A129194 -id:A129194 - cp's OEIS frontend

A168251 a(n) = n^2 if n is odd, n^2*2^(n-2) if n is even.

0, 1, 4, 9, 64, 25, 576, 49, 4096, 81, 25600, 121, 147456, 169, 802816, 225, 4194304, 289, 21233664, 361, 104857600, 441, 507510784, 529, 2415919104, 625, 11341398016, 729, 52613349376, 841, 241591910400, 961, 1099511627776, 1089, 4964982194176, 1225

Offset: 0

Paul Curtz, Nov 21 2009

This is the main diagonal of the following array defined by T(n,2k+1) = A168077(k) for odd column indices and T(n,2k) = A168077(2k)*2^n for even column indices:
0, 1, 1, 9, 4, 25, ... A168077
0, 1, 2, 9, 8, 25, ... A129194
0, 1, 4, 9, 16,25, ... A000290
0, 1, 8, 9, 32,25, ...
0, 1, 16,9, 64,25, ... A154615

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0, 15, 0, -87, 0, 245, 0, -348, 0, 240, 0, -64).

[(n^2)*2^((n-2)*(1+(-1)^n) div 2): n in [0..40]]; // Vincenzo Librandi, Jul 17 2016

A168251 := proc(n)
        if type(n,'even') then
                n^2*2^n/4 ;
        else
                n^2 ;
        end if;
end proc: # R. J. Mathar, Sep 20 2011

Table[(n^2)*2^((n - 2)*(1 + (-1)^n)/2), {n, 0, 50}] (* G. C. Greubel, Jul 16 2016 *)
Table[If[OddQ[n],n^2,n^2 2^(n-2)],{n,0,50}] (* or *) LinearRecurrence[{0,15,0,-87,0,245,0,-348,0,240,0,-64},{0,1,4,9,64,25,576,49,4096,81,25600,121},41] (* Harvey P. Dale, May 14 2022 *)

a(2n) = A128782(n). a(2n+1) = A016754(n).
a(n) = +15*a(n-2) -87*a(n-4) +245*a(n-6) -348*a(n-8) +240*a(n-10) - 64*a(n-12).
G.f.: x*(1 + 4*x - 6*x^2 + 4*x^3 - 23*x^4 - 36*x^5 + 212*x^6 + 44*x^7 - 336*x^8 - 16*x^9 - 64*x^10) / ( (1-x)^3*(2*x+1)^3*(1-2*x)^3*(1+x)^3 ). - R. J. Mathar, Sep 20 2011
a(n) = (n^2)*2^((n-2)*(1+(-1)^n)/2). - Luce ETIENNE, Feb 03 2015

A290168 If n is even then a(n) = n^2(n+2)/8, otherwise a(n) = (n-1)n*(n+1)/8.

Original entry on oeis.org

0, 0, 2, 3, 12, 15, 36, 42, 80, 90, 150, 165, 252, 273, 392, 420, 576, 612, 810, 855, 1100, 1155, 1452, 1518, 1872, 1950, 2366, 2457, 2940, 3045, 3600, 3720, 4352, 4488, 5202, 5355, 6156, 6327, 7220, 7410, 8400

Offset: 0

Jean-François Alcover and Paul Curtz, Jul 23 2017

nonn

Bisection of a(n) [0, 2, 12, 36, 80, 150, 252, ...] is A011379.
Bisection [0, 3, 15, 42, 90, 165, 273, ...] is A059270.
Considering s(n) = [0, 0, 0, 0, 1, 1, 3, 3, 6, 6, 10, 10, 15, 15, ...] (triangular numbers repeated - see A008805), a(n) = n*s(n+2) holds.
Considering the first differences of a(n), b(n) = [0, 2, 1 , 9, 3, 21, 6, 38, 10, 60, 15, 87, ...], b(n) shows bisections A000217 and A005476. In addition, b(n) begins like A249264 up to 12th term, and is an alternation of 4 multiples of 3 and 2 not multiples; b(n) is also such that b(2n) + b(2n+1) = A049450(n).
Considering the second differences c(n), c(n) shows bisections A001105(n+1) and -A000384(n+1), c(n) has 3 consecutive terms multiples of 3 alternating with 3 not multiples; in addition, c(2n) + c(2n+1) = A000027(n).
Considering a(n)/c(n) = [0, 0, 1/4, -1/2, 2/3, -1, 9/8, -3/2, 8/5, -2, 25/12, -5/2, ...], it appears that it is A129194(n)/A022998(n+1) and -A026741(n)/A000034(n) alternating.

