A155212 -id:A155212 - 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)

A051938 Truncated triangular numbers: a(n) = n*(n+1)/2 - 18.

Original entry on oeis.org

3, 10, 18, 27, 37, 48, 60, 73, 87, 102, 118, 135, 153, 172, 192, 213, 235, 258, 282, 307, 333, 360, 388, 417, 447, 478, 510, 543, 577, 612, 648, 685, 723, 762, 802, 843, 885, 928, 972, 1017, 1063, 1110, 1158, 1207, 1257, 1308, 1360, 1413, 1467, 1522, 1578

Offset: 6

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 21 1999

If a 3-set Y and a 3-set Z, having one element in common, are subsets of an n-set X then a(n+2) is the number of 3-subsets of X intersecting both Y and Z. - Milan Janjic, Oct 03 2007

Colin Barker, Table of n, a(n) for n = 6..1000
Milan Janjic, Two Enumerative Functions.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

a(n) = A000217(n) - 18 for n>5.

Cf. A155212. - Vincenzo Librandi, Jan 22 2009

Drop[Accumulate[Range[60]]-18,5] (* Harvey P. Dale, Dec 08 2017 *)

Vec(x^6*(3*x^2-x-3)/(x-1)^3 + O(x^100)) \\ Colin Barker, Mar 18 2015

a(n) = n + a(n-1) (with a(6)=3). - Vincenzo Librandi, Aug 06 2010
G.f.: x^6*(3*x^2-x-3) / (x-1)^3. - Colin Barker, Mar 18 2015
Sum_{n>=6} 1/a(n) = 4423/6120 + 2*Pi*tan(sqrt(145)*Pi/2)/sqrt(145). - Amiram Eldar, Dec 13 2022

A237519 Triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists the partial sums of the column k of A237273 starting in row k^2.

Original entry on oeis.org

1, 4, 8, 13, 2, 19, 2, 26, 7, 34, 7, 43, 13, 53, 13, 3, 64, 20, 3, 76, 20, 3, 89, 28, 10, 103, 28, 10, 118, 37, 10, 134, 37, 18, 151, 47, 18, 4, 169, 47, 18, 4, 188, 58, 27, 4, 208, 58, 27, 4, 229, 70, 27, 13, 251, 70, 37, 13, 274, 83, 37, 13, 298, 83, 37, 13, 323, 97, 48, 23, 349, 97, 48, 23, 5

Offset: 1

Omar E. Pol, Feb 08 2014

The sum of row n is A024916(n), the sum of all divisors of all positive integers <= n.
Row n has length A000196(n).
Column 1 is A034856.
Column 2 lists the elements of A155212 repeated.
Column 3 lists the elements of A051938 repeated with three copies of every element.
Column k contains k copies of every element.

			Triangle begins:
1;
4;
8;
13,   2;
19,   2;
26,   7;
34,   7;
43,  13;
53,  13,  3;
64,  20,  3;
76,  20,  3;
89,  28, 10;
103, 28, 10;
118, 37, 10;
134, 37, 18;
151, 47, 18,  4;
169, 47, 18,  4;
188, 58, 27,  4;
208, 58, 27,  4;
229, 70, 27, 13;
251, 70, 37, 13;
274, 83, 37, 13;
298, 83, 37, 13;
323, 97, 48, 23;
349, 97, 48, 23,  5;
...

Cf. A000203, A000196, A024916, A034856, A051938, A155212, A236104, A235791, A237273.

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

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

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A051938 Truncated triangular numbers: a(n) = n*(n+1)/2 - 18.

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A237519 Triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists the partial sums of the column k of A237273 starting in row k^2.

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

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Python

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