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 1-2 of 2 results.

A254594 Expansion of 1 / ((1 - x^2)^2 * (1 - x^3) * (1 - x^4)) in powers of x.

Original entry on oeis.org

1, 0, 2, 1, 4, 2, 7, 4, 11, 7, 16, 11, 23, 16, 31, 23, 41, 31, 53, 41, 67, 53, 83, 67, 102, 83, 123, 102, 147, 123, 174, 147, 204, 174, 237, 204, 274, 237, 314, 274, 358, 314, 406, 358, 458, 406, 514, 458, 575, 514, 640, 575, 710, 640, 785, 710, 865, 785, 950
Offset: 0

Views

Author

Michael Somos, Feb 02 2015

Keywords

Comments

Partitions of n into parts of size 3 and size 4 and two kinds of parts of size 2.
The number of quadruples of integers [x, u, v, w] which satisfy x > u > v > w >=0, n+5 = x+u, u+v >= x+w, and x+u+v+w is even.
Euler transform of length 4 sequence [ 0, 2, 1, 1].

Examples

			G.f. = 1 + 2*x^2 + x^3 + 4*x^4 + 2*x^5 + 7*x^6 + 4*x^7 + 11*x^8 + 7*x^9 + ...

Links

Crossrefs

Cf. A001400, A002620, A005044, A115264, A165188.
Cf. A241526, A000601, A181120.

Programs

  • Magma
    I:=[1,0,2,1,4,2,7,4,11,7,16]; [n le 11 select I[n] else 2*Self(n-2)+Self(n-3)-2*Self(n-5)-2*Self(n-6)+Self(n-8)+2*Self(n-9)-Self(n-11): n in [1..60]]; // Vincenzo Librandi, Feb 03 2015
  • Mathematica
    a[ n_] := Quotient[ n^3 + If[ OddQ[n], 12 n^2 + 33 n + 54, 21 n^2 + 132 n + 288], 288];
a[ n_] := Module[{s = 1, m = n}, If[ n < 0, s = -1; m = -11 - n]; s SeriesCoefficient[ 1 / ((1 - x^2)^2 (1 - x^3) (1 - x^4)), {x, 0, m}]];
a[ n_] := Length @ FindInstance[ {x > u, u > v, v > w, w >= 0, x + u == n + 5, u + v >= x + w, x + u + v + w == 2 k}, {x, u, v, w, k}, Integers, 10^9];
CoefficientList[Series[1 / (1 - 2 x^2 - x^3 + 2 x^5 + 2 x^6 - x^8 - 2 x^9 + x^11), {x, 0, 60}], x] (* Vincenzo Librandi, Feb 03 2015 *)
  • PARI
    {a(n) = (n^3 + if(n%2, 12*n^2 + 33*n + 54, 21*n^2 + 132*n + 288)) \ 288};
  • PARI
    {a(n) = my(s=1); if( n<0, s=-1; n=-11-n); s * polcoeff( 1 / ((1 - x^2)^2 * (1 - x^3) * (1 - x^4)) + x * O(x^n), n)};

Formula

G.f.: 1 / (1 - 2*x^2 - x^3 + 2*x^5 + 2*x^6 - x^8 - 2*x^9 + x^11).
a(n) = -a(-11-n) for all n in Z.
a(n+3) - a(n) = 0 if n even else floor((n+7)^2 / 16).
0 = a(n) - 2*a(n+2) - a(n+3) + 2*a(n+5) + 2*a(n+6) - a(n+8) - 2*a(n+9) + a(n+11) for all n in Z.
a(n) - a(n-2) = A005044(n+3) for all n in Z.
a(n) + a(n-1) = A001400(n) for all n in Z.
a(n) + a(n-2) = A165188(n+1) for all n in Z.
a(n) = A115264(n) - A115264(n-1) for all n in Z.
a(2*n) - a(2*n-6) = a(2*n+3) - a(2*n-3) = A002620(n+2) for all n in Z. - Michael Somos, Feb 11 2015
a(n) = (2*n^3+33*n^2+181*n+234+3*(3*n^2+33*n+86)*(-1)^n+84*(-1)^((2*n+1-(-1)^n)/4)-96*((1+(-1)^n)*floor(((2*n+9+(-1)^n-6*(-1)^((2*n+3+(-1)^n)/4))/24))+(1-(-1)^n)*floor(((2*n+5+(-1)^n-6*(-1)^((2*n-1+(-1)^n)/4))/24))))/576. - Luce ETIENNE, May 22 2015

A258440 Number of squares of all sizes in 3*n*(n+1)/2-ominoes in form of three-quarters of Aztec diamonds.

Original entry on oeis.org

3, 11, 25, 49, 84, 132, 196, 278, 379, 503, 651, 825, 1028, 1262, 1528, 1830, 2169, 2547, 2967, 3431, 3940, 4498, 5106, 5766, 6481, 7253, 8083, 8975, 9930, 10950, 12038, 13196, 14425, 15729, 17109, 18567, 20106, 21728, 23434, 25228, 27111, 29085, 31153, 33317, 35578, 37940, 40404, 42972, 45647, 48431
Offset: 1

Views

Author

Luce ETIENNE, May 30 2015

Keywords

Comments

These polyominoes are 6*n-gons, and thus their number of vertices is n*(3*n+7).
Schäfli's notation for figure corresponding to a(1): 4.4.4.

Examples

			a(1)=3, a(2)=9+2=11, a(3)=18+7=25, a(4)=30+15+4=49, a(5)=45+26+11+2=84.

Links

Crossrefs

Cf. A000217, A045943, A111746, A140090, A173196, A241526.

Programs

  • Magma
    [(52*n^3+186*n^2+212*n-3*(32*Floor(n/3)+3*(1-(-1)^n)))/144: n in [1..50]]; // Vincenzo Librandi, Jun 02 2015
  • Maple
    A258440:=n->(52*n^3+186*n^2+212*n-3*(32*floor(n/3)+3*(1-(-1)^n)))/144: seq(A258440(n), n=1..100); # Wesley Ivan Hurt, Jul 10 2015
  • Mathematica
    Table[(52 n^3 + 186 n^2 + 212 n - 3 (32 Floor[n/3] + 3 (1 - (-1)^n)))/144, {n, 45}] (* Vincenzo Librandi, Jun 02 2015 *)
  • PARI
    Vec(x*(2*x^3+3*x^2+5*x+3)/((x-1)^4*(x+1)*(x^2+x+1)) + O(x^100)) \\ Colin Barker, Jun 01 2015

Formula

a(n) = 2*A241526(n) - A173196(n+1).
a(n) = (1/8)*(Sum_{i=0..(n-1-floor(n/3)}(4*n+1-6*i-(-1)^i)*(4*n+3-6*i+(-1)^i)- Sum_{j=0..(2*n-1+(-1)^n)}(2*n+1+(-1)^n-4*j)*(2*n+1-(-1)^n-4*j)).
a(n) = (52*n^3+186*n^2+212*n-3*(32*floor(n/3)+3*(1-(-1)^n)))/144.
a(n) = 2*a(n-1)-a(n-3)-a(n-4)+2*a(n-6)-a(n-7) for n>7. - Colin Barker, Jun 01 2015
G.f.: x*(2*x^3+3*x^2+5*x+3) / ((x-1)^4*(x+1)*(x^2+x+1)). - Colin Barker, Jun 01 2015

Extensions

Typo in data fixed by Colin Barker, Jun 01 2015
Name edited by Michel Marcus, Dec 22 2020
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.