A022856 -id:A022856 - cp's OEIS frontend

A055998 a(n) = n*(n+5)/2.

0, 3, 7, 12, 18, 25, 33, 42, 52, 63, 75, 88, 102, 117, 133, 150, 168, 187, 207, 228, 250, 273, 297, 322, 348, 375, 403, 432, 462, 493, 525, 558, 592, 627, 663, 700, 738, 777, 817, 858, 900, 943, 987, 1032, 1078, 1125, 1173, 1222, 1272

Offset: 0

Barry E. Williams, Jun 14 2000

If X is an n-set and Y a fixed (n-3)-subset of X then a(n-3) is equal to the number of 2-subsets of X intersecting Y. - Milan Janjic, Aug 15 2007
Bisection of A165157. - Jaroslav Krizek, Sep 05 2009
a(n) is the number of (w,x,y) having all terms in {0,...,n} and w=x+y-1. - Clark Kimberling, Jun 02 2012
Numbers m >= 0 such that 8m+25 is a square. - Bruce J. Nicholson, Jul 26 2017
a(n-1) = 3*(n-1) + (n-1)*(n-2)/2 is the number of connected, loopless, non-oriented, multi-edge vertex-labeled graphs with n edges and 3 vertices. Labeled multigraph analog of A253186. There are 3*(n-1) graphs with the 3 vertices on a chain (3 ways to label the middle graph, n-1 ways to pack edges on one of connections) and binomial(n-1,2) triangular graphs (one way to label the graphs, pack 1 or 2 or ...n-2 on the 1-2 edge, ...). - R. J. Mathar, Aug 10 2017
a(n) is also the number of vertices of the quiver for PGL_{n+1} (see Shen). - Stefano Spezia, Mar 24 2020
Starting from a(2) = 7, this is the 4th column of the array: natural numbers written by antidiagonals downwards. See the illustration by Kival Ngaokrajang and the cross-references. - Andrey Zabolotskiy, Dec 21 2021

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, p. 193.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Karl Dilcher and Larry Ericksen, Polynomials and algebraic curves related to certain binary and b-ary overpartitions, arXiv:2405.12024 [math.CO], 2024. See p. 10.
Milan Janjic, Two Enumerative Functions.
Kival Ngaokrajang, Illustration from A000027 (contains errors).
Linhui Shen, Duals of semisimple Poisson-Lie groups and cluster theory of moduli spaces of G-local systems, arXiv:2003.07901 [math.RT], 2020. See p. 8.
Leo Tavares, Illustration: Truncated Point Triangles.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

a(n) = A095660(n+1, 2): third column of (1, 3)-Pascal triangle.
Cf. A000096, A000217, A001477, A002522.
Row n=2 of A255961.
Cf. other rows, columns and diagonals of A000027 written as a table: A034856, A046691, A052905, A055999, A155212, A051936, A056000, A183897, A056115, A051938; A000124, A022856, A152950, A145018, A077169, A166136, A167487, A173036; A059993, A090288, A054000, A142463, A056220, A001105, A001844, A058331, A051890, A097080, A093328, A137882.

[n*(n+5)/2: n in [0..50]]; // G. C. Greubel, Apr 05 2019

f[n_]:=n*(n+5)/2; f[Range[0,50]] (* Vladimir Joseph Stephan Orlovsky, Feb 10 2011 *)

