A000217 -id:A000217 - cp's OEIS frontend

A285088 Numbers n such that the number of partitions of n(n+1)/2 (=A000041(A000217(n))) is prime.

2, 3, 8, 3947, 43968, 61681

Offset: 1

Serge Batalov, Apr 09 2017

Because asymptotically A000041(n*(n+1)/2) ~ exp(Pi*sqrt(2/3*(n*(n+1)/2))) / (4*sqrt(3)*(n*(n+1)/2)), the sum of the prime probabilities ~1/log(A000041(n*(n+1)/2)) is diverging and there are no obvious restrictions on primality; therefore, this sequence may be conjectured to be infinite.

			a(3) = 8 is in the sequence because A000041(8*9/2) = 17977 is a prime.

Chris K. Caldwell, Top twenty prime partition numbers, The Prime Pages.
Eric Weisstein's World of Mathematics, Partition Function P
Eric Weisstein's World of Mathematics, Integer Sequence Primes

Cf. A000041, A046063, A072213, A284594, A285086, A285087.

for(n=1,2000,if(ispseudoprime(numbpart(n*(n+1)/2)),print1(n,", ")))

A346873 Triangle read by rows in which row n lists the row A000217(n) of A237591, n >= 1.

Original entry on oeis.org

1, 2, 1, 4, 1, 1, 6, 2, 1, 1, 8, 3, 2, 1, 1, 11, 4, 3, 1, 1, 1, 15, 5, 3, 2, 1, 1, 1, 19, 6, 4, 2, 2, 1, 1, 1, 23, 8, 5, 2, 2, 2, 1, 1, 1, 28, 10, 5, 3, 3, 2, 1, 1, 1, 1, 34, 11, 6, 4, 3, 2, 2, 1, 1, 1, 1, 40, 13, 7, 5, 3, 2, 2, 2, 1, 1, 1, 1, 46, 16, 8, 5, 4, 2, 3

Offset: 1

Omar E. Pol, Aug 06 2021

The characteristic shape of the symmetric representation of sigma(A000217(n)) consists in that in the main diagonal of the diagram the smallest Dyck path has a valley and the largest Dyck path has a peak, or vice versa, the smallest Dyck path has a peak and the largest Dyck path has valley.
So knowing this characteristic shape we can know if a number is a triangular number (or not) just by looking at the diagram, even ignoring the concept of triangular number.
Therefore we can see a geometric pattern of the distribution of the triangular numbers in the stepped pyramid described in A245092.
T(n,k) is also the length of the k-th line segment of the largest Dyck path of the symmetric representation of sigma(A000217(n)), from the border to the center, hence the sum of the n-th row of triangle is equal to A000217(n).
T(n,k) is also the difference between the total number of partitions of all positive integers <= n-th triangular number into exactly k consecutive parts, and the total number of partitions of all positive integers <= n-th triangular number into exactly k + 1 consecutive parts.