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

Cf. A000027, A000034, A000217, A000384, A001105, A005476, A008805, A011379, A022998, A026741, A049450, A059270, A129194, A135713, A161680, A249264.

a[n_] := If[EvenQ[n], n^2*(n + 2)/8, (n - 1)*n*(n + 1)/8]; Table[a[n], {n, 0, 40}]

a(n) = if(n%2==0, n^2*(n+2)/8, (n-1)*n*(n+1)/8) \\ Felix Fröhlich, Jul 23 2017

G.f.: x^2*(2 + x + 3*x^2)/((x-1)^4*(x+1)^3).
a(n) = (1/16)*(-1)^n*n*(1 + (-1)^(n+1) + 2*(1 + (-1)^n)*n + 2*(-1)^n*n^2).
Sum_{n>=2} 1/a(n) = 5 + Pi^2/6 - 8*log(2). - Amiram Eldar, Sep 17 2022

A168037 Period length 18: repeat 0,1,2,0,8,7,0,4,5,0,5,4,0,7,8,0,2,1.

Original entry on oeis.org

0, 1, 2, 0, 8, 7, 0, 4, 5, 0, 5, 4, 0, 7, 8, 0, 2, 1, 0, 1, 2, 0, 8, 7, 0, 4, 5, 0, 5, 4, 0, 7, 8, 0, 2, 1, 0, 1, 2, 0, 8, 7, 0, 4, 5, 0, 5, 4, 0, 7, 8, 0, 2, 1

Offset: 0

Paul Curtz, Nov 17 2009

Represents also the decimal expansion of 447668335336223/37037037037037037.

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

Cf. A154811.

PadRight[{},60,{0,1,2,0,8,7,0,4,5,0,5,4,0,7,8,0,2,1}] (* Harvey P. Dale, Jan 17 2021 *)

a(n) = A129194(n) mod 9.
a(n) = a(n-18).
a(n+1) = a(17-n), 0<=n<= 16, palindromic.
G.f. ( -x*(1+2*x+8*x^3+7*x^4+4*x^6+5*x^7+5*x^9+4*x^10+7*x^12+8*x^13+2*x^15+x^16) ) / ( (x-1)*(1+x+x^2)*(1+x^3+x^6)*(1+x)*(1-x+x^2)*(1-x^3+x^6) ). - R. J. Mathar, Jan 22 2011

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A168251 a(n) = n^2 if n is odd, n^2*2^(n-2) if n is even.

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A290168 If n is even then a(n) = n^2(n+2)/8, otherwise a(n) = (n-1)n*(n+1)/8.

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A168037 Period length 18: repeat 0,1,2,0,8,7,0,4,5,0,5,4,0,7,8,0,2,1.

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cp's OEIS Frontend

A168251 a(n) = n^2 if n is odd, n^2*2^(n-2) if n is even.

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Keywords

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Magma

Maple

Mathematica

Formula

A290168 If n is even then a(n) = n^2*(n+2)/8, otherwise a(n) = (n-1)*n*(n+1)/8.

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PARI

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A168037 Period length 18: repeat 0,1,2,0,8,7,0,4,5,0,5,4,0,7,8,0,2,1.

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A290168 If n is even then a(n) = n^2(n+2)/8, otherwise a(n) = (n-1)n*(n+1)/8.