a(n)=n*(n+5)/2 \\ Charles R Greathouse IV, Sep 24 2015

[n*(n+5)/2 for n in (0..50)] # G. C. Greubel, Apr 05 2019

G.f.: x*(3-2*x)/(1-x)^3.
a(n) = A027379(n), n > 0.
a(n) = A126890(n,2) for n > 1. - Reinhard Zumkeller, Dec 30 2006
a(n) = A000217(n) + A005843(n). - Reinhard Zumkeller, Sep 24 2008
If we define f(n,i,m) = Sum_{k=0..n-i} binomial(n,k)*Stirling1(n-k,i)*Product_{j=0..k-1} (-m-j), then a(n) = -f(n,n-1,3), for n >= 1. - Milan Janjic, Dec 20 2008
a(n) = A167544(n+8). - Philippe Deléham, Nov 25 2009
a(n) = a(n-1) + n + 2 with a(0)=0. - Vincenzo Librandi, Aug 07 2010
a(n) = Sum_{k=1..n} (k+2). - Gary Detlefs, Aug 10 2010
a(n) = A034856(n+1) - 1 = A000217(n+2) - 3. - Jaroslav Krizek, Sep 05 2009
Sum_{n>=1} 1/a(n) = 137/150. - R. J. Mathar, Jul 14 2012
a(n) = 3*n + A000217(n-1) = 3*n - floor(n/2) + floor(n^2/2). - Wesley Ivan Hurt, Jun 15 2013
a(n) = Sum_{i=3..n+2} i. - Wesley Ivan Hurt, Jun 28 2013
a(n) = 3*A000217(n) - 2*A000217(n-1). - Bruno Berselli, Dec 17 2014
a(n) = A046691(n) + 1. Also, a(n) = A052905(n-1) + 2 = A055999(n-1) + 3 for n>0. - Andrey Zabolotskiy, May 18 2016
E.g.f.: x*(6+x)*exp(x)/2. - G. C. Greubel, Apr 05 2019
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/5 - 47/150. - Amiram Eldar, Jan 10 2021
From Amiram Eldar, Feb 12 2024: (Start)
Product_{n>=1} (1 - 1/a(n)) = -5*cos(sqrt(33)*Pi/2)/(4*Pi).
Product_{n>=1} (1 + 1/a(n)) = 15*cos(sqrt(17)*Pi/2)/(2*Pi). (End)

A152948 a(n) = (n^2 - 3*n + 6)/2.

Original entry on oeis.org

2, 2, 3, 5, 8, 12, 17, 23, 30, 38, 47, 57, 68, 80, 93, 107, 122, 138, 155, 173, 192, 212, 233, 255, 278, 302, 327, 353, 380, 408, 437, 467, 498, 530, 563, 597, 632, 668, 705, 743, 782, 822, 863, 905, 948, 992, 1037, 1083, 1130, 1178, 1227, 1277, 1328, 1380

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 15 2008

a(1) = 2; then add 0 to the first number, then 1, 2, 3, 4, ... and so on.
Essentially the same as A022856, A089071 and A133263. - R. J. Mathar, Dec 19 2008
First differences are A001477.
From Vladimir Shevelev, Jan 20 2014: (Start)
If we ignore the zero polygonal numbers, then for n >= 3, a(n) is the minimal k such that the k-th n-gonal number is a sum of two n-gonal numbers (see formula and example).
If the zero polygonal numbers are ignored, then for n >= 4, the a(n)-th n-gonal number is a sum of the (a(n)-1)-th n-gonal number and the (n-1)-th n-gonal number. (End)
Numbers m such that 8m - 15 is a square. - Bruce J. Nicholson, Jul 24 2017

			a(7)=17. This means that the 17th (positive) heptagonal number 697 (cf. A000566) is the smallest heptagonal number which is a sum of two (positive) heptagonal numbers. We have 697 = 616 + 81 with indices 17, 16, 6 in A000566. - _Vladimir Shevelev_, Jan 20 2014

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Marilena Barnabei, Flavio Bonetti, Niccolò Castronuovo, and Matteo Silimbani, Permutations avoiding a simsun pattern, The Electronic Journal of Combinatorics (2020) Vol. 27, Issue 3, P3.45.
E. R. Berlekamp, A contribution to mathematical psychometrics, Unpublished Bell Labs Memorandum, Feb 08 1968 [Annotated scanned copy]
Kyu-Hwan Lee and Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.
Ângela Mestre and José Agapito, Square Matrices Generated by Sequences of Riordan Arrays, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000124, A000217, A000566, A001477, A022856, A089071, A133263, A152947.