			Triangle begins:
   1;
   2,  1;
   4,  1, 1;
   6,  2, 1, 1;
   8,  3, 2, 1, 1;
  11,  4, 3, 1, 1, 1;
  15,  5, 3, 2, 1, 1, 1;
  19,  6, 4, 2, 2, 1, 1, 1;
  23,  8, 5, 2, 2, 2, 1, 1, 1;
  28, 10, 5, 3, 3, 2, 1, 1, 1, 1;
  34, 11, 6, 4, 3, 2, 2, 1, 1, 1, 1;
  40, 13, 7, 5, 3, 2, 2, 2, 1, 1, 1, 1;
  46, 16, 8, 5, 4, 2, 3, 1, 2, 1, 1, 1, 1;
...
Illustration of initial terms:
Column T gives the triangular numbers (A000217).
Column S gives A074285, the sum of the divisors of the triangular numbers which equals the area (and the number of cells) of the associated diagram.
-------------------------------------------------------------------------
  n    T    S   Diagram
-------------------------------------------------------------------------
                 _   _     _       _         _           _             _
  1    1    1   |_| | |   | |     | |       | |         | |           | |
               1 _ _|_|   | |     | |       | |         | |           | |
  2    3    4   |_ _|  _ _| |     | |       | |         | |           | |
                  2  1|    _|     | |       | |         | |           | |
                 _ _ _|  _|    _ _| |       | |         | |           | |
  3    6   12   |_ _ _ _| 1   |  _ _|       | |         | |           | |
                    4    1 _ _|_|           | |         | |           | |
                          |  _|1       _ _ _|_|         | |           | |
                 _ _ _ _ _| | 1    _ _| |               | |           | |
  4   10   18   |_ _ _ _ _ _|2    |    _|               | |           | |
                      6          _|  _|          _ _ _ _|_|           | |
                                |_ _|1 1        | |                   | |
                                | 2            _| |                   | |
                 _ _ _ _ _ _ _ _|4            |  _|          _ _ _ _ _| |
  3   15   24   |_ _ _ _ _ _ _ _|          _ _|_|           |  _ _ _ _ _|
                        8              _ _|  _|1            | |
                                      |_ _ _|1 1         _ _| |
                                      |  3           _ _|  _ _|
                                      |4            |    _|
                 _ _ _ _ _ _ _ _ _ _ _|            _|  _|
  4   21   32   |_ _ _ _ _ _ _ _ _ _ _|      _ _ _|  _|1 1
                          11                |  _ _ _|2
                                            | |  3
                                            | |
                                            | |5
                 _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
  5   28   56   |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
                              15
.

Row sums give A000217, n >= 1.
Column 1 gives A039823.
For the characteristic shape of sigma(A000040(n)) see A346871.
For the characteristic shape of sigma(A000079(n)) see A346872.
For the visualization of Mersenne numbers A000225 see A346874.
For the characteristic shape of sigma(A000384(n)) see A346875.
For the characteristic shape of sigma(A000396(n)) see A346876.
For the characteristic shape of sigma(A008588(n)) see A224613.
Cf. A000203, A074285, A237591, A237593, A245092, A249351, A262626.

T(n,k) = A237591(A000217(n),k). - Omar E. Pol, Feb 06 2023

Name corrected by Omar E. Pol, Feb 06 2023

A360010 First part of the n-th weakly decreasing triple of positive integers sorted lexicographically. Each n > 0 is repeated A000217(n) times.

Original entry on oeis.org

1, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8

Offset: 1

Gus Wiseman, Feb 11 2023

nonn

			Triples begin: (1,1,1), (2,1,1), (2,2,1), (2,2,2), (3,1,1), (3,2,1), (3,2,2), (3,3,1), (3,3,2), (3,3,3), ...

For pairs instead of triples we have A002024.
Positions of first appearances are A050407(n+2) = A000292(n)+1.
The zero-based version is A056556.
The triples have sums A070770.
The second instead of first part is A194848.
The third instead of first part is A333516.
Concatenating all the triples gives A360240.
Cf. A000217, A003056, A056557, A056558, A069905, A158842, A331195.

nn=9;First/@Select[Tuples[Range[nn],3],GreaterEqual@@#&]

from math import comb
from sympy import integer_nthroot
def A360010(n): return (m:=integer_nthroot(6*n,3)[0])+(n>comb(m+2,3)) # Chai Wah Wu, Nov 04 2024

a(n) = A056556(n) + 1 = A331195(3n) + 1.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/8 + log(2)/4. - Amiram Eldar, Feb 18 2024
a(n) = m+1 if n>binomial(m+2,3) and a(n) = m otherwise where m = floor((6n)^(1/3)). - Chai Wah Wu, Nov 04 2024

A115364 a(n) = A000217(A001511(n)), where A001511 is one more than the 2-adic valuation of n, and A000217(n) is the n-th triangular number, binomial(n+1, 2).

