A152947 -id:A152947 - cp's OEIS frontend

A080855 a(n) = (9n^2 - 3n + 2)/2.

1, 4, 16, 37, 67, 106, 154, 211, 277, 352, 436, 529, 631, 742, 862, 991, 1129, 1276, 1432, 1597, 1771, 1954, 2146, 2347, 2557, 2776, 3004, 3241, 3487, 3742, 4006, 4279, 4561, 4852, 5152, 5461, 5779, 6106, 6442, 6787, 7141, 7504, 7876, 8257, 8647, 9046

Offset: 0

Paul Barry, Feb 23 2003

The old definition of this sequence was "Generalized polygonal numbers".
Row T(3,n) of A080853.
Equals binomial transform of [1, 3, 9, 0, 0, 0, ...] - Gary W. Adamson, Apr 30 2008
a(n) is also the least weight of self-conjugate partitions having n different parts such that each part is congruent to 2 modulo 3. The first such self-conjugate partitions, corresponding to a(n)=1,2,3,4, are 2+2, 5+5+2+2+2, 8+8+5+5+5+2+2+2, 11+11+8+8+8+5+5+5+2+2+2. - Augustine O. Munagi, Dec 18 2008
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=3, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n >= 3, a(n-1) = -coeff(charpoly(A,x), x^(n-2)). - Milan Janjic, Jan 27 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Milan Janjić, Hessenberg Matrices and Integer Sequences, J. Int. Seq. 13 (2010) # 10.7.8.
A. O. Munagi, Pairing conjugate partitions by residue classes, Discrete Math., 308 (2008), 2492-2501.
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
Leo Tavares, Illustration: Hexagonal Tri-Rays
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A027468, A038764.
Cf. A283394 (see Crossrefs section).
Cf. A003215, A000217, A227776.

List([0..50],n->(9*n^2-3*n+2)/2); # Muniru A Asiru, Nov 02 2018

[(9*n^2 - 3*n +2)/2: n in [0..50]]; // G. C. Greubel, Nov 02 2018

seq((9*n^2-3*n+2)/2,n=0..50); # Muniru A Asiru, Nov 02 2018

s = 1; lst = {s}; Do[s += n + 2; AppendTo[lst, s], {n, 1, 500, 9}]; lst (* Zerinvary Lajos, Jul 11 2009 *)
Table[(9n^2-3n+2)/2,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1}, {1,4,16}, 50] (* Harvey P. Dale, Jul 24 2013 *)

a(n)=binomial(3*n,2)+1 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: (1 + x + 7*x^2)/(1 - x)^3.
a(n) = 9*n + a(n-1) - 6 with n > 0, a(0)=1. - Vincenzo Librandi, Aug 08 2010
a(n) = n*A005448(n+1) - (n-1)*A005448(n), with A005448(0)=1. - Bruno Berselli, Jan 15 2013
a(0)=1, a(1)=4, a(2)=16; for n > 2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Jul 24 2013
a(n) = A152947(3*n+1). - Franck Maminirina Ramaharo, Jan 10 2018
E.g.f.: (2 + 6*x + 9*x^2)*exp(x)/2. - G. C. Greubel, Nov 02 2018
From Leo Tavares, Feb 20 2022: (Start)
a(n) = A003215(n-1) + 3*A000217(n). See Hexagonal Tri-Rays illustration in links.
a(n) = A227776(n) - 3*A000217(n). (End)

Definition replaced with the closed form by Bruno Berselli, Jan 15 2013

A112465 Riordan array (1/(1+x), x/(1-x)).

Original entry on oeis.org

1, -1, 1, 1, 0, 1, -1, 1, 1, 1, 1, 0, 2, 2, 1, -1, 1, 2, 4, 3, 1, 1, 0, 3, 6, 7, 4, 1, -1, 1, 3, 9, 13, 11, 5, 1, 1, 0, 4, 12, 22, 24, 16, 6, 1, -1, 1, 4, 16, 34, 46, 40, 22, 7, 1, 1, 0, 5, 20, 50, 80, 86, 62, 29, 8, 1, -1, 1, 5, 25, 70, 130, 166, 148, 91, 37, 9, 1, 1, 0, 6, 30, 95, 200, 296, 314, 239, 128, 46, 10, 1

Offset: 0

Paul Barry, Sep 06 2005

Inverse is A112466. Note that C(n,k) = Sum_{j = 0..n-k} C(j+k-1, j).

