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A282513 a(n) = floor((3*n + 2)^2/24 + 1/3). 2
0, 1, 3, 5, 8, 12, 17, 22, 28, 35, 43, 51, 60, 70, 81, 92, 104, 117, 131, 145, 160, 176, 193, 210, 228, 247, 267, 287, 308, 330, 353, 376, 400, 425, 451, 477, 504, 532, 561, 590, 620, 651, 683, 715, 748, 782, 817, 852, 888, 925, 963 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

List of quadruples: 2*n*(3*n+1), (2*n+1)*(3*n+1), 6*n^2+8*n+3, (n+1)*(6*n+5). These terms belong to the sequences A033580, A033570, A126587 and A049452, respectively. See links for all the permutations.

After 0, subsequence of A025767.

LINKS

Table of n, a(n) for n=0..50.

Luce ETIENNE, Permutations

Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,1).

FORMULA

G.f.: x*(1 + x + x^3)/((1 + x)*(1 + x^2)*(1 - x)^3).

a(n) = 2*a(n-1) - a(n-2) + a(n-4) - 2*a(n-5) + a(n-6) for n>5.

a(n) = floor((3*n + 2)^2/24 + 2/3).

a(n) = (6*n^2 + 8*n + 3 + (-1)^n - 2*((-1)^((2*n - 1 + (-1)^n)/4) + (-1)^((2*n + 1 - (-1)^n)/4)))/16. Therefore:

a(2*k)   = (6*k^2 + 4*k + 1 - (-1)^k)/4,

a(2*k+1) = (k + 1)*(3*k + 2)/2.

a(n) = (6*n^2 + 8*n + 3 + cos(n*Pi) - 4*cos(n*Pi/2))/16.

a(n) = (3*n + 2)^2/24 + 1/3 + (-6 + (1 + (-1)^n)*(1 + 2*i^((n+1)*(n+2))))/16, where i=sqrt(-1).

EXAMPLE

Rectangular array with four columns:

.   0,   1,   3,   5;

.   8,  12,  17,  22;

.  28,  35,  43,  51;

.  60,  70,  81,  92;

. 104, 117, 131, 145, etc.

MATHEMATICA

Table[Floor[(3 n + 2)^2/24 + 1/3], {n, 0, 50}] (* or *) CoefficientList[Series[x (1 + x + x^3)/((1 + x) (1 + x^2) (1 - x)^3), {x, 0, 50}], x] (* or *) Table[(6 n^2 + 8 n + 3 + Cos[n Pi] - 4 Cos[n Pi/2])/16, {n, 0, 50}] (* or *) Table[(3 n + 2)^2/24 + 1/3 + (-6 + (1 + (-1)^n) (1 + 2 I^((n + 1) (n + 2))))/16, {n, 0, 50}] (* Michael De Vlieger, Feb 17 2017 *)

PROG

(PARI) a(n)=(3*n^2 + 4*n + 4)\8 \\ Charles R Greathouse IV, Feb 17 2017

(MAGMA) [(3*n^2+4*n+4) div 8: n in [0..50]]; // Bruno Berselli, Feb 17 2017

CROSSREFS

Cf. A033436: floor((3*n)^2/24 + 1/3).

Cf. A000326, A000567, A025767, A033570, A033580, A049452, A064412, A126587, A222017, A269064, A274221.

Sequence in context: A020678 A310034 A014811 * A241567 A131674 A095173

Adjacent sequences:  A282510 A282511 A282512 * A282514 A282515 A282516

KEYWORD

nonn,easy

AUTHOR

Luce ETIENNE, Feb 17 2017

EXTENSIONS

Corrected and extended by Bruno Berselli, Feb 17 2017

STATUS

approved

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Last modified January 26 16:58 EST 2020. Contains 331280 sequences. (Running on oeis4.)