Magma
```
[ (n^2-3*n+6)/2: n in [1..60] ];
```

Array[(#^2 - 3 # + 6)/2 &, 54] (* or *) Rest@ CoefficientList[Series[-x (2 - 4 x + 3 x^2)/(x - 1)^3, {x, 0, 54}],x] (* Michael De Vlieger, Mar 25 2020 *)

a(n)=(n^2-3*n+6)/2 \\ Charles R Greathouse IV, Sep 28 2015

[2+binomial(n,2) for n in range(0, 54)] # Zerinvary Lajos, Mar 12 2009

a(n) = a(n-1) + n-2 (with a(1)=2). - Vincenzo Librandi, Nov 26 2010
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: -x*(2 - 4*x + 3*x^2) /  (x-1)^3. - R. J. Mathar, Oct 30 2011
Sum_{n>=1} 1/a(n) = 1/2 + 2*Pi*tanh(sqrt(15)*Pi/2)/sqrt(15). - Amiram Eldar, Dec 13 2022
E.g.f.: exp(x)*(6 - 2*x + x^2)/2 - 3. - Stefano Spezia, Nov 14 2024

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

A089071 Number of liberties a big eye of size n gives in the game of Go.

Original entry on oeis.org

1, 2, 3, 5, 8, 12, 17, 23, 30, 38, 47, 57, 68, 80, 93, 107, 122, 138, 155, 173, 192, 212, 233, 255, 278, 302, 327, 353, 380, 408, 437, 467, 498, 530, 563, 597, 632, 668, 705, 743, 782, 822, 863, 905, 948, 992, 1037, 1083, 1130, 1178, 1227, 1277, 1328, 1380

Offset: 1

André Engels, Dec 03 2003

The terms after the seventh are considered to be of only theoretical importance, since the largest dead shape is six spaces.

			A 5-space big eye can be almost filled in 4 moves, after which one takes and has a 4-space big eye (5 liberties) left. This gives a total of 4 + 5 moves for the opponent and 1 for oneself, for de facto 8 liberties.

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

Cf. A022856.

[n eq 1 select 1 else Binomial(n-1,2) +2: n in [1..65]]; // G. C. Greubel, Oct 31 2022

Join[{1}, Binomial[Range[65],2] +2] (* G. C. Greubel, Oct 31 2022 *)

[binomial(n-1,2)+2-int(n==1) for n in range(1,65)] # G. C. Greubel, Oct 31 2022

a(n) = a(n-1) + n - 2 for n>=3.
From Paul Barry, Dec 07 2009: (Start)
G.f.: (1 - x + x^3)/(1-x)^3.
a(n) = n + 1 - 0^n + C(n-1,2). (End)
a(n) = A022856(n+2). - R. J. Mathar, Oct 30 2011

More terms from David Wasserman, Aug 29 2005

A229445 T(n,k)=Number of nXk 0..2 arrays with horizontal differences mod 3 never 1, vertical differences mod 3 never -1, and rows and columns lexicographically nondecreasing.

Original entry on oeis.org

3, 4, 5, 5, 7, 8, 6, 10, 13, 12, 7, 14, 22, 25, 17, 8, 19, 37, 53, 47, 23, 9, 25, 60, 109, 128, 84, 30, 10, 32, 93, 212, 324, 293, 142, 38, 11, 40, 138, 387, 753, 915, 625, 228, 47, 12, 49, 197, 665, 1609, 2546, 2402, 1244, 350, 57, 13, 59, 272, 1083, 3184, 6374, 8024

