A123167 -id:A123167 - cp's OEIS frontend

A176743 a(n) = gcd(A000217(n+1), A002378(n+2)).

1, 3, 2, 10, 3, 7, 4, 18, 5, 11, 6, 26, 7, 15, 8, 34, 9, 19, 10, 42, 11, 23, 12, 50, 13, 27, 14, 58, 15, 31, 16, 66, 17, 35, 18, 74, 19, 39, 20, 82, 21, 43, 22, 90, 23, 47, 24, 98, 25, 51, 26

Offset: 0

Paul Curtz, Apr 25 2010

These greatest common divisors of (n+1)*(n+2)/2 and (n+2)*(n+3) appear in the second row of the table discussed in A177427: 0, -1/6, -1/4, -3/10, -1/3, -5/14, -3/8, -7/18, -2/5, ... These fractions can be written as A000217(n+1)/A002378(n+2), n >= 0, and the current sequence shows the common factors that reduces the fractions to the standard format with coprime numerator and denominator.

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

Cf. A000217 (triangular), A002378 (oblong), A176662.

A176743 := proc(n)
    if type(n,'even') then
        n/2+1 ;
    else
        A123167((n-1)/2+2) ;
    end if;
end proc: # R. J. Mathar, Jul 25 2013

Table[GCD[Plus@@Range[n + 1], (n + 2)(n + 3)], {n, 0, 49}] (* Alonso del Arte, Jan 16 2011 *)

a(n)=(n+2)*gcd((n+1)/2, n+3) \\ Charles R Greathouse IV, Sep 02 2015

a(2n) = n+1, a(2n+1) = A123167(n+2).
a(n) = 2*a(n-4) - a(n-8).
G.f.: (1 + 3*x + 2*x^2 + 10*x^3 + x^4 + x^5 - 2*x^7) / ( (x-1)^2*(1+x)^2*(x^2+1)^2 ). - R. J. Mathar, Jan 16 2011
a(n) = (8*n + 16 - 4*(n+2)*(-1)^n + (2*n + 5 + (-1)^n)*((1-(-1)^n)*(-1)^((2*n + 3 + (-1)^n)/4)))/8. - Luce ETIENNE, Feb 03 2015

A176662 a(0)=2, a(1)=7, and a(n) = (3n+1)2^(n-1) if n > 1.

Original entry on oeis.org

2, 7, 14, 40, 104, 256, 608, 1408, 3200, 7168, 15872, 34816, 75776, 163840, 352256, 753664, 1605632, 3407872, 7208960, 15204352, 31981568, 67108864, 140509184, 293601280, 612368384, 1275068416, 2650800128, 5502926848, 11408506880, 23622320128, 48855252992

Offset: 0

Paul Curtz, Apr 23 2010

The sequence appears on the main diagonal of the array defined by A123167 in the first row and successive differences in followup rows:
2, 3, 10, 7, 18, 11, 26, 15, 34, 19, ... A123167
1, 7, -3, 11, -7, 15, -11, 19, -15, 23, ... first diff
6, -10, 14, -18, 22 -26, 30, -34, 38, ... second diff
-16, 24, -32, 40, -48, 56, -64, 72, -80, ... third diff

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-4).

LinearRecurrence[{4,-4},{2,7,14,40},40] (* or *) Join[{2,7},Table[ (3n+1) 2^(n-1),{n,2,40}]] (* Harvey P. Dale, Oct 05 2019 *)

a(n) mod 9 = A010710(n-1), n > 2.
a(2n) + a(2n+1) = 9, 54, 360, 2016, ...
a(n) - 2*a(n-1) = 12*A131577(n-2), n > 1.
a(n) = 4*a(n-1) - 4*a(n-2), n > 3.
G.f.: (-6*x^2+12*x^3+2-x)/(1-2*x)^2.

Edited by R. J. Mathar, Jun 30 2010

A185138 a(4n) = n(4n-1); a(2n+1) = n(n+1)/2; a(4n+2) = (2n+1)(4*n+1).

Original entry on oeis.org

0, 0, 1, 1, 3, 3, 15, 6, 14, 10, 45, 15, 33, 21, 91, 28, 60, 36, 153, 45, 95, 55, 231, 66, 138, 78, 325, 91, 189, 105, 435, 120, 248, 136, 561, 153, 315, 171, 703, 190, 390, 210, 861, 231, 473, 253, 1035, 276, 564, 300, 1225, 325