Original entry on oeis.org

1, 3, 1, 6, 1, 3, 1, 10, 1, 3, 1, 6, 1, 3, 1, 15, 1, 3, 1, 6, 1, 3, 1, 10, 1, 3, 1, 6, 1, 3, 1, 21, 1, 3, 1, 6, 1, 3, 1, 10, 1, 3, 1, 6, 1, 3, 1, 15, 1, 3, 1, 6, 1, 3, 1, 10, 1, 3, 1, 6, 1, 3, 1, 28, 1, 3, 1, 6, 1, 3, 1, 10, 1, 3, 1, 6, 1, 3, 1, 15, 1, 3, 1, 6, 1, 3, 1, 10, 1, 3, 1, 6, 1

Offset: 1

Paul Barry, Jan 21 2006

Row sums of A115363. In general, the row sums of ((1,x) - m(x,x^2))^(-2) are obtained by following the ruler function A001511(n) by the solution of the recurrence a(1)=1, a(n) = n*m^(n-1) + a(n-1), n > 1.

The Stephan formula says this is the Dirichlet convolution of A000012 with A104117. - R. J. Mathar, Feb 07 2011

Antti Karttunen, Table of n, a(n) for n = 1..16383

Cf. A000217, A001511, A007814, A104117, A115363.

Cf. also A094290.

Array[PolygonalNumber[IntegerExponent[#, 2] + 1] &, 93] (* Michael De Vlieger, Nov 02 2018 *)

A115364(n) = binomial(valuation(n,2)+2,2); \\ Antti Karttunen, Nov 02 2018

def A115364(n): return (m:=((~n & n-1).bit_length()+1))*(m+1)>>1 # Chai Wah Wu, Jul 02 2022

a(n) = binomial(A007814(n)+2, 2) = binomial(A001511(n)+1, 2).
Dirichlet g.f.: zeta(s)*(2^s/(2^s-1))^2. - Ralf Stephan, Jun 17 2007
Multiplicative with a(2^k) = A000217(k+1), a(p^k) = 1 for odd primes p. - Antti Karttunen, Nov 02 2018
O.g.f.: Sum_{k >= 1} k*x^(2^(k-1))/(1 - x^(2^(k-1))). More generally, if f(n) is an arithmetic function and g(n) := Sum_{k = 1..n} f(k), then Sum_{k >= 1} f(k)*x^(2^(k-1))/(1 - x^(2^(k-1))) = Sum_{n >= 1} g(A001511(n))*x^n. This is the case f(n) = n. - Peter Bala, Mar 26 2019
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 4. - Amiram Eldar, Oct 22 2022
More precise asymptotics: Sum_{k=1..n} a(k) ~ 4*n - log(n)*(log(n) + 2*log(4*Pi))/(4*log(2)^2). - Vaclav Kotesovec, Jun 25 2024

Formula corrected and the name changed by Antti Karttunen, Nov 02 2018

A129307 Intersection of A000217 and A005098.

Original entry on oeis.org

1, 3, 10, 15, 28, 45, 78, 105, 153, 190, 253, 300, 325, 435, 465, 528, 595, 630, 780, 903, 1128, 1275, 1830, 2145, 2415, 2485, 2628, 3160, 3403, 3570, 3655, 3828, 4095, 4753, 4950, 5050, 5253, 5460, 5995, 6105, 6670, 7503, 8515, 9180, 9453, 9730, 10440, 11175

Offset: 1

Zak Seidov, May 26 2007

nonn

Triangular numbers T(m)=m(m+1)/2 indices m of which are in A027861. T(m) such that m^2+(m+1)^2 is prime.

Cf. A000217, A005098, A027861.

select(x-> isprime(4*x+1), [i*(i+1)/2$i=0..400])[];  # Alois P. Heinz, Feb 24 2024

Select[Table[n(n+1)/2, {n, 0, 200}], PrimeQ[4#+1]&] (* Jean-François Alcover, Feb 24 2024 *)

a(n) = A027861(n)*(A027861(n)+1)/2.

a(n) = A000217(A027861(n)).

A130703 a(n) = smallest k such that A000217(n+1) = A000217(n) + (A000217(n) mod k), or 0 if no such k exists.

