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.

A102214 Expansion of (1 + 4*x + 4*x^2)/((1+x)*(1-x)^3).

Original entry on oeis.org

1, 6, 16, 30, 49, 72, 100, 132, 169, 210, 256, 306, 361, 420, 484, 552, 625, 702, 784, 870, 961, 1056, 1156, 1260, 1369, 1482, 1600, 1722, 1849, 1980, 2116, 2256, 2401, 2550, 2704, 2862, 3025, 3192, 3364, 3540, 3721, 3906, 4096, 4290, 4489, 4692, 4900
Offset: 0

Views

Author

Creighton Dement, Feb 17 2005

Keywords

Comments

A floretion-generated sequence.
a(n) gives the number of triples (x,y,x+y) with positive integers satisfying x < y and x + y <= 3*n. - Marcus Schmidt (marcus-schmidt(AT)gmx.net), Jan 13 2006
Number of different partitions of numbers x + y = z such that {x,y,z} are integers {1,2,3,...,3n} and z > y > x. - Artur Jasinski, Feb 09 2010
Second bisection preceded by zero is A152743. - Bruno Berselli, Oct 25 2011
a(n) has no final digit 3, 7, 8. - Paul Curtz, Mar 04 2020
One odd followed by three evens.
From Paul Curtz, Mar 06 2020: (Start)
b(n) = 0, 1, 6, 16, 30, 49, ... = 0, a(n).
( 25, 12, 4, 0, 1, 6, 16, 30, ...
-13, -8, -4 1, 5, 10, 14, 19, ...
5, 4, 5, 4, 5, 4, 5, 4, ... .)
b(-n) = 0, 4, 12, 25, 42, 64, 90, 121, ... .
A154589(n) are in the main diagonal of b(n) and b(-n). (End)

Links

Crossrefs

Cf. A016778, A038764, A001859, A006578, A069905, A077043, A274221, A330707.
Cf. A000326, A016789, A152743, A001651, A032766, A047237, A047251, A154589.

Programs

  • Magma
    [(6*n*(3*n+4)+(-1)^n+7)/8: n in [0..60]]; // Vincenzo Librandi, Oct 26 2011
  • Mathematica
    aa = {}; Do[i = 0; Do[Do[Do[If[x + y == z, i = i + 1], {x, y + 1, 3 n}], {y, 1, 3 n}], {z, 1, 3 n}]; AppendTo[aa, i], {n, 1, 20}]; aa (* Artur Jasinski, Feb 09 2010 *)
  • PARI
    a(n)=(6*n*(3*n+4)+(-1)^n+7)/8 \\ Charles R Greathouse IV, Apr 16 2020

Formula

G.f.: -(4*x^2 + 4*x + 1)/((x+1)*(x-1)^3) = (1+2*x)^2/((1+x)*(1-x)^3).
a(2n) = A016778(n) = (3n+1)^2.
a(n) + a(n+1) = A038764(n+1).
a(n) = floor( (3*n+2)/2 ) * ceiling( (3*n+2)/2 ). - Marcus Schmidt (marcus-schmidt(AT)gmx.net), Jan 13 2006
a(n) = (6*n*(3*n+4) + (-1)^n+7)/8. - Bruno Berselli, Oct 25 2011
a(n) = A198392(n) + A198392(n-1). - Bruno Berselli, Nov 06 2011
From Paul Curtz, Mar 04 2020: (Start)
a(n) = A006578(n) + A001859(n) + A077043(n+1).
a(n) = A274221(2+2*n).
a(20+n) - a(n) = 30*(32+3*n).
a(1+2*n) = 3*(1+n)*(2+3*n).
a(n) = A047237(n) * A047251(n).
a(n) = A001651(n+1) * A032766(n).(End)
E.g.f.: ((4 + 21*x + 9*x^2)*cosh(x) + 3*(1 + 7*x + 3*x^2)*sinh(x))/4. - Stefano Spezia, Mar 04 2020

Extensions

Definition rewritten by Bruno Berselli, Oct 25 2011

A331952 a(n) = (-7 + (-1)^(1+n) + 6*n^2) / 8.

Original entry on oeis.org

-1, 0, 2, 6, 11, 18, 26, 36, 47, 60, 74, 90, 107, 126, 146, 168, 191, 216, 242, 270, 299, 330, 362, 396, 431, 468, 506, 546, 587, 630, 674, 720, 767, 816, 866, 918, 971, 1026, 1082, 1140, 1199, 1260, 1322, 1386, 1451, 1518, 1586, 1656, 1727, 1800, 1874, 1950, 2027
Offset: 0

Views

Author

Paul Curtz, Feb 02 2020

Keywords

Comments

a(n+1) is once in the hexagonal spiral in A330707. a(n+2) is twice in the same spiral.
a(n) has one odd followed by three evens.
Difference table:
-1, 0, 2, 6, 11, 18, 26, 36, ... = a(n)
1, 2, 4, 5, 7, 8, 10, 11, ... = A001651(n+1)
1, 2, 1, 2, 1, 2, 1, 2, ... = A000034.

