A253145 -id:A253145 - cp's OEIS frontend

A271511 a(n) = (p+1)*(p+2)/2 where p is the n-th prime.

6, 10, 21, 36, 78, 105, 171, 210, 300, 465, 528, 741, 903, 990, 1176, 1485, 1830, 1953, 2346, 2628, 2775, 3240, 3570, 4095, 4851, 5253, 5460, 5886, 6105, 6555, 8256, 8778, 9591, 9870, 11325, 11628, 12561, 13530, 14196, 15225, 16290, 16653, 18528, 18915, 19701

Offset: 1

Vincenzo Librandi, Apr 09 2016

Subsequence of A253145.

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

Cf. A253145.

[(p+1)*(p+2) / 2: p in PrimesUpTo(200)];

Table[(Prime[n]^2 + 3 Prime[n] + 2) / 2, {n, 50}]

lista(nn) = forprime(p=2, nn, print1((p+1)*(p+2)/2, ", ")); \\ Altug Alkan, Apr 09 2016

A335927 a(n+1) = Sum_{k=1..n} (a(k) + k*(n-k)), with a(1)=1.

Original entry on oeis.org

1, 2, 7, 20, 50, 115, 251, 530, 1096, 2237, 4529, 9124, 18326, 36743, 73591, 147302, 294740, 589633, 1179437, 2359064, 4718338, 9436907, 18874067, 37748410, 75497120, 150994565, 301989481, 603979340, 1207959086

Offset: 1

Daniel Cieslinski, Jul 01 2020

nonn

First column of matrix given by:
C(1,1) = 1,
C(n,k+1) = C(n,k) + n,
C(n+1,1) = Sum_{k=1..n-1} C(k, n-k+1);
where C(i,j) denotes cell at row i and column j
1  2  3  4  5  6  ..
2  4  6  8  10 ...
7  10 13 16 ...
20 24 28 ...
50 55 ...
115...
-------
Can also be seen as diagonal of the following triangle, which is obtained by shifting n-th row of the earlier mentioned matrix, by n-1 cells:
1 2 3  4  5   6 ...
  2 4  6  8  10 ...
    7  10 13 16 ...
       20 24 28 ...
          50 55 ...
             115...

Index entries for linear recurrences with constant coefficients, signature (5,-9,7,-2).

Cf. A253145.

a[1] = 1; a[n_] := a[n] = Sum[a[k] + k*(n - k), {k, 1, n - 1}]; Array[a, 30] (* Amiram Eldar, Jul 02 2020 *)
LinearRecurrence[{5,-9,7,-2},{1,2,7,20,50},30] (* Harvey P. Dale, Sep 27 2024 *)

def a(n):
    if n == 1: return 1
    return sum([a(k) + k*(n-k) for k in range(1,n)])

a(1) = 1, a(n+1) = Sum_{k=1..n} (a(k) + k*(n-k)); for n>1.
a(n) = 1/4 * (9*2^n - 2*n^2  - 6*n - 8); for n > 1.
a(n+1) = 2 * a(n) + A253145(n-1).
From Stefano Spezia, Jul 02 2020: (Start)
G.f.: x*(1 - 3*x + 6*x^2 - 4*x^3 + x^4)/((1 - x)^3*(1 - 2*x)).
a(n) = 5*a(n-1) - 9*a(n-2) + 7*a(n-3) - 2*a(n-4) for n > 5. (End)

cp's OEIS Frontend

A271511 a(n) = (p+1)*(p+2)/2 where p is the n-th prime.

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