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A191567 Four interlaced 2nd order polynomials: a(4*k) = k*(1+2*k); a(1+2*k) = 2*(1+2*k)*(3+2*k); a(2+4*k) = 4*(1+k)*(1+2*k). 3
0, 6, 4, 30, 3, 70, 24, 126, 10, 198, 60, 286, 21, 390, 112, 510, 36, 646, 180, 798, 55, 966, 264, 1150, 78, 1350, 364, 1566, 105, 1798, 480, 2046, 136, 2310, 612, 2590, 171, 2886, 760, 3198, 210, 3526, 924, 3870, 253, 4230, 1104, 4606, 300, 4998, 1300, 5406, 351 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n) = T(0,n) and differences T(n,k) = T(n-1,k+1) - T(n-1,k) define the array

0,   6,  4,  30,    3,  70,   24,  126,   10,  198,   60,  286,   21,  390,  ..

6,  -2, 26, -27,   67, -46,  102, -116,  188, -138,  226, -265,  369, -278, ..

-8, 28 -53,  94, -113, 148, -218,  304, -326,  364, -491,  634, -647,  676, ...

T(3,n) mod 9 is the sequence 1, 1, 1, 4, 4, 4, 7, 7, 7, 4, 4, 4 (and periodically repeated with period 12).

A064680(2+n) divides a(n), where b(n) = a(n)/A064680(2+n) = 0, 1, 2, 3, 1, 5, 6, 7, 2,... for n>=0, obeys b(4*k) = k and has recurrence b(n) = 2*b(n-4) - b(n-8).

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (0,0,0,3,0,0,0,-3,0,0,0,1).

FORMULA

a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12).

a(n) = A061037(n+2) + A181318(n). - Paul Curtz, Jul 19 2011

a(n) = A060819(n) * A145979(n). - Paul Curtz, Sep 06 2011

G.f.: x*(-6-4*x-30*x^2-3*x^3-52*x^4-12*x^5-36*x^6-x^7-6*x^8+2*x^10) / ( (x-1)^3 *(1+x)^3 *(x^2+1)^3 ). - R. J. Mathar, Jun 17 2011

Let BEB(n) = a(n)/A061038(n+2) = A060819(n)/A145979(n). Then (BEB(n))^2 = A181318(n)/A061038(n+2) = BEB(n) - A061037(n+2)/A061038(n+2). - Paul Curtz, Jul 19 2011, index corrected by R. J. Mathar, Sep 09 2011

From Luce ETIENNE, Apr 18 2017: (Start)

a(n) = n*(n + 2)*(37 - 27*(-1)^n - 3*((-1)^((2*n + 1 - (-1)^n)/4) + (-1)^((2*n - 1 + (-1)^n)/4)))/32.

a(n) = n*(n+2)*(37-27*cos(n*Pi) - 6*cos(n*Pi/2))/32.

a(n) = n*(n + 2)*(37 - 27*(-1)^n - 3*(i^n + (-i)^n))/32, where i=sqrt(-1). (End)

MATHEMATICA

Table[Which[OddQ@ n, 2 (1 + 2 #) (3 + 2 #) &[(n - 1)/2], Mod[n, 4] == 0, # (1 + 2 #) &[n/4], True, 4 (1 + #) (1 + 2 #) &[(n - 2)/4]], {n, 0, 36}] (* or *)

CoefficientList[Series[x (-6 - 4 x - 30 x^2 - 3 x^3 - 52 x^4 - 12 x^5 - 36 x^6 - x^7 - 6 x^8 + 2 x^10)/((x - 1)^3*(1 + x)^3*(x^2 + 1)^3), {x, 0, 36}], x] (* Michael De Vlieger, Apr 22 2017 *)

LinearRecurrence[{0, 0, 0, 3, 0, 0, 0, -3, 0, 0, 0, 1}, {0, 6, 4, 30, 3, 70, 24, 126, 10, 198, 60, 286}, 80] (* Vincenzo Librandi, Apr 23 2017 *)

PROG

(MAGMA) I:=[0, 6, 4, 30, 3, 70, 24, 126, 10, 198, 60, 286]; [n le 12 select I[n] else 3*Self(n-4)-3*Self(n-8)+Self(n-12): n in [1..60]]; // Vincenzo Librandi, Apr 23 2017

CROSSREFS

Cf. A014105, A000466, A000384, A177427.

Sequence in context: A211945 A121682 A237425 * A274707 A163934 A163939

Adjacent sequences:  A191564 A191565 A191566 * A191568 A191569 A191570

KEYWORD

nonn,easy

AUTHOR

Paul Curtz, Jun 12 2011

STATUS

approved

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Last modified August 17 07:59 EDT 2017. Contains 290635 sequences.