Examples

			G.f. = -1 + 2*x^2 + 6*x^3 + 11*x^4 + 18*x^5 + 26*x^6 + 36*x^7 + 47*x^8 + ... - _Michael Somos_, Sep 08 2023

Links

Crossrefs

Cf. A000034, A001651, A158463, 6*A014105, 2*A003154, A152746(n+1), A008585(1+n).
Equals 2 less than A084684, 1 less than A077043, and 1 more than A276382(n-1). - Greg Dresden, Feb 22 2020

Programs

  • Magma
    a:=[-1,0,2,6]; [n le 4 select a[n] else 2*Self(n-1)-2*Self(n-3)+Self(n-4): n in [1..45]]; // Marius A. Burtea, Feb 02 2020
  • Mathematica
    LinearRecurrence[{2, 0, -2, 1}, {-1, 0, 2, 6}, 100] (* Amiram Eldar, Feb 02 2020 *)
a[n_] := Floor[(n^2 - 1)*3/4]; (* Michael Somos, Sep 08 2023 *)
  • PARI
    Vec(-(1 - 2*x - 2*x^2) / ((1 - x)^3*(1 + x)) + O(x^40)) \\ Colin Barker, Feb 03 2020
  • PARI
    {a(n) = (n^2 - 1)*3\4}; /* Michael Somos, Sep 08 2023 */

Formula

a(-n) = a(n).
a(20+n) - a(n) = 30*(10+n).
a(2+n) = a(n) + 3*(1+n), a(0)=-1 and a(1)=0.
a(4*n) = 12*n^2 - 1, a(1+4*n) = 6*n*(1+2*n), a(2+4*n) = 2 + 12*n*(1+n), a(3+4*n) = 6*(1+n)*(1+2*n) for n>= 0.
From Colin Barker, Feb 02 2020: (Start)
G.f.: -(1 - 2*x - 2*x^2) / ((1 - x)^3*(1 + x)).
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n>3.
a(n) = (-7 + (-1)^(1+n) + 6*n^2) / 8.
(End)
E.g.f.: (1/8)*(exp(x)*(6*x^2 + 6*x - 7) - exp(-x)). - Stefano Spezia, Feb 02 2020 after Colin Barker
a(n) = floor((n^2 - 1)*3/4). - Michael Somos, Sep 09 2023

Extensions

a(42)-a(52) from Stefano Spezia, Feb 02 2020

A332410 a(n) = 2*a(n-1) - a(n-2) + a(n-5) - 2*a(n-6) + a(n-7) with a(0)=0, a(1)=1, a(2)=3, a(3)=6, a(4)=11, a(5)=17, a(6)=24.

Original entry on oeis.org

0, 1, 3, 6, 11, 17, 24, 32, 41, 52, 64, 77, 91, 106, 123, 141, 160, 180, 201, 224, 248, 273, 299, 326, 355, 385, 416, 448, 481, 516, 552, 589, 627, 666, 707, 749, 792, 836, 881, 928, 976, 1025, 1075, 1126, 1179
Offset: 0

Views

Author

Paul Curtz, Feb 11 2020

Keywords

Comments

This sequence occurs twice as a linear spoke in the hexagonal spiral constructed from A002266:
17 17 17 17 17 18 18
16 11 11 11 11 12 12 18
16 11 6 6 7 7 7 12 18
16 10 6 3 3 3 3 7 12 18
16 10 6 3 1 1 1 4 7 12 19
16 10 6 2 0 0 0 1 4 8 13 19
15 10 5 2 0 0 1 4 8 13 19
15 10 5 2 2 2 4 8 13 19
15 9 5 5 5 4 8 13 19
15 9 9 9 9 8 13 20
15 14 14 14 14 14 20
a(-1-n) = 0, 1, 4, 8, 13, 19, 26, 35, 45, ... also occurs twice in the same spiral.
Difference table:
0, 1, 3, 6, 11, 17, 24, 32, 41, 52, ... = a(n)
1, 2, 3, 5, 6, 7, 8, 9, 11, 12, ... = A047256(n+1)
1, 1, 2, 1, 1, 1, 1, 2, 1, 1, ... = A130782.
There is no linear spoke with three copies in this spiral. Compare with the spiral illustrated in sequence A330707 and constructed from A002265 where the same spokes occur three times: A006578, A001859 and A077043, essentially. Strictly, three times from 1, 1, 1 for A006578, from 2, 2, 2 for A001859 and from 7, 7, 7 for A077043.

Links

Crossrefs

Cf. A002266, A008602, A047256, A130782, A330707.
Cf. A001859, A006578, A077043.

Programs

  • Mathematica
    LinearRecurrence[{2, -1, 0, 0, 1, -2, 1}, {0, 1, 3, 6, 11, 17, 24}, 45] (* Amiram Eldar, Feb 12 2020 *)
  • PARI
    concat(0, Vec(x*(1 + x)*(1 + x^2 + x^3) / ((1 - x)^3*(1 + x + x^2 + x^3 + x^4)) + O(x^50))) \\ Colin Barker, Feb 11 2020, Apr 24 2020

Formula

a(8+n) - a(8-n) = 20*n.
G.f.: x*(1 + x)*(1 + x^2 + x^3) / ((1 - x)^3*(1 + x + x^2 + x^3 + x^4)). - Colin Barker, Feb 11 2020
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