			Triangle starts
   1;
  -1, 1;
   1, 0, 1;
  -1, 1, 1,  1;
   1, 0, 2,  2,  1;
  -1, 1, 2,  4,  3,  1;
   1, 0, 3,  6,  7,  4,  1;
  -1, 1, 3,  9, 13, 11,  5, 1;
   1, 0, 4, 12, 22, 24, 16, 6, 1;
Production matrix begins
  -1, 1;
   0, 1, 1;
   0, 0, 1, 1;
   0, 0, 0, 1, 1;
   0, 0, 0, 0, 1, 1;
   0, 0, 0, 0, 0, 1, 1;
   0, 0, 0, 0, 0, 0, 1, 1;
   0, 0, 0, 0, 0, 0, 0, 1, 1; - _Paul Barry_, Apr 08 2011

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
Roland Bacher, Chebyshev polynomials, quadratic surds and a variation of Pascal's triangle, arXiv:1509.09054 [math.CO], 2015.
E. Deutsch, L. Ferrari and S. Rinaldi, Production Matrices, Advances in Mathematics, 34 (2005) pp. 101-122.
Index entries for triangles and arrays related to Pascal's triangle

Cf. A059260, A108561, A112466, A112468.
Columns: A033999(n) (k=0), A000035(n) (k=1), A004526(n) (k=2), A002620(n-1) (k=3), A002623(n-4) (k=4), A001752(n-5) (k=5), A001753(n-6) (k=6), A001769(n-7) (k=7), A001779(n-8) (k=8), A001780(n-9) (k=9), A001781(n-10) (k=10), A001786(n-11) (k=11), A001808(n-12) (k=12).
Diagonals: A000012(n) (k=n), A023443(n) (k=n-1), A152947(n-1) (k=n-2), A283551(n-3) (k=n-3).
Main diagonal: A072547.
Sums: A078008 (row), A078024 (diagonal), A092220 (signed diagonal), A280560 (signed row).

a112465 n k = a112465_tabl !! n !! k
a112465_row n = a112465_tabl !! n
a112465_tabl = iterate f [1] where
   f xs'@(x:xs) = zipWith (+) ([-x] ++ xs ++ [0]) ([0] ++ xs')
-- Reinhard Zumkeller, Jan 03 2014

A112465:= func< n,k | (-1)^(n+k)*(&+[(-1)^j*Binomial(j+k-1,j): j in [0..n-k]]) >;
[A112465(n,k): k in [0..n], n in [0..13]]; // G. C. Greubel, Apr 18 2025

T[n_, k_]:= Sum[Binomial[j+k-1, j]*(-1)^(n-k-j), {j, 0, n-k}];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* Jean-François Alcover, Jul 23 2018 *)

def A112465(n,k): return (-1)^(n+k)*sum((-1)^j*binomial(j+k-1,j) for j in range(n-k+1))
print(flatten([[A112465(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Apr 18 2025

Number triangle T(n, k) = Sum_{j=0..n-k} (-1)^(n-k-j)*C(j+k-1, j).
T(2*n, n) = A072547(n) (main diagonal). - Paul Barry, Apr 08 2011
From Reinhard Zumkeller, Jan 03 2014: (Start)
T(n, k) = T(n-1, k-1) + T(n-1, k), 0 < k < n, with T(n, 0) = (-1)^n and T(n, n) = 1.
T(n, k) = A108561(n, n-k). (End)
T(n, k) = T(n-1, k-1) + T(n-2, k) + T(n-2, k-1), T(0, 0) = 1, T(1, 0) = -1, T(1, 1) = 1, T(n, k) = 0 if k < 0 or if k > n. - Philippe Deléham, Jan 11 2014
exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(-1 + x + x^2/2! + x^3/3!) = -1 + 2*x^2/2! + 6*x^3/3! + 13*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). - Peter Bala, Dec 21 2014

A261070 Irregular triangle read by rows: T(n,k) is the number of arrangements of n circles with 2k intersections (using the same rules as A250001).

Original entry on oeis.org

1, 1, 2, 1, 4, 4, 2, 4, 9, 15, 15, 31, 24, 35, 44, 20, 50

Offset: 0

Benoit Jubin, Aug 08 2015

Length of n-th row: 1 + (n-1)n/2 (for a configuration for T(n,(n-1)n/2), consider n circles of radius 1 and centers at (k/n,0) for 1<=k<=n).