Offset: 1

R. H. Hardin Sep 23 2013

Table starts
..3...4....5....6.....7.....8......9.....10......11......12......13.......14
..5...7...10...14....19....25.....32.....40......49......59......70.......82
..8..13...22...37....60....93....138....197.....272.....365.....478......613
.12..25...53..109...212...387....665...1083....1684....2517....3637.....5105
.17..47..128..324...753..1609...3184...5890...10281...17075...27176....41696
.23..84..293..915..2546..6374..14536..30571...59969..110816..194535...326723
.30.142..625.2402..8024.23610..62205.149031..329106..677706.1314145..2419348
.38.228.1244.5843.23428.81177.247607.676983.1685570.3873314.8307126.16784531

			Some solutions for n=4 k=4
..0..2..2..2....0..2..2..2....0..0..2..2....0..0..2..2....0..2..2..2
..1..0..0..2....1..0..0..0....0..0..2..2....1..1..0..0....0..2..2..2
..2..1..1..0....2..1..1..1....1..1..0..0....1..1..1..1....0..2..2..2
..2..1..1..1....2..2..2..2....1..1..1..1....2..2..1..1....1..0..0..2

R. H. Hardin, Table of n, a(n) for n = 1..480

Column 1 is A022856(n+4)

Row 2 is A145018(n+1)

Empirical for column k:
k=1: a(n) = (1/2)*n^2 + (1/2)*n + 2
k=2: a(n) = (1/24)*n^4 + (1/12)*n^3 - (1/24)*n^2 + (23/12)*n + 2
k=3: [polynomial of degree 6]
k=4: [polynomial of degree 8]
k=5: [polynomial of degree 10]
k=6: [polynomial of degree 12]
k=7: [polynomial of degree 14]
Empirical for row n:
n=1: a(n) = n + 2
n=2: a(n) = (1/2)*n^2 + (1/2)*n + 4
n=3: a(n) = (1/3)*n^3 + (8/3)*n + 5
n=4: a(n) = (1/4)*n^4 - (1/3)*n^3 + (13/4)*n^2 + (11/6)*n + 7
n=5: a(n) = (11/60)*n^5 - (1/2)*n^4 + (15/4)*n^3 - n^2 + (257/30)*n + 6
n=6: [polynomial of degree 6]
n=7: [polynomial of degree 7]

A047080 Triangular array T read by rows: T(h,k)=number of paths from (0,0) to (k,h-k) using step-vectors (0,1), (1,0), (1,1) with no right angles between pairs of consecutive steps.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 3, 3, 1, 1, 4, 5, 5, 4, 1, 1, 5, 8, 9, 8, 5, 1, 1, 6, 12, 15, 15, 12, 6, 1, 1, 7, 17, 24, 27, 24, 17, 7, 1, 1, 8, 23, 37, 46, 46, 37, 23, 8, 1, 1, 9, 30, 55, 75, 83, 75, 55, 30, 9, 1, 1, 10, 38, 79, 118, 143, 143, 118, 79, 38, 10, 1

Offset: 0

Clark Kimberling

T(n,k) equals the number of reduced alignments between a string of length n and a string of length k. See Andrade et. al. - Peter Bala, Feb 04 2018

			E.g., row 3 consists of T(3,0)=1; T(3,1)=2; T(3,2)=2; T(3,3)=1.
Triangle begins:
  1;
  1,  1;
  1,  1,  1;
  1,  2,  2,  1;
  1,  3,  3,  3,  1;
  1,  4,  5,  5,  4,  1;
  1,  5,  8,  9,  8,  5,  1;
  1,  6, 12, 15, 15, 12,  6,  1;

Muniru A Asiru, Table of n, a(n) for n = 0..1325
H. Andrade, I. Area, J. J. Nieto, and A. Torres, The number of reduced alignments between two DNA sequences, BMC Bioinformatics (2014) Vol. 15: 94.

Cf. A047081, A047082, A047083, A047084, A047085, A047086, A047087, A047088.

Cf. A001590, A022856, A028310, A078042, A089071, A091337, A171155.

