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A226023 A142705 (numerators of 1/4-1/(4n^2)) sorted to natural order. 4
0, 2, 3, 6, 12, 15, 20, 30, 35, 42, 56, 63, 72, 90, 99, 110, 132, 143, 156, 182, 195, 210, 240, 255, 272, 306, 323, 342, 380, 399, 420, 462, 483, 506, 552, 575, 600, 650, 675, 702, 756, 783, 812, 870, 899 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

A198442(n) without indices 4*n+2.

a(n)/A130823(n+1) = 0, 2,3,2, 4,5,4, 6,7,6, 8,9,8, ... (equal to A133310+1, after 0; see also A008611).

-1,   0,   2,   3, is divisible by  1 (for a(-1)=-1),

3,    6,  12,  15,                  3,

15,  20,  30,  35                   5,

35,  42,  56,  63                   7,

63,  72,  90,  99                   9,

99, 110, 132, 143,                 11, etc.

First column:  A000466(n),

second column: A002943(n),

third column:  A002939(n+1),

fourth column: A000466(n+1).

a(n) is also the numerator of 1/4-1/(4*n+2)^2: 0/1, 2/9, 3/16, 6/25, 12/49, 15/64, 20/81, 30/121, 35/144, 42/169, 56/225,...

The n-th denominator is equal to 4*a(n) + A146325(n+2).

Note that the differences of a(n-1): 1, 2, 1, 3, 6, 3, 5, 10, 5, 7, 14, 7, 9, 18, 9, 11, 22,... (from A043547 by pairs and 2*n+1) has the same recurrence.

(Of course every sequence which obeys a linear recurrence with constant coefficients has first differences that obey the same linear recurrence. - R. J. Mathar, Jun 14 2013)

LINKS

Table of n, a(n) for n=0..44.

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

FORMULA

a(n) = floor( (2*n + 1)/3 ) * floor( (2*n + 5)/3 ) = A004396(n) * A004396(n+2).

Recurrences: a(n) = 3*a(n-3) -3*a(n-6) +a(n-9) = a(n-1) +2*a(n-3) -2*a(n-4) -a(n-6) +a(n-7).

a(n+15)  - a(n) = 10*A042968(n+8).

a(n+1) - a(n-2) = 2*A042968(n) with a(-2)=0, a(-1)=-1.

G.f.: x*(2+x+3*x^2+2*x^3+x^4-x^5)/((1-x)^3 * (1+x+x^2)^2). [Ralf Stephan, May 24 2013]

MAPLE

A226023 := proc(n)

    option remember;

    if n <=6 then

        op(n+1, [0, 2, 3, 6, 12, 15, 20]) ;

    else

        procname(n-1)+2*procname(n-3)-2*procname(n-4)-procname(n-6)+procname(n-7) ;

    end if;

end proc: # R. J. Mathar, Jun 28 2013

CROSSREFS

Trisections: A002939, A000466, A002943.

Sequence in context: A111271 A217647 A070926 * A124485 A200175 A286906

Adjacent sequences:  A226020 A226021 A226022 * A226024 A226025 A226026

KEYWORD

nonn,easy

AUTHOR

Paul Curtz, May 23 2013

STATUS

approved

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Last modified September 20 18:52 EDT 2019. Contains 327245 sequences. (Running on oeis4.)