Original entry on oeis.org

0, 0, 0, 0, 9, 14, 10, 27, 35, 22, 18, 65, 77, 18, 26, 119, 27, 38, 34, 27, 209, 46, 28, 55, 299, 36, 35, 377, 45, 62, 58, 45, 527, 40, 54, 629, 95, 54, 74, 779, 63, 86, 82, 63, 989, 94, 54, 161, 235, 68, 91, 265, 81, 65, 106, 81, 145, 118, 90, 1769, 1829

Offset: 1

Rémi Eismann, Aug 16 2007 - Jan 10 2011

nonn

a(n) is the weight of triangular numbers.

The decomposition of triangular numbers into weight * level + gap is A000217(n) = a(n) * A184219(n) + (n + 1) if a(n) > 0.

			For n = 1 we have A000217(n) = 1, A000217(n+1) = 3; there is no k such that 3 - 1 = 2 = (1 mod k), hence a(1) = 0.
For n = 5 we have A000217(n) = 15, A000217(n+1) = 21; 9 is the smallest k such that 21 - 15 = 6 = (15 mod k), hence a(5) = 9.
For n = 22 we have A000217(n) = 253, A000217(n+1) = 276; 46 is the smallest k such that 276 - 253 = 23 = (253 mod k), hence a(22) = 46.

Remi Eismann, Table of n, a(n) for n=1..9999

Cf. A020639, A117078, A117563, A001223, A118534, A090369, A090368.

A210838 Coordinates (x,y) of the endpoint of a structure (or curve) formed by Q-toothpicks of size = 1..n. The inflection points are the n-th nodes if n is a triangular number A000217.

Original entry on oeis.org

0, 0, 1, 1, 3, 3, 0, 6, -4, 10, 1, 15, 7, 9, 14, 2, 22, 10, 13, 19, 3, 9, -8, -2, -20, 10, -7, 23, 7, 9, -8, -6, -24, -22, -7, -39, 11, -21, -8, -2, -28, -22, -7, -43, 15, -65, -8, -88, -32, -64, -7, -39, 19, -65, -8, -92, -36, -64, -65, -35, -95, -65, -64, -96

Offset: 0

Omar E. Pol, Mar 28 2012

It appears there is an infinite family of this type of curves or structures in which the terms of a sequence of positive integers are represented as inflection points and the gaps between them are essentially represented as nodes of spirals. For example: consider a structure formed by Q-toothpicks of size = Axxxxxa connected by their endpoints in which the inflection points are the exposed endpoints at stage Axxxxxb(n), where both Axxxxxa and Axxxxxb are sequences with positive integers. Also instead of Q-toothpicks we can use semicircumferences or also 3/4 of circumferences. For the definition of Q-toothpicks see A187210.
We start at stage 0 with no Q-toothpicks.
At stage 1 we place a Q-toothpick of size 1 centered at (1,0) with its endpoints at (0,0) and (1,1). Since 1 is a positive triangular number we have that the end of the curve is also an inflection point.
At stage 2 we place a Q-toothpick of size 2 centered at (1,3) with its endpoints at (1,1) and (3,3).
At stage 3 we place a Q-toothpick of size 3 centered at (0,3) with its endpoints at (3,3) and (0,6). Since 3 is a positive triangular number we have that the end of the curve is also an inflection point.
At stage 4 we place a Q-toothpick of size 4 centered at (0,10) with its endpoints at (0,6) and (-4,10).
And so on...

