A316224 - OEIS

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

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OFFSET

0,2

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).

LINKS

Table of n, a(n) for n=0..39.

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

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

EXAMPLE

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.

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}]

PROG

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

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

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

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

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

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

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

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).

Sequence in context: A164541 A145789 A010822 * A022707 A323334 A151974

Adjacent sequences: A316221 A316222 A316223 * A316225 A316226 A316227

KEYWORD

nonn,easy

AUTHOR

Bruno Berselli, Jun 27 2018

STATUS

approved