The generating function down the column k=1 is 1+z^2 *C^2(z) *[C^2(z)+C(z^2)]/ (2*[1-z*C(z)]) = 1+ z^2 +4*z^3 +15*z^4+ 50*z^5+...where C(z) = 1+z+2*z^2+4*z^3+... is the g.f. of A000081 divided by z; eq. (78) in arXiv:1603.00077. - R. J. Mathar, Mar 05 2016

			n\k 0  1  2  3  4  5  6
0   1
1   1
2   2  1
3   4  4  2  4
4   9 15 15 31 24 35 44
5  20 50  .  .  .  .  .  .  .  .  .

R. J. Mathar, Topologically Distinct Sets of Non-intersecting Circles in the Plane, arXiv:1603.00077 [math.CO], 2016.

Row sums give A250001.

Cf. A000081, A152947, A249752, A252158, A280786 (column k=1)

A250001(n) = Sum_{k>=0} T(n,k).

A000081(n+1) = T(n,0).

T(4,2)..T(5,0) (6 terms) from Travis Vasquez, Nov 28 2024

A300401 Array T(n,k) = n(binomial(k, 2) + 1) + k(binomial(n, 2) + 1) read by antidiagonals.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 3, 4, 4, 3, 4, 7, 8, 7, 4, 5, 11, 14, 14, 11, 5, 6, 16, 22, 24, 22, 16, 6, 7, 22, 32, 37, 37, 32, 22, 7, 8, 29, 44, 53, 56, 53, 44, 29, 8, 9, 37, 58, 72, 79, 79, 72, 58, 37, 9, 10, 46, 74, 94, 106, 110, 106, 94, 74, 46, 10, 11, 56, 92, 119

Offset: 0

Franck Maminirina Ramaharo, Mar 05 2018

Antidiagonal sums are given by 2*A055795.
Rows/columns n are binomial transform of {n, A152947(n+1), n, 0, 0, 0, ...}.
Some primes in the array are
n = 1: {2, 7, 11, 29, 37, 67, 79, 137, 191, 211, 277, 379, ...} = A055469, primes of the form k*(k + 1)/2 + 1;
n = 3: {3, 7, 37, 53, 479, 653, 1249, 1619, 2503, 3727, 4349, 5737, 7109, 8179, 9803, 11839, 12107, ...};
n = 4: {11, 37, 79, 137, 211, 821, 991, 1597, 1831, 2081, 2347, ...} = A188382, primes of the form 8*(2*k - 1)^2 + 2*(2*k - 1) + 1.

			The array T(n,k) begins
0     1    2    3    4     5     6     7     8     9    10    11  ...
1     2    4    7   11    16    22    29    37    46    56    67  ...
2     4    8   14   22    32    44    58    74    92   112   134  ...
3     7   14   24   37    53    72    94   119   147   178   212  ...
4    11   22   37   56    79   106   137   172   211   254   301  ...
5    16   32   53   79   110   146   187   233   284   340   401  ...
6    22   44   72  106   146   192   244   302   366   436   512  ...
7    29   58   94  137   187   244   308   379   457   542   634  ...
8    37   74  119  172   233   302   379   464   557   658   767  ...
9    46   92  147  211   284   366   457   557   666   784   911  ...
10   56  112  178  254   340   436   542   658   784   920  1066  ...
11   67  134  212  301   401   512   634   767   911  1066  1232  ...
12   79  158  249  352   467   594   733   884  1047  1222  1409  ...
13   92  184  289  407   538   682   839  1009  1192  1388  1597  ...
14  106  212  332  466   614   776   952  1142  1346  1564  1796  ...
15  121  242  378  529   695   876  1072  1283  1509  1750  2006  ...
16  137  274  427  596   781   982  1199  1432  1681  1946  2227  ...
17  154  308  479  667   872  1094  1333  1589  1862  2152  2459  ...
18  172  344  534  742   968  1212  1474  1754  2052  2368  2702  ...
19  191  382  592  821  1069  1336  1622  1927  2251  2594  2956  ...
20  211  422  653  904  1175  1466  1777  2108  2459  2830  3221  ...
...
The inverse binomial transforms of the columns are
0     1    2    3    4     5     6     7     8     9    10    11  ...  A001477
1     1    2    4    7    11    22    29    37    45    56    67  ...  A152947
0     1    2    3    4     5     6     7     8     9    10    11  ...  A001477
0     0    0    0    0     0     0     0     0     0     0     0  ...
0     0    0    0    0     0     0     0     0     0     0     0  ...
0     0    0    0    0     0     0     0     0     0     0     0  ...
...

