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.

Showing 1-3 of 3 results.

A045947 Triangles in open triangular matchstick arrangement (triangle minus one side) of side n.

Original entry on oeis.org

0, 0, 2, 7, 17, 33, 57, 90, 134, 190, 260, 345, 447, 567, 707, 868, 1052, 1260, 1494, 1755, 2045, 2365, 2717, 3102, 3522, 3978, 4472, 5005, 5579, 6195, 6855, 7560, 8312, 9112, 9962, 10863, 11817, 12825, 13889, 15010, 16190, 17430, 18732, 20097, 21527
Offset: 0

Views

Author

R. K. Guy

Keywords

Links

Crossrefs

First differences of A082289.
Cf. A000447, A002412, A051895.

Programs

  • Magma
    [Floor((4*n^3+2*n^2-4*n)/16): n in [0..50]]; // Vincenzo Librandi, Aug 29 2011
  • Mathematica
    LinearRecurrence[{3, -2, -2, 3, -1}, {0, 0, 2, 7, 17}, 45] (* Jean-François Alcover, Dec 12 2016 *)
CoefficientList[Series[(2x^2+x^3)/((1-x)^3(1-x^2)),{x,0,50}],x] (* Harvey P. Dale, Jun 26 2021 *)
  • PARI
    a(n)=(4*n^3+2*n^2-4*n)\16

Formula

G.f.: (2*x^2+x^3)/((1-x)^3*(1-x^2)). - Michael Somos
a(n) = (1/16)*(2*n*(2*n^2+n-2)+(-1)^n-1). - Bruno Berselli, Aug 29 2011
a(2*n) = A000447(n)+A002412(n); a(2*n+1) = A051895(n). - J. M. Bergot, Apr 12 2018
E.g.f.: (x*(1 + 7*x + 2*x^2)*cosh(x) - (1 - x - 7*x^2 - 2*x^3)*sinh(x))/8. - Stefano Spezia, Aug 22 2023

A070893 Let r, s, t be three permutations of the set {1,2,3,..,n}; a(n) = value of Sum_{i=1..n} r(i)*s(i)*t(i), with r={1,2,3,..,n}; s={n,n-1,..,1} and t={n,n-2,n-4,...,1,...,n-3,n-1}.

Original entry on oeis.org

1, 6, 19, 46, 94, 172, 290, 460, 695, 1010, 1421, 1946, 2604, 3416, 4404, 5592, 7005, 8670, 10615, 12870, 15466, 18436, 21814, 25636, 29939, 34762, 40145, 46130, 52760, 60080, 68136, 76976, 86649, 97206, 108699, 121182, 134710, 149340
Offset: 1

Views

Author

Wouter Meeussen, May 22 2002

Keywords

Comments

See A070735 for the minimal values for these products. This sequence is an upper bound. The third permutation 't'= ceiling(abs(range(n-1/2,-n,-2))) is such that it associates its smallest factor with the largest factor of the product 'r'*'s'.
We observe that is the transform of A002717 by the following transform T: T(u_0,u_1,u_2,u_3,...) = (u_0,u_0+u_1, u_0+u_1+u_2, u_0+u_1+u_2+u_3+u_4,...). In other words, v_p = Sum_{k=0..p} u_k and the g.f. phi_v of v is given by phi_v = phi_u/(1-z). - Richard Choulet, Jan 28 2010

Examples

			{1,2,3,4,5,6,7}*{7,6,5,4,3,2,1}*{7,5,3,1,2,4,6} gives {49,60,45,16,30,48,42}, with sum 290, so a(7)=290.

