print[T]

A356754 Triangle read by rows: T(n,k) = ((n-1)(n+2))/2 + 2k.

Original entry on oeis.org

2, 4, 6, 7, 9, 11, 11, 13, 15, 17, 16, 18, 20, 22, 24, 22, 24, 26, 28, 30, 32, 29, 31, 33, 35, 37, 39, 41, 37, 39, 41, 43, 45, 47, 49, 51, 46, 48, 50, 52, 54, 56, 58, 60, 62, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87

Offset: 1

Torlach Rush, Aug 25 2022

The first column of the triangle is the Lazy Caterer's sequence A000124.
Each subsequent column starts with A000124(n) + (2 * (n-1)).
The first downward diagonal is A046691(n).
Columns and downward diagonals of the triangle identify many sequences (possibly shifted) in the database. Examples can be found in crossrefs below.
The sum of the n-th upward diagonal of the triangle is A356288(n).

			Triangle T(n,k) begins:
  n\k   1   2   3   4   5   6   7   8   9  10  11  ...
   1:   2
   2:   4   6
   3:   7   9  11
   4:  11  13  15  17
   5:  16  18  20  22  24
   6:  22  24  26  28  30  32
   7:  29  31  33  35  37  39  41
   8:  37  39  41  43  45  47  49  51
   9:  46  48  50  52  54  56  58  60  62
  10:  56  58  60  62  64  66  68  70  72  74
  11:  67  69  71  73  75  77  79  81  83  85  87
  ...

Cf. A000124, A004120, A046691, A051938, A055999, A056000, A155212, A167487, A167499, A167614, A246172, A334563, A356288.

Table[((n-1)(n+2))/2+2k,{n,20},{k,n}]//Flatten (* Harvey P. Dale, May 26 2023 *)

def T(n, k): return ((n-1) * (n+2))//2 + 2*k
for n in range(1, 12):
    for k in range(1,(n+1)): print(T(n,k), end = ', ')

# Indexed as a linear sequence.
def a000124(n): return n*(n+1)//2 + 1
def a(n):
    l = m = 0
    for k in range(1,n):
        lc = a000124(k - 1)
        if n >= lc:
            l = lc
            m = k
        else: break
    return n + m + (n - l)

T(n,k) = ((n-1) * (n+2))/2 + 2*k.

T(n,k+1) = T(n,k) + 2, k < n.

cp's OEIS Frontend

A134227 Row sums of triangle A134226.

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A185732 Accumulation array of the polygonal number array (A086270), by antidiagonals.

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A174183 a(n) is the period k such that binomial(m, n) (mod 10) = binomial(m + k, n) (mod 10).

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A193002 Triangle T(n,k)=0 (k odd), T(0,0)=-3, T(n,0)=1 (n > 0) and T(n,k) = T(n-1,k) - T(n-2,k-2).

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A214859 Triangle read by rows, T(n,k) = n^2 - k*(k+1)/2 if k*(k+1)/2 <= n^2.

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A279895 a(n) = n*(5*n + 11)/2.

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A095668 Sixth column (m=5) of (1,4)-Pascal triangle A095666.

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A214859 Triangle read by rows, T(n,k) = n^2 - k(k+1)/2 if k(k+1)/2 <= n^2.

A279895 a(n) = n(5n + 11)/2.

A356754 Triangle read by rows: T(n,k) = ((n-1)(n+2))/2 + 2k.