Miklós Bóna, Introduction to Enumerative Combinatorics, McGraw-Hill, 2007.
L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Reidel Publishing Company, 1974.
R. P. Stanley, Enumerative Combinatorics, second edition, Cambridge University Press, 2011.

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened).
Cheyne Homberger, Patterns in Permutations and Involutions: A Structural and Enumerative Approach, arXiv preprint 1410.2657 [math.CO], 2014.
Franck Ramaharo, A generating polynomial for the two-bridge knot with Conway's notation C(n,r), arXiv:1902.08989 [math.CO], 2019.

Cf. A000124, A001477, A006000, A008815, A014206, A051601, A055469, A077028, A081436, A084849, A131074, A134394, A139600, A141387, A179000, A188377, A188382, A273465.

T := (n, k) -> n*(binomial(k, 2) + 1) + k*(binomial(n, 2) + 1);
for n from 0 to 20 do seq(T(n, k), k = 0 .. 20) od;

T[n_, k_] := n (Binomial[k, 2] + 1) + k (Binomial[n, 2] + 1);
Table[T[n - k, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 07 2018 *)

T(n, k) := n*(binomial(k, 2) + 1) + k*(binomial(n, 2) + 1)$
for n:0 thru 20 do
  print(makelist(T(n, k), k, 0, 20));

T(n, k) = n*(binomial(k,2) + 1) + k*(binomial(n,2) + 1);
tabl(nn) = for (n=0, nn, for (k=0, nn, print1(T(n, k), ", ")); print); \\ Michel Marcus, Mar 12 2018

T(n,k) = T(k,n) = n*A152947(k+1) + k*A152947(n+1).
T(n,0) = A001477(n).
T(n,1) = A000124(n).
T(n,2) = A014206(n).
T(n,3) = A273465(3*n+2).
T(n,4) = A084849(n+1).
T(n,n) = A179000(n-1,n), n >= 1.
T(2*n,2*n) = 8*A081436(n-1), n >= 1.
T(2*n+1,2*n+1) = 2*A006000(2*n+1).
T(n,n+1) = A188377(n+3).
T(n,n+2) = A188377(n+2), n >= 1.
Sum_{k=0..n} T(k,n-k) = 2*(binomial(n, 4) + binomial(n, 2)).
G.f.: -((2*x*y - y - x)*(2*x*y - y - x + 1))/(((x - 1)*(y - 1))^3).
E.g.f.: (1/2)*(x + y)*(x*y + 2)*exp(x + y).

A350684 Number T(n,k) of partitions of [n] such that the sum of elements i contained in block i equals k when blocks are ordered with decreasing largest elements; triangle T(n,k), n>=0, 0<=k<=max(0,A008805(n-1)), read by rows.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 1, 2, 1, 6, 3, 4, 2, 16, 7, 8, 14, 3, 3, 1, 73, 25, 26, 51, 12, 12, 4, 298, 91, 92, 164, 116, 56, 30, 21, 4, 4, 1, 1453, 390, 391, 601, 676, 256, 163, 147, 28, 28, 7, 7366, 1797, 1798, 2484, 3228, 1927, 897, 876, 307, 307, 87, 31, 31, 5, 5, 1

Offset: 0

Alois P. Heinz, Jan 11 2022

			T(4,0) = 6: 432|1, 42|31, 42|3|1, 4|31|2, 4|3|21, 4|3|2|1.
T(4,1) = 3: 432(1), 42(1)|3, 4(1)|3|2.
T(4,2) = 4: 43|(2)1, 43|(2)|1, 4|3(2)1, 4|3(2)|1,
T(4,3) = 2: 43(1)|(2), 4(1)|3(2).
Triangle T(n,k) begins:
     1;
     0,   1;
     1,   1;
     1,   1,   2,   1;
     6,   3,   4,   2;
    16,   7,   8,  14,   3,   3,   1;
    73,  25,  26,  51,  12,  12,   4;
   298,  91,  92, 164, 116,  56,  30,  21,  4,  4, 1;
  1453, 390, 391, 601, 676, 256, 163, 147, 28, 28, 7;
  ...