			-------------------------------------
Stage n also              The end as
the size of     Pair      inflection
Q-toothpick   (x    y)      point
-------------------------------------
.    0         0,   0,        -
.    1         1,   1,       Yes
.    2         3,   3,        -
.    3         0,   6,       Yes
.    4        -4,  10,        -
.    5         1,  15,        -
.    6         7,   9,       Yes
.    7        14,   2,        -
.    8        22,  10,        -
.    9        13,  19,        -
.   10         3,   9,       Yes
.   11        -8,  -2,        -
.   12       -20,  10,        -
.   13        -7,  23,        -
.   14         7,   9,        -
.   15        -8,  -6,       Yes

Paolo Xausa, Table of n, a(n) for n = 0..9999
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Paolo Xausa, Animation of terms n = 0..41 (first 21 coordinate pairs), where orange dots are toothpick endpoints (hollow dots are inflection points) and blue dots are toothpick centers
Paolo Xausa, Animation of terms n = 0..507 (first 254 coordinate pairs)
Paolo Xausa, Scatterplot of 10^5 coordinate pairs
Index entries for sequences related to toothpick sequences

Cf. A210841 (the same idea for primes).

Cf. A000217, A005132, A187210, A210606, A211000, A211011.

A210838[nmax_]:=Module[{ep={0, 0}, angle=3/4Pi, turn=Pi/2, infl=0}, Join[{ep}, Table[If[n>1&&IntegerQ[Sqrt[8(n-1)+1]], infl++, If[Mod[infl, 2]==1, turn*=-1]; angle-=turn; infl=0]; ep=AngleVector[ep, {Sqrt[2]n, angle}], {n, nmax}]]];
A210838[100] (* Generates 101 coordinate pairs *) (* Paolo Xausa, Jan 12 2023 *)

A210838(nmax) = my(ep=vector(nmax+1), turn=1, infl=0, ep1, ep2); ep[1]=[0, 0]; if(nmax==0, return(ep)); ep[2]=[1, 1]; for(n=2, nmax, ep1=ep[n-1]; ep2=ep[n]; if(issquare((n-1)<<3+1), infl++; ep[n+1]=[ep2[1]+n*sign(ep2[1]-ep1[1]), ep2[2]+n*sign(ep2[2]-ep1[2])], if(infl%2, turn*=-1); infl=0; ep[n+1]=[ep2[1]-turn*n*sign(ep1[2]-ep2[2]), ep2[2]+turn*n*sign(ep1[1]-ep2[1])])); ep;
A210838(100) \\ Generates 101 coordinate pairs - Paolo Xausa, Jan 12 2023

from numpy import sign
from sympy import integer_nthroot
def A210838(nmax):
    ep, turn, infl = [(0, 0), (1, 1)], 1, 0
    for n in range(2, nmax + 1):
        ep1, ep2 = ep[-2], ep[-1]
        if integer_nthroot(((n - 1) << 3) + 1, 2)[1]: # Continue straight
            infl += 1
            dx = n * sign(ep2[0] - ep1[0])
            dy = n * sign(ep2[1] - ep1[1])
        else: # Turn
            if infl % 2: turn *= -1
            infl = 0
            dx = turn * n * sign(ep2[1] - ep1[1])
            dy = turn * n * sign(ep1[0] - ep2[0])
        ep.append((ep2[0] + dx, ep2[1] + dy))
    return ep[:nmax+1]
print(A210838(100)) # Generates 101 coordinate pairs - Paolo Xausa, Jan 12 2023

a(30)-a(33) corrected and more terms by Paolo Xausa, Jan 12 2023

A212652 a(n) is the least positive integer M such that n = T(M) - T(k), for k an integer, 0 <= k <= M, where T(r) = A000217(r) is the r-th triangular number.

Original entry on oeis.org

1, 2, 2, 4, 3, 3, 4, 8, 4, 4, 6, 5, 7, 5, 5, 16, 9, 6, 10, 6, 6, 7, 12, 9, 7, 8, 7, 7, 15, 8, 16, 32, 8, 10, 8, 8, 19, 11, 9, 10, 21, 9, 22, 9, 9, 13, 24, 17, 10, 12, 11, 10, 27, 10, 10, 11, 12, 16, 30, 11, 31, 17, 11, 64, 11, 11, 34

