cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A141516 The main diagonal of the array of A141425 and its higher order differences.

Original entry on oeis.org

1, 2, 1, -7, -23, -1, 7, -103, -251, -133, -149, -1387, -3143, -3001, -4913, -19663, -42611, -55693, -101549, -291667, -612863, -960001, -1831433, -4460023, -9185771, -15980053, -31162949, -69500347, -141392183, -261261001
Offset: 0

Views

Author

Paul Curtz, Aug 11 2008

Keywords

Comments

The sequence A141425 and higher order differences in subsequent rows starts (see A141533):
1, 2, 4, 5, 7, 8, 1, 2, 4, 5, 7, 8, 1, 2, 4,...
1, 2, 1, 2, 1,-7, 1, 2, 1, 2, 1,-7, 1, 2, 1, 2,...
1,-1, 1, -1, -8, 8, 1,-1, 1, -1, -8, 8, 1, -1,..
-2, 2,-2, -7, 16,-7,-2, 2,-2, -7, 16,-7, -2,..
4,-4,-5, 23,-23, 5, 4,-4,-5, 23,-23, 5, 4,..
-8,-1,28,-46, 28,-1,-8,-1,28,-46, 28,-1,..
Reading downwards the main diagonal of this array defines the sequence.

Links

Programs

  • Maple
    A108411 := proc(n) 3^floor(n/2) ; end proc:
A141516 := proc(n) if n = 0 then 1; else (-3*(-1)^n-2^n+3*(-1)^(floor((n-1)/2))*A108411(n))/2 ; end if; end proc: # R. J. Mathar, Mar 08 2011
  • Mathematica
    LinearRecurrence[{1,-1,3,6},{1,2,1,-7,-23},30] (* Harvey P. Dale, Nov 23 2022 *)

Formula

a(n) = ( -3*(-1)^n -2^n +3*(-1)^(floor((n-1)/2))*A108411(n) )/2, n>0. - R. J. Mathar, Mar 08 2011
a(2n)+a(2n+1)= -A002023(n-1) = -3*A081294(n), n>0.
a(4n)+a(4n+1)+a(4n+2)+a(4n+3) = -120*16^(n-1), n>0.
a(4n+2)+a(4n+3)+a(4n+4)+a(4n+5) = -30*A001025(n).
G.f. x*(-2+x+6*x^2+21*x^3) / ( (2*x-1)*(1+x)*(3*x^2+1) ). - R. J. Mathar, Mar 08 2011

A143769 Expansion of 3*x*(3*x+1)*(2*x-1) / ( (1+x)*(3*x^2+1) ).

Original entry on oeis.org

-3, 0, 27, -18, -63, 36, 207, -126, -603, 360, 1827, -1098, -5463, 3276, 16407, -9846, -49203, 29520, 147627, -88578, -442863, 265716, 1328607, -797166, -3985803, 2391480, 11957427, -7174458, -35872263, 21523356, 107616807, -64570086, -322850403, 193710240
Offset: 1

Views

Author

Paul Curtz, Aug 31 2008

Keywords

Comments

All entries are multiples of 3.

Links

Programs

  • Magma
    I:=[-3,0,27]; [n le 3 select I[n] else -Self(n-1)-3*Self(n-2) -3*Self(n-3): n in [1..50]]; // Vincenzo Librandi, Aug 17 2013
  • Mathematica
    CoefficientList[Series[3 (3 x + 1) (2 x - 1) / ((1 + x) (3 x^2 + 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Aug 17 2013 *)
LinearRecurrence[{-1,-3,-3},{-3,0,27},40] (* Harvey P. Dale, Dec 26 2014 *)
  • PARI
    a(n)=3/2*((2*(-1)^n-3)*(-3)^floor(n/2)-3*(-1)^n); \\ Ralf Stephan, Aug 17 2013

Formula

a(n) = A141533(n+1)-2*A141533(n).
a(2n)+a(2n+1) = (-1)^n*A000244(n).
a(n) = (3/2) * ((2*(-1)^n - 3) * (-3)^floor(n/2) - 3*(-1)^n ). - Ralf Stephan, Aug 17 2013
Showing 1-2 of 2 results.

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