Alois P. Heinz, Rows n = 0..100, flattened
Wikipedia, Partition of a set

Columns k=0-1 give: A350649, A350650.
Row sums give A000110.
Cf. A000217, A008805, A152947, A350647, A350683.

b:= proc(n, m) option remember; expand(`if`(n=0, 1, add(
     `if`(n=j, x^j, 1)*b(n-1, max(m, j)), j=1..m+1)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
seq(T(n), n=0..10);

b[n_, m_] := b[n, m] = Expand[If[n == 0, 1, Sum[
     If[n == j, x^j, 1]*b[n - 1, Max[m, j]], {j, 1, m + 1}]]];
T[n_] := CoefficientList[b[n, 0], x];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Mar 18 2022, after Alois P. Heinz *)

Sum_{k=1..max(0,A008805(n-1))} k * T(n,k) = A350683(n).
T(2n,A000217(n)) = A152947(n+1).
T(2n-1,A000217(n)) = 1 for n>=1.
T(n,2) - T(n,1) = 1 for n>=3.

A076263 Triangle read by rows: T(n,k) = number of nonisomorphic connected graphs with n vertices and k edges (n >= 1, n-1 <= k <= n(n-1)/2).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 1, 3, 5, 5, 4, 2, 1, 1, 6, 13, 19, 22, 20, 14, 9, 5, 2, 1, 1, 11, 33, 67, 107, 132, 138, 126, 95, 64, 40, 21, 10, 5, 2, 1, 1, 23, 89, 236, 486, 814, 1169, 1454, 1579, 1515, 1290, 970, 658, 400, 220, 114, 56, 24, 11, 5, 2, 1, 1, 47, 240, 797, 2075, 4495

Offset: 1

Arne Ring (arne.ring(AT)epost.de), Oct 03 2002

The index of the T(n,k) in the sequence is ((n-2)^3 - n + 6*k + 8)/6.

T(n,k)=1 for k = n*(n-1)/2-1 and k = n*(n-1)/2 (therefore {1,1} separates sublists for given numbers of vertices (n > 2)).

			There are 2 connected graphs with 4 vertices and 3 edges up to isomorphy (first graph: ((1,2),(2,3),(3,4)); second graph: ((1,2),(1,3),(1,4))). Index within the sequence is ((4-2)^3 - 4 + 6*3 + 8)/6 = 5.
Triangle begins:
   1;
   1;
   1,  1;
   2,  2,  1,   1;
   3,  5,  5,   4,   2,   1,   1;
   6, 13, 19,  22,  20,  14,   9,  5,  2,  1,  1;
  11, 33, 67, 107, 132, 138, 126, 95, 64, 40, 21, 10, 5, 2, 1, 1;

T. D. Noe, Rows 1 to 16 of triangle, flattened (from Gordon Royle's website)
Keith M. Briggs, Combinatorial Graph Theory.
Sriram V. Pemmaraju, The Combinatorica Project
Marko R. Riedel, Number of distinct connected digraphs, Math StackExchange.
Marko Riedel, Maple code for sequences A008406 and A076263 including cycle indices.
Eric Weisstein's World of Mathematics, Connected Graph.

Row lengths (excluding first row): A000124. Number of connected graphs for given number of vertices: A001349. Number of connected graphs for given number of edges: A002905.
Number of entries in the n-th row is A152947. Row sums give A001349.
Starting each row from k=0 gives A054924, which is the main entry for this triangle.

NumberOfConnectedGraphs[vertices_, edges_] := Plus @@ ConnectedQ /@ ListGraphs[vertices, edges] /. {True->1, False ->0}
(* first do *) Needs["DiscreteMath`Combinatorica`"] (* then *) Table[Plus @@ ConnectedQ /@ ListGraphs[Vert, i] /. {True -> 1, False -> 0}, {Vert, 8}, {i, Vert - 1, Vert*(Vert - 1)/2}]

Corrected by Keith Briggs and Robert G. Wilson v, May 01 2005
Rows 5, 6 & 7 from Robert G. Wilson v, Jun 21 2005
More terms from Keith Briggs, Jun 28 2005
Name corrected by Andrey Zabolotskiy, Nov 20 2017

A121635 Number of deco polyominoes of height n, having no 2-cell columns starting at level 0. A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column.