F:=Factorial;
p:= func< n,k | (&+[ (-1)^j*F(n+k-3*j)/(F(j)*F(n-2*j)*F(k-2*j)): j in [0..Min(Floor(n/2), Floor(k/2))]]) >;
q:= func< n,k | n eq 0 or k eq 0 select 0 else (&+[ (-1)^j*F(n+k-3*j-2)/(F(j)*F(n-2*j-1)*F(k-2*j-1)) : j in [0..Min(Floor((n-1)/2), Floor((k-1)/2))]]) >;
A:= func< n,k | p(n,k) - q(n,k) >;
A047080:= func< n,k | n eq 0 select 1 else A(n-k, k) >;
[[A(n,k): k in [1..6]]: n in [1..6]];
[A047080(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 30 2022

T := proc(n, k) option remember; if n < 0 or k > n then return 0 fi;
if n < 3 then return 1 fi; if k < iquo(n,2) then return T(n, n-k) fi;
T(n-1, k-1) + T(n-1, k) - T(n-4, k-2)  end:
seq(seq(T(n,k), k=0..n), n=0..11); # Peter Luschny, Feb 11 2018

T[n_, k_] := T[n, k] = Which[n<0 || k>n, 0, n<3, 1, kJean-François Alcover, Jul 30 2018 *)

f=factorial
def p(n,k): return sum( (-1)^j*f(n+k-3*j)/(f(j)*f(n-2*j)*f(k-2*j)) for j in range(1+min((n//2), (k//2))) )
def q(n,k): return sum( (-1)^j*f(n+k-3*j-2)/(f(j)*f(n-2*j-1)*f(k-2*j-1)) for j in range(1+min(((n-1)//2), ((k-1)//2))) )
def A(n,k): return p(n,k) - q(n,k)
def A047080(n,k): return A(n-k, k)
flatten([[A047080(n,k) for k in range(n+1)] for n in range(14)]) # G. C. Greubel, Oct 30 2022

T(h, k) = T(h-1, k-1) + T(h-1, k) - T(h-4, k-2);
Writing T(h, k) = F(h-k, k), generating function for F is (1-xy)/(1-x-y+x^2y^2).
From Peter Bala, Feb 04 2018: (Start)
T(n, k) = (Sum_{i = 0..A} (-1)^i*(n+k-3*i)!/(i!*(n-2*i)!*(k-2*i)!)) - (Sum_{i = 0..B} (-1)^i*(n+k-3*i-2)!/(i!*(n-2*i-1)!*(k-2*i-1)!)), where A = min{floor(n/2), floor(k/2)} and B = min{floor((n-1)/2), floor((k-1)/2)}.
T(2*n, n) = A171155(n). (End)
From G. C. Greubel, Oct 30 2022: (Start) (formulas for triangle T(n,k))
T(n, n-k) = T(n, k).
T(n, n) = A000012(n).
T(n, n-1) = A028310(n-1).
T(n, n-2) = A089071(n-1) = A022856(n+1).
T(2*n, n-1) = A047087(n).
T(2*n+1, n-1) = A047088(n).
Sum_{k=0..n} T(n, k) = (-1)^n*A078042(n) = A001590(n+3).
Sum_{k=0..n} (-1)^k*T(n, k) = A091337(n+1).
Sum_{k=0..floor(n/2)} T(n, k) = A047084(n). (End)

Sequence recomputed to correct terms from 23rd onward, and recurrence and generating function added by Michael L. Catalano-Johnson (mcj(AT)pa.wagner.com), Jan 14 2000

A229428 T(n,k) = Number of n X k 0..2 arrays with horizontal differences mod 3 never 1, vertical differences mod 3 never -1, rows lexicographically nondecreasing, and columns lexicographically nonincreasing.