Offset: 1

L. Edson Jeffery, Feb 14 2013

nonn

n = A000217(a(n)) - A000217(a(n) - A109814(n)).
Conjecture: n appears in row a(n) of A209260.
From Daniel Forgues, Jan 06 2016: (Start)
n = Sum_{i=k+1..M} i = T(M) - T(k) = (M-k)*(M+k+1)/2.
n = 2^m, m >= 0, iff M = n = 2^m and k = n - 1 = 2^m - 1. (Points on line with slope 1.) (Powers of 2 can't be the sum of consecutive numbers.)
n is odd prime iff k = M-2. Thus M = (n+1)/2 when n is odd prime. (Points on line with slope 1/2.) (Odd primes can't be the sum of more than 2 consecutive numbers.) (End)
If n = 2^m*p where p is an odd prime, then a(n) = 2^m + (p-1)/2. - Robert Israel, Jan 14 2016
This also expresses the following geometry: along a circle having (n) points on its circumference, a(n) expresses the minimum number of hops from a start point, in a given direction (CW or CCW), when each hop is increased by one, before returning to a visited point. For example, on a clock (n=12), starting at 12 (same as zero), the hops would lead to the points 1, 3, 6, 10 and then 3, which was already visited: 5 hops altogether, so a(12) = 5. - Joseph Rozhenko, Dec 25 2023
Conjecture: a(n) is the smallest of the largest parts of the partitions of n into consecutive parts. - Omar E. Pol, Jan 07 2025

			For n = 63, we have D(63) = {1,3,7,9,21,63}, B_63 = {11,12,13,22,32,63} and a(63) = min(11,12,13,22,32,63) = 11. Since A109814(63) = 9, T(11) - T(11-9) = T(11) - T(2) = 66 - 3 = 63.

David W. Wilson, Table of n, a(n) for n = 1..10000
Max Alekseyev, is this sequence interesting?, Sequence Fans Mailing List, Mar 31 2008.

Cf. A000217, A109814, A118235, A138796, A141419, A209260, A218621.

f:= n ->  min(map(t -> n/t + (t-1)/2,
numtheory:-divisors(n/2^padic:-ordp(n,2)))):
map(f, [$1..100]); # Robert Israel, Jan 14 2016

Table[Min[n/# + (# - 1)/2 &@ Select[Divisors@ n, OddQ]], {n, 67}] (* Michael De Vlieger, Dec 11 2015 *)

{ A212652(n) = my(m); m=2*n+1; fordiv(n/2^valuation(n,2), d, m=min(m,d+(2*n)\d)); (m-1)\2; } \\ Max Alekseyev, Mar 31 2008

a(n) = Min_{odd d|n} (n/d + (d-1)/2).
a(n) = A218621(n) + (n/A218621(n) - 1)/2.
a(n) = A109814(n) + A118235(n) - 1.

Reference to Max Alekseyev's 2008 proposal of this sequence added by N. J. A. Sloane, Nov 01 2014

A257293 Numbers n such that T(n) + T(n+1) + ... + T(n+12) is a square, where T = A000217 (triangular numbers).

Original entry on oeis.org

3, 29, 75, 432, 998, 3624, 8310, 44717, 102443, 370269, 848195, 4561352, 10448838, 37764464, 86508230, 465213837, 1065679683, 3851605709, 8822991915, 47447250672, 108688879478, 392826018504, 899858667750, 4839154355357, 11085200027723, 40064402282349

Offset: 1

M. F. Hasler, May 04 2015

It is well known that T(n)+T(n+1) is always a square. T(n)+T(n+1)+T(n+2) is a square for n in A165517. T(n)+T(n+1)+T(n+2)+T(n+3) is a square for n in A202391. There is no sequence of 5, 6, 7, 8, 9 or 10 consecutive T(i)'s which sum to a square, cf. A176541. The next possible length is 11, see A116476. Then comes this sequence, corresponding to length 13.

Positive integers y in the solutions to 2*x^2-13*y^2-169*y-728 = 0. - Colin Barker, May 04 2015

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,102,-102,0,0,-1,1).

Cf. A176541, A176542, A000217, A000292, A001110, A077415.