Original entry on oeis.org

1, 1, 2, 8, 42, 264, 1920, 15840, 146160, 1491840, 16692480, 203212800, 2674425600, 37841126400, 572885913600, 9240898867200, 158228598528000, 2866422214656000, 54775863926784000, 1101208277385216000, 23234214178086912000, 513342323725271040000

Offset: 1

Emeric Deutsch, Aug 13 2006

			a(2)=1 because the deco polyominoes of height 2 are the horizontal and vertical dominoes and the horizontal one has no 2-cell column starting at level 0.

Alois P. Heinz, Table of n, a(n) for n = 1..450 (n=2..101 from Muniru A Asiru)
E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.
Mark Dukes, Chris D White, Web Matrices: Structural Properties and Generating Combinatorial Identities, arXiv:1603.01589 [math.CO], 2016.
Mark Dukes, Chris D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, Electronic Journal Of Combinatorics, 23(1) (2016), #P1.45.

Cf. A121634, A001710.

a:= n-> `if`(n=1, 1, (n^2-3*n+4)*(n-2)!/2): seq(a(n), n=1..23);

a(n) = A121634(n,0).
a(1)=1, a(n) = (n-2)!(n^2-3*n+4)/2 = A000142(n-2)*A152947(n) for n>=2.
a(1)=1, a(2)=1, a(n) = (n-2)*[(n-2)! + a(n-1)] for n>=3.
D-finite with recurrence a(n) +(-n-2)*a(n-1) +2*(n-1)*a(n-2) +2*(-n+4)*a(n-3)=0. - R. J. Mathar, Jul 26 2022

Missing a(1) inserted by Alois P. Heinz, Nov 25 2018

A152949 a(n) = 3 + binomial(n-1,2).

Original entry on oeis.org

3, 3, 4, 6, 9, 13, 18, 24, 31, 39, 48, 58, 69, 81, 94, 108, 123, 139, 156, 174, 193, 213, 234, 256, 279, 303, 328, 354, 381, 409, 438, 468, 499, 531, 564, 598, 633, 669, 706, 744, 783, 823, 864, 906, 949, 993, 1038, 1084, 1131, 1179, 1228, 1278, 1329, 1381

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 15 2008

a(1)=3; then add 0 to the first number, then 1,2,3,4,... and so on.

Muniru A Asiru, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A016028, A152947, A000124, A152948, A152950.

List([1..55],n->3+Binomial(n-1,2)); # Muniru A Asiru, Oct 28 2018

seq(coeff(series(x*(4*x^2-6*x+3)/(1-x)^3,x,n+1), x, n), n = 1 .. 55); # Muniru A Asiru, Oct 28 2018

s=3;lst={3};Do[s+=n;AppendTo[lst,s],{n,0,5!}];lst
Table[Binomial[n-1,2],{n,60}]+3 (* Harvey P. Dale, Feb 27 2013 *)

Vec( x*(3-6*x+4*x^2)/(1-x)^3 + O(x^66) ) \\ Joerg Arndt, Jul 24 2013

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

a(n) = a(n-1) + n - 2 (with a(1)=3). - Vincenzo Librandi, Nov 27 2010
G.f.: x*(3-6*x+4*x^2)/(1-x)^3. - Nikita Gogin, Jul 24 2013
a(n) = A016028(n+1) for n >= 2. - Georg Fischer, Oct 28 2018
Sum_{n>=1} 1/a(n) = 1/3 + 2*Pi*tanh(sqrt(23)*Pi/2)/sqrt(23). - Amiram Eldar, Dec 13 2022

A245796 T(n,k) is the number of labeled graphs of n vertices and k edges that have endpoints, where an endpoint is a vertex with degree 1.

Original entry on oeis.org

0, 1, 3, 3, 6, 15, 16, 12, 10, 45, 110, 195, 210, 120, 20, 15, 105, 435, 1320, 2841, 4410, 4845, 3360, 1350, 300, 30, 21, 210, 1295, 5880, 19887, 51954, 106785, 171360, 208565, 186375, 120855, 56805, 19110, 4410, 630, 42

Offset: 1

Chai Wah Wu, Aug 01 2014

The length of the rows are 1,1,2,4,7,11,16,22,...: (1+(n-1)*(n-2)/2) = A152947(n).
T(n,k) = 0 if k > (n-1)*(n-2)/2 + 1.
Let j = (n-1)*(n-2)/2.  For i >=0, n >= 4+i, T(n,j-i+1) = n*(n-1)*binomial(j,i).
For k <= 3, T(n,k) is equal to the number of labeled bipartite graphs with n vertices and k edges.  In particular, T(n,1) = A000217(n-1), T(n,2) = A050534(n) and T(n,3) = A053526(n).