Original entry on oeis.org

3, 5, 5, 8, 12, 8, 12, 27, 27, 12, 17, 55, 83, 55, 17, 23, 102, 222, 222, 102, 23, 30, 175, 524, 754, 524, 175, 30, 38, 282, 1116, 2204, 2204, 1116, 282, 38, 47, 432, 2187, 5700, 7816, 5700, 2187, 432, 47, 57, 635, 4005, 13345, 24126, 24126, 13345, 4005, 635, 57

Offset: 1

R. H. Hardin, Sep 22 2013

Table starts
..3...5....8....12.....17.....23......30.......38.......47........57........68
..5..12...27....55....102....175.....282......432......635.......902......1245
..8..27...83...222....524...1116....2187.....4005.....6936.....11465.....18219
.12..55..222...754...2204...5700...13345....28794....58053....110550....200533
.17.102..524..2204...7816..24126...66503...166972...387738....842802...1731129
.23.175.1116..5700..24126..87648..281016...812352..2152643...5297329..12231874
.30.282.2187.13345..66503.281016.1037193..3420692.10260128..28379127..73192023
.38.432.4005.28794.166972.812352.3420692.12768612.43042290.132960319.380811699

			Some solutions for n=4 k=4
..2..2..1..0....1..1..0..0....1..1..0..0....2..1..1..1....1..0..0..0
..2..2..1..1....2..1..0..0....2..1..0..0....2..1..1..1....1..1..1..1
..2..2..1..1....2..1..1..1....2..1..1..1....2..2..2..1....2..2..1..1
..2..2..2..1....2..1..1..1....2..2..1..1....2..2..2..2....2..2..2..1

R. H. Hardin, Table of n, a(n) for n = 1..480

Column 1 is A022856(n+4).

Main diagonal is A229421.

Empirical for column k:
k=1: a(n) = (1/2)*n^2 + (1/2)*n + 2
k=2: a(n) = (1/24)*n^4 + (5/12)*n^3 + (11/24)*n^2 + (25/12)*n + 2, A229422
k=3: [polynomial of degree 6], A229423
k=4: [polynomial of degree 8], A229424
k=5: [polynomial of degree 10], A229425
k=6: [polynomial of degree 12], A229426
k=7: [polynomial of degree 14]

A188553 T(n,k) = Number of n X k binary arrays without the pattern 0 1 diagonally, vertically, antidiagonally or horizontally.

Original entry on oeis.org

2, 3, 3, 4, 5, 4, 5, 8, 7, 5, 6, 12, 12, 9, 6, 7, 17, 20, 16, 11, 7, 8, 23, 32, 28, 20, 13, 8, 9, 30, 49, 48, 36, 24, 15, 9, 10, 38, 72, 80, 64, 44, 28, 17, 10, 11, 47, 102, 129, 112, 80, 52, 32, 19, 11, 12, 57, 140, 201, 192, 144, 96, 60, 36, 21, 12, 13, 68, 187, 303, 321, 256, 176

Offset: 1

R. H. Hardin, Apr 04 2011

From Miquel A. Fiol, Feb 06 2024: (Start)
Also, T(n,k) is the number of words of length k, x(1)x(2)...x(k), on the alphabet {0,1,...,n}, such that, for i=2,...,k, x(i)=either x(i-1) or x(i)=x(i-1)-1.
For the bijection between arrays and sequences, notice that the i-th column consists of 1's and then 0's, and there are x(i)=0 to n of 1's.
Such a bijection implies that all the empirical/conjectured formulas in A188554, A188555, A188556, A188557, A188558, and A188559 become correct.
(End)