Cf. A116476 (length 11).

I:=[3,29,75,432,998,3624,8310,44717,102443]; [n le 9 select I[n] else Self(n-1)+102*Self(n-4)-102*Self(n-5)-Self(n-8)+Self(n-9): n in [1..40]]; // Vincenzo Librandi, May 05 2015

Select[Range[10^5],IntegerQ[Sqrt[(#^2+13*#+56)*13/2]]&] (* Ivan N. Ianakiev, May 04 2015 *)
LinearRecurrence[{1, 0, 0, 102, -102, 0, 0, -1, 1}, {3, 29, 75, 432, 998, 3624, 8310, 44717, 102443}, 50] (* Vincenzo Librandi, May 05 2015 *)

for(n=0,10^8,issquare(binomial(n+14,3)-binomial(n+1,3))&&print1(n","))

Vec(x*(3*x^8+7*x^7+6*x^6+26*x^5-260*x^4-357*x^3-46*x^2-26*x-3) / ((x-1)*(x^4-10*x^2-1)*(x^4+10*x^2-1)) + O(x^100)) \\ Colin Barker, May 04 2015

G.f.: x*(3*x^8+7*x^7+6*x^6+26*x^5-260*x^4-357*x^3-46*x^2-26*x-3) / ((x-1)*(x^4-10*x^2-1)*(x^4+10*x^2-1)). - Colin Barker, May 04 2015

A257711 Triangular numbers (A000217) that are the sum of seven consecutive triangular numbers.

Original entry on oeis.org

210, 3486, 51681, 883785, 13125126, 224476266, 3333728685, 57016086141, 846753959226, 14481861401910, 215072171913081, 3678335779997361, 54627484911961710, 934282806257926146, 13875166095466359621, 237304154453733242085, 3524237560763543380386

Offset: 1

Colin Barker, May 05 2015

			210 is in the sequence because T(20) = 210 = 10+15+21+28+36+45+55 = T(4)+ ... +T(10).

Colin Barker, Table of n, a(n) for n = 1..830
Index entries for linear recurrences with constant coefficients, signature (1,254,-254,-1,1).

Cf. A000217, A001110, A129803, A131557, A257712, A257713, A259413, A259414, A259415.

LinearRecurrence[{1, 254, -254, -1, 1}, {210, 3486, 51681, 883785, 13125126}, 30] (* Vincenzo Librandi, Jun 27 2015 *)

Vec(-21*x*(x^4-245*x^2+156*x+10) / ((x-1)*(x^2-16*x+1)*(x^2+16*x+1)) + O(x^100))

G.f.: -21*x*(x^4-245*x^2+156*x+10) / ((x-1)*(x^2-16*x+1)*(x^2+16*x+1)).

16*a(n) = 104 +225*A157456(n+1) +7*(-1)^n*A159678(n+1). - R. J. Mathar, Apr 28 2020

cp's OEIS Frontend

A285088 Numbers n such that the number of partitions of n(n+1)/2 (=A000041(A000217(n))) is prime.

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A346873 Triangle read by rows in which row n lists the row A000217(n) of A237591, n >= 1.

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A360010 First part of the n-th weakly decreasing triple of positive integers sorted lexicographically. Each n > 0 is repeated A000217(n) times.

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A115364 a(n) = A000217(A001511(n)), where A001511 is one more than the 2-adic valuation of n, and A000217(n) is the n-th triangular number, binomial(n+1, 2).

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A129307 Intersection of A000217 and A005098.

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A130703 a(n) = smallest k such that A000217(n+1) = A000217(n) + (A000217(n) mod k), or 0 if no such k exists.

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A210838 Coordinates (x,y) of the endpoint of a structure (or curve) formed by Q-toothpicks of size = 1..n. The inflection points are the n-th nodes if n is a triangular number A000217.

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A212652 a(n) is the least positive integer M such that n = T(M) - T(k), for k an integer, 0 <= k <= M, where T(r) = A000217(r) is the r-th triangular number.

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