A316224 a(n) = n*(2*n + 1)*(4*n + 1).

0, 15, 90, 273, 612, 1155, 1950, 3045, 4488, 6327, 8610, 11385, 14700, 18603, 23142, 28365, 34320, 41055, 48618, 57057, 66420, 76755, 88110, 100533, 114072, 128775, 144690, 161865, 180348, 200187, 221430, 244125, 268320, 294063, 321402, 350385, 381060, 413475, 447678, 483717

Offset: 0

Author

Bruno Berselli, Jun 27 2018

Keywords

• nonn
• easy

Comments

Sums of the consecutive integers from A000384(n) to A000384(n+1)-1. This is the case s=6 of the formula n*(n*(s-2) + 1)*(n*(s-2) + 2)/2 related to s-gonal numbers.

The inverse binomial transform is 0, 15, 60, 48, 0, ... (0 continued).

Examples

			Row sums of the triangle:
|  0 |  ................................................................. 0
|  1 |  2  3  4  5  .................................................... 15
|  6 |  7  8  9 10 11 12 13 14  ........................................ 90
| 15 | 16 17 18 19 20 21 22 23 24 25 26 27  ........................... 273
| 28 | 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44  ............... 612
| 45 | 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65  .. 1155
...
where:
. first column is A000384,
. second column is A130883 (without 1),
. third column is A033816,
. diagonal is A014106,
. 0, 2, 8, 18, 32, 50, ... are in A001105.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

First bisection of A059270 and subsequence of A034828, A047866, A109900, A290168.

Sums of the consecutive integers from P(s,n) to P(s,n+1)-1, where P(s,k) is the k-th s-gonal number: A027480 (s=3), A055112 (s=4), A228888 (s=5).

Programs

GAP

List([0..40], n -> n*(2*n+1)*(4*n+1));

Julia

[n*(2*n+1)*(4*n+1) for n in 0:40] |> println

Magma

[n*(2*n+1)*(4*n+1): n in [0..40]];

Maple

seq(n*(2*n+1)*(4*n+1),n=0..40); # Muniru A Asiru, Jun 27 2018

Mathematica

Table[n (2 n + 1) (4 n + 1), {n, 0, 40}]

Maxima

makelist(n*(2*n+1)*(4*n+1), n, 0, 40);

PARI

vector(40, n, n--; n*(2*n+1)*(4*n+1))

Python

[n*(2*n+1)*(4*n+1) for n in range(40)]

Sage

[n*(2*n+1)*(4*n+1) for n in (0..40)]

Formula

O.g.f.: 3*x*(5 + 10*x + x^2)/(1 - x)^4.

E.g.f.: x*(15 + 30*x + 8*x^2)*exp(x).

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).

a(n) = 3*A258582(n).

a(n) = -3*A100157(-n).

Sum_{n>0} 1/a(n) = 2*(3 - log(4)) - Pi.

Sum_{n>=1} (-1)^(n+1)/a(n) = log(2) + 2*sqrt(2)*log(1+sqrt(2)) + (sqrt(2)-1/2)*Pi - 6. - Amiram Eldar, Sep 17 2022