			Triangle starts:
..0
..1
..3......3
..6.....15.....16.....12
.10.....45....110....195....210....120.....20
.15....105....435...1320...2841...4410...4845...3360...1350....300.....30
...

Chai Wah Wu, Graphs whose normalized Laplacian matrices are separable as density matrices in quantum mechanics, arXiv:1407.5663 [quant-ph], 2014.

Sum of n-th row is A245797(n).

Cf. A000217, A050534, A053526, A152947.

A279610 a(n) = concatenate n consecutive integers, starting with the last number of the previous batch.

Original entry on oeis.org

1, 12, 234, 4567, 7891011, 111213141516, 16171819202122, 2223242526272829, 293031323334353637, 37383940414243444546, 4647484950515253545556, 565758596061626364656667, 67686970717273747576777879, 7980818283848586878889909192

Offset: 1

José de Jesús Camacho Medina, Dec 09 2016, and Paolo Iachia, Dec 15 2016

A variant of A053067. The first number of the concatenation a(n) is A152947(n) = (n-2)*(n-1)/2+1 and the last is (n-1)*n/2+1.
The fourth term, 4567, is a prime. When is the next prime, if there is another? - N. J. A. Sloane, Dec 16 2016
a(n) is the concatenation of the terms of the n-th row of A122797 when seen as a triangle. - Michel Marcus, Dec 17 2016

			a(4) is the concatenation of 4 numbers beginning with the last number (4) that was used to build a(3), so a(4) = 4 5 6 7 = 4567. Then a(5) is the concatenation of 5 numbers beginning with the last number of a(4), which is 7, so a(5) = 7 8 9 10 11 = 7891011. And so on.
For n = 3, n^2/2 - n/2 + 1 = 4; a(3) = 4 + 3*10^1 + 2*10^(1+1) = 234.

Chai Wah Wu, Table of n, a(n) for n = 1..200

Cf. A053067, A122797, A152947.

A subsequence of A035333. For primes in latter, see A052087.

Table[FromDigits[Flatten[IntegerDigits /@ Range[(n(n - 1))/2 + 1, (n(n + 1))/2 + 1 ]]], {n, 0, 20}]

from _future_ import division
def A279610(n):
    return int(''.join(str(d) for d in range((n-1)*(n-2)//2+1,n*(n-1)//2+2))) # Chai Wah Wu, Dec 17 2016

a(n) = n^2/2 - n/2 + 1 + Sum{k=1..n-1} ((n^2/2 - n/2 + 1 - k)*10^Sum{j=0..k-1} (floor(1+log_10(n^2/2 - n/2 + 1 - j)))).

cp's OEIS Frontend

A080855 a(n) = (9*n^2 - 3*n + 2)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Formula

Extensions

A112465 Riordan array (1/(1+x), x/(1-x)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

SageMath

Formula

A261070 Irregular triangle read by rows: T(n,k) is the number of arrangements of n circles with 2k intersections (using the same rules as A250001).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A300401 Array T(n,k) = n*(binomial(k, 2) + 1) + k*(binomial(n, 2) + 1) read by antidiagonals.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Maxima

PARI

Formula

A350684 Number T(n,k) of partitions of [n] such that the sum of elements i contained in block i equals k when blocks are ordered with decreasing largest elements; triangle T(n,k), n>=0, 0<=k<=max(0,A008805(n-1)), read by rows.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A076263 Triangle read by rows: T(n,k) = number of nonisomorphic connected graphs with n vertices and k edges (n >= 1, n-1 <= k <= n(n-1)/2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A121635 Number of deco polyominoes of height n, having no 2-cell columns starting at level 0. A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column.

Views

Author

A080855 a(n) = (9n^2 - 3n + 2)/2.

A300401 Array T(n,k) = n(binomial(k, 2) + 1) + k(binomial(n, 2) + 1) read by antidiagonals.