			Table starts
..2..3..4..5...6...7...8...9...10...11...12....13....14....15....16.....17
..3..5..8.12..17..23..30..38...47...57...68....80....93...107...122....138
..4..7.12.20..32..49..72.102..140..187..244...312...392...485...592....714
..5..9.16.28..48..80.129.201..303..443..630...874..1186..1578..2063...2655
..6.11.20.36..64.112.192.321..522..825.1268..1898..2772..3958..5536...7599
..7.13.24.44..80.144.256.448..769.1291.2116..3384..5282..8054.12012..17548
..8.15.28.52..96.176.320.576.1024.1793.3084..5200..8584.13866.21920..33932
..9.17.32.60.112.208.384.704.1280.2304.4097..7181.12381.20965.34831..56751
.10.19.36.68.128.240.448.832.1536.2816.5120..9217.16398.28779.49744..84575
.11.21.40.76.144.272.512.960.1792.3328.6144.11264.20481.36879.65658.115402
Some solutions for 5 X 3:
  1 1 1   1 0 0   0 0 0   1 1 1   1 1 1   1 1 1   1 1 1
  1 1 1   0 0 0   0 0 0   1 1 1   1 1 1   1 1 1   1 1 1
  1 1 1   0 0 0   0 0 0   1 1 1   1 0 0   1 1 0   1 1 1
  1 1 1   0 0 0   0 0 0   1 1 0   0 0 0   1 0 0   1 1 1
  1 1 1   0 0 0   0 0 0   1 0 0   0 0 0   0 0 0   1 1 0
Some solutions for T(5,3): By taking the sums of the columns in the above arrays we get 555, 100, 000, 543, 322, 432, 554. - _Miquel A. Fiol_, Feb 04 2024

R. H. Hardin, Table of n, a(n) for n = 1..9934

Diagonal is A045623.
Column 4 is A086570.
Rows 2..8 are A022856(n+4), A188554, A188555, A188556, A188557, A188558, A188559.
Upper diagonals T(n,n+i) for i=1..8 give: A001792, A001787(n+1), A000337(n+1), A045618, A045889, A034009, A055250, A055251.
Lower diagonals T(n+i,n) for i=1..7 give: A045891(n+1), A034007(n+2), A111297(n+1), A159694(n-1), A159695(n-1), A159696(n-1), A159697(n-1).
Antidiagonal sums give A065220(n+5).

T:= (n,k)-> `if`(k<=n+1, (2*n+3-k)*2^(k-2), (n+1-k)*binomial(k-1, n) * add(binomial(n, j-1)/(k-j)*T(n, j)*(-1)^(n-j), j=1..n+1)): seq(seq(T(n, 1+d-n), n=1..d), d=1..15); #Alois P. Heinz in the Sequence Fans Mailing List, Apr 04 2011 [We do not permit programs based on conjectures, but this program is now justified by Fiol's comment. - N. J. A. Sloane, Mar 09 2024]

Empirical: T(n,k) = (n+1)*2^(k-1) + (1-k)*2^(k-2) for k < n+3, and then the entire row n is a polynomial of degree n in k.
From Miquel A. Fiol, Feb 06 2024: (Start)
The above empirical formula is correct.
It can be proved that T(n,k) satisfies the recurrence
  T(n,k) = Sum_{r=1..n+1} (-1)^(r+1)*binomial(n+1,r)*T(n,k-r)
with initial values
  T(n,k) = Sum_{r=0..k-1} (n+1-r)*binomial(k-1,r) for k = 1..n+1. (End)

A055472 Primes of the form k(k+1)/2+2 (i.e., two more than a triangular number).

Original entry on oeis.org

2, 3, 5, 17, 23, 47, 107, 173, 233, 353, 467, 563, 743, 863, 1277, 1433, 1487, 2213, 2417, 2777, 3083, 3323, 4007, 4373, 5153, 7877, 8387, 10733, 11177, 11783, 13043, 13697, 14537, 15053, 15227, 17207, 17393, 17957, 18917, 21323, 22157, 23873

Offset: 1

Henry Bottomley, Jun 27 2000

nonn

Equal to primes of the form (k^2+15)/8. Also equal to primes p such that 8*p-15 is a square. - Chai Wah Wu, Jul 14 2014

Primes of A152948. - Klaus Purath, Jan 03 2021

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000040, A000217, A022856, A152948.