Offset: 0

Paul Curtz, Mar 12 2012

a(n) is divisible by the n-th term of the sequence 3, 3, 1, 1, 3, 3 (periodically repeated with period 6).
a(n) is divisible by b(floor((n-1)/3)), where b(n) = 1, 3, 2, 3, 7, 3, 5, 3, 13, 3, 8, 3, 19, 3,... , n>=0, is defined by inserting a 3 after each entry of A165355.
(n+1)*(n+2)*(n+3)/2=3*A000292(n+1) is divisible by a(n+2), so there is an integer sequence c(n)= 3*A000292(n+1)/a(n+2) = 3, 12, 10, 20, 7, 28, 18,... with c(2*n)=A123167(n+1) and c(n)/A109613(n+2)=A176895(n).
The sequence of denominators of a(n+2)/n has period length 8: 1, 2, 1, 4, 1, 1, 1, 4.
A table T(k,c) = a(1+c*(1+2k)) of (2*k+1)-sections starts as follows:
0  1  1   3   3   15...
0  3  6  45  21   60...
0 15 15  60  55  325...
0 14 28 231 105  315...
0 45 45 189 171 1035...
The table of T'(k,c) = T(k,c)/(2k+1), columns c>=0, looks as follows, construction similar to A165943:
0 1 1  3  3  15  6  14   k=0
0 1 2 15  7  20 15  77   k=1
0 3 3 12 11  65 24  63   k=2
0 2 4 33 15  45 33 175   k=3
0 5 5 21 19 115 42 112   k=4
0 3 6 51 23  70 51 273   k=5
The entries T'(k,c) are divisible by A060819(c).
Differences are T'(2,c)-T'(0,c) = T'(4,c)-T'(2,c) = 0, 2, 2, 9, 8, 50, 18, 49, 32, ... which is A168077(c) multiplied by the c-th term of the period-4 sequence 2, 2, 2, 1.
Differences are T'(3,c)- T'(1,c) = T'(5,c)-T'(3,c) = 0, 1, 2, 18, 8, 25, 18, 98, 32,... which is  A168077(c) multiplied by the period-4 sequence 2, 1, 2, 2.
The reduced fractions T'(0,c)/T'(1,c) = 1, 1/2, 1/5, 3/7, 3/4, 2/5, 2/11, 5/13, 5/7, 3/8, 3/17, 7/19, .., c>=1, have a numerator sequence A026741(floor(c/2)+1). The denominator sequence is f(c) = 1, 2, 5, 7, 4, 5,.. = A001651(c+1)/A130658(c+1), with f(2*c+1) +f(2*c+2) = 3, 12, 9, 24 .. =3*A022998(c).

G. C. Greubel, 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).

Cf. A000217, A014634, A026741, A033991, A064038, A060819, A165355, A208950.

A185138 := proc(n)
        if n mod 4 = 0 then
                return n/4*(n-1) ;
        elif n mod 2 = 1 then
                return (n-1)*(n+1)/8 ;
        else
                return (n-1)*n/2 ;
        end if;
end proc: # R. J. Mathar, Apr 05 2012

Clear[b];b[1] = 0; b[2] = 0; b[3] = 1; b[4] = 1; b[5] = 3; b[6] = 3; b[7] = 15;b[8] = 6;b[n_Integer] := b[n] = ((-2 + n) (-4 (-4 + n) (-3 + n) (-2 + n) (8 + n (-9 + 2 n)) b[-3 + n] + (-5 + n) ((-3 +n) ((-4 + n) (211 + 2 n (-215 + n (147 + n (-41 + 4 n)))) - 4 (-1 + n) (19 + n (-13 + 2 n)) b[-2 + n]) - 4 (-4 + n)^2 (8 + n (-9 + 2 n)) b[-1 + n])))/(4 (-5 + n) (-4 + n) (-3 + n)^2 (19 + n (-13 + 2 n)))
a = Table[b[n], {n, 1, 52}] (* Roger L. Bagula, Mar 14 2012 *)
LinearRecurrence[{0,0,0,3,0,0,0,-3,0,0,0,1},{0,0,1,1,3,3,15,6,14,10,45,15},60] (* Harvey P. Dale, Nov 23 2015 *)

x='x+O('x^50); concat([0,0], Vec(-x^2*(3*x^8+x^7+5*x^6+3*x^5+12*x^4+3*x^3+3*x^2+x+1)/ ((x-1)^3*(x+1)^3*(x^2+1)^3))) \\ G. C. Greubel, Jun 23 2017

a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12).
a(2*n) = A064038(2*n), a(2*n+1) = A000217(n).
a(n) = 3*A208950(n)/A109613(n).
a(n+1) = A060819(n) * A026741(n+2)(floor(n/2)).
G.f.: -x^2*(3*x^8+x^7+5*x^6+3*x^5+12*x^4+3*x^3+3*x^2+x+1)/ ((x-1)^3*(x+1)^3*(x^2+1)^3). - R. J. Mathar, Mar 22 2012
a(n) = (4*n^2-3*n-1+(2*n^2-3*n+1)*(-1)^n + n*(n-1)*(1+(-1)^n)*(-1)^((2*n-3-(-1)^n)/4))/16. - Luce ETIENNE, May 13 2016
Sum_{n>=2} 1/a(n) = 2 - Pi/4 + 7*log(2)/2. - Amiram Eldar, Aug 12 2022

cp's OEIS Frontend

A176743 a(n) = gcd(A000217(n+1), A002378(n+2)).

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A176662 a(0)=2, a(1)=7, and a(n) = (3n+1)2^(n-1) if n > 1.

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A185138 a(4n) = n(4n-1); a(2n+1) = n(n+1)/2; a(4n+2) = (2n+1)(4*n+1).

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cp's OEIS Frontend

A176743 a(n) = gcd(A000217(n+1), A002378(n+2)).

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Maple

Mathematica

PARI

Formula

A176662 a(0)=2, a(1)=7, and a(n) = (3*n+1)*2^(n-1) if n > 1.

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A185138 a(4*n) = n*(4*n-1); a(2*n+1) = n*(n+1)/2; a(4*n+2) = (2*n+1)*(4*n+1).

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A176662 a(0)=2, a(1)=7, and a(n) = (3n+1)2^(n-1) if n > 1.

A185138 a(4n) = n(4n-1); a(2n+1) = n(n+1)/2; a(4n+2) = (2n+1)(4*n+1).