Links

Crossrefs

Cf. A070735, A082289. a(n)=A082290(2n-2).
Cf. A000034, A032766, A006578, A002717. - Richard Choulet, Jan 28 2010
Cf. A002717 (first differences). - Bruno Berselli, Aug 26 2011
Column k=3 of A166278. - Alois P. Heinz, Nov 02 2012

Programs

  • Magma
    [(1/96)*(2*n*(n+2)*(3*n^2+10*n+4)+3*(-1)^n-3): n in [1..40]]; // Vincenzo Librandi, Aug 26 2011
  • Mathematica
    Table[Plus@@(Range[n]*Range[n, 1, -1]*Ceiling[Abs[Range[n-1/2, -n, -2]]]), {n, 49}];
(* or *)
CoefficientList[Series[ -(1+2x)/(-1+x)^5/(1+x), {x, 0, 48}], x]//Flatten
  • PARI
    a(n)=sum(i=1,n,i*(n+1-i)*ceil(abs(n+3/2-2*i)))
  • PARI
    a(n)=polcoeff(if(n<0,x^4*(2+x)/((1+x)*(1-x)^5),x*(1+2*x)/((1+x)*(1-x)^5))+x*O(x^abs(n)),abs(n))

Formula

G.f.: x*(1+2*x)/((1+x)*(1-x)^5). - Michael Somos, Apr 07 2003
a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5) + 3. If sequence is also defined for n <= 0 by this equation, then a(n)=0 for -3 <= n <= 0 and a(n)=A082289(-n) for n <= -4. - Michael Somos, Apr 07 2003
a(n) = (1/96)*(2*n*(n+2)*(3*n^2+10*n+4)+3*(-1)^n-3). a(n) - a(n-2) = A002411(n). - Bruno Berselli, Aug 26 2011

A082290 Expansion of (1+x+x^2)/((1+x^2)*(1+x)^4*(1-x)^5).

Original entry on oeis.org

1, 2, 6, 9, 19, 26, 46, 59, 94, 116, 172, 206, 290, 340, 460, 530, 695, 790, 1010, 1135, 1421, 1582, 1946, 2149, 2604, 2856, 3416, 3724, 4404, 4776, 5592, 6036, 7005, 7530, 8670, 9285, 10615, 11330, 12870, 13695, 15466, 16412, 18436, 19514, 21814, 23036
Offset: 0

Views

Author

Michael Somos, Apr 07 2003

Keywords

Examples

			G.f. = 1 + 2*x + 6*x^2 + 9*x^3 + 19*x^4 + 26*x^5 + 46*x^6 + 59*x^7 + ...

Links

Crossrefs

Cf. A070893, A082289.

Programs

  • Magma
    [(6*n^4 +108*n^3 +666*n^2 +1620*n +1251 +(4*n^3 +54*n^2 +236*n +333)*(-1)^n -48*(-1)^Floor((6*n -1 +(-1)^n)/4))/1536: n in [0..50]]; // Vincenzo Librandi, Oct 23 2014
  • Mathematica
    Table[(6 n^4 + 108 n^3 + 666 n^2 + 1620 n + 1251 + (4 n^3 + 54 n^2 + 236 n + 333) (-1)^n - 48 (-1)^((6 n - 1 + (-1)^n)/4))/1536, {n, 0, 50}] (* after Luce ETIENNE; or, by definition: *) CoefficientList[Series[(1 + x + x^2)/((1 + x^2)*(1 + x)^4*(1 - x)^5), {x, 0, 50}], x] (* Bruno Berselli, Oct 26 2014 *)
  • PARI
    {a(n) = if( n<-8, a(-9-n), polcoeff( (1 + x + x^2) / ((1 + x^2) *(1 + x)^4 * (1 - x)^5) + x * O(x^n), n))};

Formula

Euler transform of length 4 sequence [ 2, 3, -1, 1]. - Michael Somos, Feb 15 2006
G.f.: (1 + x + x^2) / ((1 + x^2) * (1 + x)^4 * (1 - x)^5).
a(n) = 3*a(n-2) - 2*a(n-4) - 2*a(n-6) + 3*a(n-8) - a(n-10) + 3.
a(n) = a(-9-n) for all n in Z.
a(2*n) = A070893(n+1). a(2*n + 1) = A082289(n+4).
a(n) = (6*n^4+108*n^3+666*n^2+1620*n+1251+(4*n^3+54*n^2+236*n+333)*(-1)^n-48*(-1)^((6*n-1+(-1)^n)/4))/1536. - Luce ETIENNE, Oct 23 2014
Showing 1-3 of 3 results.

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