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.

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

A165190 G.f.: 1/((1-x^4)*(1-x^5)).

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 3, 2, 2, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 4, 4, 3, 3, 4, 4, 4, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 4, 4, 4, 5, 5, 4, 4, 5, 5, 5, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 5, 5, 5, 6
Offset: 0

Views

Author

Alford Arnold, Sep 24 2009

Keywords

Comments

A121262 convolved with A079998. The two sequences have very simple generating functions and can be mapped to the numeric partitions 4=4 and 5=5 respectively.
Number of partitions of n into parts 4 and 5. - Joerg Arndt, Aug 28 2015

Links

Crossrefs

Cf. A057077, A079998, A117444, A121262.

Programs

  • Magma
    [Floor((n+4)/4) - Floor((n+4)/5) : n in [0..100]]; // Wesley Ivan Hurt, Aug 27 2015
  • Maple
    A165190:=n->floor((n+4)/4) - floor((n+4)/5): seq(A165190(n), n=0..100); # Wesley Ivan Hurt, Aug 27 2015
  • Mathematica
    CoefficientList[Series[1/((1-x^4)(1-x^5)),{x,0,110}],x] (* or *) LinearRecurrence[{0,0,0,1,1,0,0,0,-1},{1,0,0,0,1,1,0,0,1},110] (* Harvey P. Dale, Aug 16 2012 *)
Table[Floor[(n + 4)/4] - Floor[(n + 4)/5], {n, 0, 100}] (* Wesley Ivan Hurt, Aug 27 2015 *)

Formula

1 followed by the Euler transform of the finite sequence [0,0,0,1,1].
G.f.: 1/((1-x)^2*(1+x)*(1+x^2)*(1+x+x^2+x^3+x^4)). [R. J. Mathar, Oct 07 2009]
a(n) = A117444(n+2)/5 + n/20 + 9/40 + (-1)^n/8 + A057077(n)/4. [R. J. Mathar, Oct 07 2009]
a(0)=1, a(1)=0, a(2)=0, a(3)=0, a(4)=1, a(5)=1, a(6)=0, a(7)=0, a(8)=1, a(n) = a(n-4)+a(n-5)-a(n-9), n>8. - Harvey P. Dale, Aug 16 2012
a(n) = floor((n+4)/4) - floor((n+4)/5). - Wesley Ivan Hurt, Aug 27 2015
a(n)+a(n-2) = A008616(n). - R. J. Mathar, Jun 23 2021

Extensions

Removed duplicate of comment in A165188; Euler transform formula corrected - R. J. Mathar, Oct 07 2009

A251260 Expansion of (1 + 2*x + x^2 + x^3) / ((1 - x^2)^2 * (1 - x^3) * (1 - x^4)) in powers of x.

Original entry on oeis.org

1, 2, 3, 6, 8, 13, 16, 24, 28, 40, 45, 61, 68, 89, 97, 124, 134, 167, 179, 219, 233, 281, 297, 353, 372, 437, 458, 533, 557, 642, 669, 765, 795, 903, 936, 1056, 1093, 1226, 1266, 1413, 1457, 1618, 1666, 1842, 1894, 2086, 2142, 2350, 2411, 2636, 2701, 2944
Offset: 0

Views

Author

Michael Somos, Mar 20 2015

Keywords

Examples

			G.f. = 1 + 2*x + 3*x^2 + 6*x^3 + 8*x^4 + 13*x^5 + 16*x^6 + 24*x^7 + ...

Links

Crossrefs

Cf. A008763, A165188, A254594, A254708, A254875.

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1 + 2*x+x^2+x^3)/((1-x^2)^2*(1-x^3)*(1-x^4)))); // G. C. Greubel, Aug 03 2018
  • Mathematica
    a[ n_] := Quotient[ 5 n^3 + If[ OddQ[n], 66 n^2 + 249 n, 57 n^2 + 204 n] + 288, 288];
a[ n_] := Length @ FindInstance[ {x > u, u > v, v > w, w >= 0, x + u == n + 6, (u + v < x + w && k == 0) || (u + v > x + w && x + u + v + w == 2 k + 1)}, {x, u, v, w, k}, Integers, 10^9];
LinearRecurrence[{0,2,1,0,-2,-2,0,1,2,0,-1},{1,2,3,6,8,13,16,24,28,40,45},60] (* Harvey P. Dale, Jul 16 2025 *)
  • PARI
    {a(n) = (5*n^3 + if( n%2, 66*n^2 + 249*n, 57*n^2 + 204*n) + 288) \ 288};
  • PARI
    {a(n) = polcoeff( if( n<0, n = -8-n; -(1 + x + 2*x^2 + x^3), 1 + 2*x + x^2 + x^3) / ((1 - x^2)^2 * (1 - x^3) * (1 - x^4)) + x * O(x^n), n)};

Formula

a(n) = A165188(n+1) + A254708(n-1) = A254594(n-1) + A008763(n+4) for all n in Z.
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(2*n) = A254875(n) for all n in Z.
G.f.: (1 + 2*x + x^2 + x^3) / ((1 - x^2)^2 * (1 - x^3) * (1 - x^4)).
Showing 1-3 of 3 results.

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