Select[Table[(n^2-n+4)/2,{n,3000}],PrimeQ] (* Vincenzo Librandi, Jul 14 2012 *)
Select[Accumulate[Range[0,300]]+2,PrimeQ] (* Harvey P. Dale, Feb 05 2019 *)

import sympy
[n*(n+1)/2+2 for n in range(10**6) if sympy.ntheory.primetest.isprime(n*(n+1)/2+2)] # Chai Wah Wu, Jul 14 2014

A133263 Binomial transform of (1, 2, 0, 1, -1, 1, -1, 1, ...).

Original entry on oeis.org

1, 3, 5, 8, 12, 17, 23, 30, 38, 47, 57, 68, 80, 93, 107, 122, 138, 155, 173, 192, 212, 233, 255, 278, 302, 327, 353, 380, 408, 437, 467, 498, 530, 563, 597, 632, 668, 705, 743, 782, 822, 863, 905, 948, 992, 1037, 1083, 1130, 1178, 1227, 1277, 1328, 1380, 1433

Offset: 0

Gary W. Adamson, Oct 15 2007

A007318 * [1, 2, 0, 1, -1, 1, -1, 1, ...]. Left column of A134249.

			a(3) = 8 = (1, 3, 3, 1) dot (1, 2 0, 1) = (1 + 6 + 0 + 1).

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

Cf. A000124, A007318, A022856, A134249, A228446, A238531.

1, seq((n^2+n+4)*1/2,n=1..50); # Emeric Deutsch, Nov 12 2007
a:=n->add((Stirling2(j+1,n)), j=0..n): seq(a(n)+1, n=0..50); # Zerinvary Lajos, Apr 12 2008

Join[{1},Table[(n^2+n+4)/2,{n,50}]] (* or *) Join[{1}, LinearRecurrence[ {3,-3,1},{3,5,8},50]] (* Harvey P. Dale, Feb 13 2012 *)

a(n)=n*(n+1)/2+2 \\ Charles R Greathouse IV, Mar 26 2014

From Emeric Deutsch, Nov 12 2007: (Start)
a(n) = (n^2 + n + 4)/2 for n > 0.
G.f.: (1 - x^2 + x^3)/(1-x)^3. (End)
a(n) = A000124(n) + 1, n >= 1. - Zerinvary Lajos, Apr 12 2008
a(0)=1, a(1)=3; for n >= 2, a(n) = a(n-1) + n. - Philippe Lallouet (philip.lallouet(AT)orange.fr), May 27 2008; corrected by Michel Marcus, Nov 03 2018
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=1, a(1)=3, a(2)=5, a(3)=8. - Harvey P. Dale, Feb 13 2012
a(n) = A238531(n+1) if n >= 0. - Michael Somos, Feb 28 2014
For n > 0: A228446(a(n)) = 5. - Reinhard Zumkeller, Mar 12 2014
a(n) = A022856(n+4) for n >= 1. - Georg Fischer, Nov 02 2018
Sum_{n>=0} 1/a(n) = 1/2 + 2*Pi*tanh(sqrt(15)*Pi/2)/sqrt(25). - Amiram Eldar, Jun 02 2025

More terms from Emeric Deutsch, Nov 12 2007

cp's OEIS Frontend

A055998 a(n) = n*(n+5)/2.

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A152948 a(n) = (n^2 - 3*n + 6)/2.

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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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A089071 Number of liberties a big eye of size n gives in the game of Go.

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A229445 T(n,k)=Number of nXk 0..2 arrays with horizontal differences mod 3 never 1, vertical differences mod 3 never -1, and rows and columns lexicographically nondecreasing.

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A047080 Triangular array T read by rows: T(h,k)=number of paths from (0,0) to (k,h-k) using step-vectors (0,1), (1,0), (1,1) with no right angles between pairs of consecutive steps.

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A229428 T(n,k) = Number of n X k 0..2 arrays with horizontal differences mod 3 never 1, vertical differences mod 3 never -1, rows lexicographically nondecreasing, and columns lexicographically nonincreasing.