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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A157982 Triangle T(n,m) read by rows which contains the coefficients [x^m] of the polynomial generating the numerators of the column A140825(.,n).

Original entry on oeis.org

1, 1, 2, -1, 0, 6, 1, 0, -6, 4, -19, 0, 120, -120, 30, 27
Offset: 0

Views

Author

Paul Curtz, Mar 10 2009

Keywords

Comments

The first five polynomials describing the first five columns of A140825 are in A000012, A005408, A140811, A141530 and A157411.

Examples

			1;
1,2;     # 2n+1
-1,0,6;  # 6n^2-1
1,0,-6,4;  # 4n^3-6n^2+1, A141530
-19,0,120,-120,30;  # 30n^4-120n^3+120n^2-19, A157411

References

  • P Curtz Integration numerique des systemes differentiels a conditions initiales, C.C.S.A., Arcueil, 1969, p.36.

Crossrefs

Cf. A141417.

A157517 a(n) = 7 + 12*n - 6*n^2.

Original entry on oeis.org

7, 13, 7, -11, -41, -83, -137, -203, -281, -371, -473, -587, -713, -851, -1001, -1163, -1337, -1523, -1721, -1931, -2153, -2387, -2633, -2891, -3161, -3443, -3737, -4043, -4361, -4691, -5033, -5387, -5753, -6131, -6521, -6923, -7337, -7763, -8201, -8651
Offset: 0

Views

Author

Paul Curtz, Mar 02 2009

Keywords

Comments

From John Couch Adams multisteps integration of differential equations, 1855.

References

  • P. Curtz Integration numerique des systemes differentiels, C.C.S.A., Arcueil, 1969, p. 36.

Links

Programs

Formula

a(n) = 12*n + 6 - A140811(n) = A017593(n) - A140811(n).
Recurrences: a(n) = 2*a(n-1) - a(n-2) - 12 = 3*a(n-1) - 3*a(n-2) + a(n-3).
First differences: a(n+1) - a(n) = -A017593(n-1), n > 0. Second differences are all -12.
a(n+2) - a(n) = -A008606(n).
G.f.: (-7 + 8*x + 11*x^2)/(x-1)^3. - R. J. Mathar, Mar 15 2009

Extensions

Edited and extended by R. J. Mathar, Mar 15 2009

A158462 a(n) = 36*n^2 - 6.

Original entry on oeis.org

30, 138, 318, 570, 894, 1290, 1758, 2298, 2910, 3594, 4350, 5178, 6078, 7050, 8094, 9210, 10398, 11658, 12990, 14394, 15870, 17418, 19038, 20730, 22494, 24330, 26238, 28218, 30270, 32394, 34590, 36858, 39198, 41610, 44094, 46650, 49278, 51978, 54750, 57594, 60510
Offset: 1

Views

Author

Vincenzo Librandi, Mar 19 2009

Keywords

Comments

The identity (12*n^2 - 1)^2 - (36*n^2 - 6)*(2*n)^2 = 1 can be written as A158463(n)^2 - a(n)*A005843(n)^2 = 1.

Links

Crossrefs

Cf. A005843, A140811, A158463.

Programs

  • Magma
    I:=[30, 138, 318]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..50]]; // Vincenzo Librandi, Feb 12 2012
  • Mathematica
    LinearRecurrence[{3, -3, 1}, {30, 138, 318}, 50] (* Vincenzo Librandi, Feb 12 2012 *)
36Range[50]^2-6 (* Harvey P. Dale, Jul 19 2025 *)
  • PARI
    for(n=1, 40, print1(36*n^2-6", ")); \\ Vincenzo Librandi, Feb 12 2012

Formula

G.f.: 6*x*(5 + 8*x - x^2)/(1-x)^3. - Bruno Berselli, Aug 27 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Feb 12 2012
From Amiram Eldar, Mar 05 2023: (Start)
Sum_{n>=1} 1/a(n) = (1 - cot(Pi/sqrt(6))*Pi/sqrt(6))/12.
Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/sqrt(6))*Pi/sqrt(6) - 1)/12. (End)
From Elmo R. Oliveira, Jan 16 2025: (Start)
E.g.f.: 6*(exp(x)*(6*x^2 + 6*x - 1) + 1).
a(n) = 6*A140811(n). (End)
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