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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A305189 a(n) = 1*2 + 3 + 4*5 + 6 + 7*8 + 9 + 10*11 + 12 + ... + (up to n).

Original entry on oeis.org

1, 2, 5, 9, 25, 31, 38, 87, 96, 106, 206, 218, 231, 400, 415, 431, 687, 705, 724, 1085, 1106, 1128, 1612, 1636, 1661, 2286, 2313, 2341, 3125, 3155, 3186, 4147, 4180, 4214, 5370, 5406, 5443, 6812, 6851, 6891, 8491, 8533, 8576, 10425, 10470, 10516, 12632
Offset: 1

Views

Author

Wesley Ivan Hurt, Sep 15 2018

Keywords

Examples

			a(1) = 1;
a(2) = 1*2 = 2;
a(3) = 1*2 + 3 = 5;
a(4) = 1*2 + 3 + 4 = 9;
a(5) = 1*2 + 3 + 4*5 = 25;
a(6) = 1*2 + 3 + 4*5 + 6 = 31;
a(7) = 1*2 + 3 + 4*5 + 6 + 7 = 38;
a(8) = 1*2 + 3 + 4*5 + 6 + 7*8 = 87;
a(9) = 1*2 + 3 + 4*5 + 6 + 7*8 + 9 = 96;
a(10) = 1*2 + 3 + 4*5 + 6 + 7*8 + 9 + 10 = 106;
a(11) = 1*2 + 3 + 4*5 + 6 + 7*8 + 9 + 10*11 = 206;
a(12) = 1*2 + 3 + 4*5 + 6 + 7*8 + 9 + 10*11 + 12 = 218; etc.

Links

Crossrefs

Cf. A093361, A228958, A319014.

Programs

  • Maple
    seq(coeff(series((x*(1+x+3*x^2+x^3+13*x^4-3*x^5-2*x^6+4*x^7))/((1-x)^4*(1+x+x^2)^3),x,n+1), x, n), n = 1 .. 50); # Muniru A Asiru, Sep 16 2018
  • Mathematica
    Table[3*Floor[n/3]*(Floor[n/3] + 1)/2 + Floor[(n + 1)/3]*(3*Floor[(n + 1)/3]^2 - 1) + n*(Floor[(n - 1)/3] - Floor[(n - 2)/3]), {n, 50}]
LinearRecurrence[{1,0,3,-3,0,-3,3,0,1,-1 }, {1, 2, 5, 9, 25, 31, 38, 87, 96, 106}, 50] (* Stefano Spezia, Sep 16 2018 *)
  • PARI
    Vec(x*(1 + x + 3*x^2 + x^3 + 13*x^4 - 3*x^5 - 2*x^6 + 4*x^7) / ((1 - x)^4*(1 + x + x^2)^3) + O(x^40)) \\ Colin Barker, Sep 16 2018

Formula

a(n) = 3*floor(n/3)*(floor(n/3) + 1)/2 + floor((n+1)/3)*(3*floor((n+1)/3)^2 - 1) + n*(floor((n-1)/3) - floor((n-2)/3)).
From Colin Barker, Sep 16 2018: (Start)
G.f.: x*(1 + x + 3*x^2 + x^3 + 13*x^4 - 3*x^5 - 2*x^6 + 4*x^7) / ((1 - x)^4*(1 + x + x^2)^3).
a(n) = a(n-1) + 3*a(n-3) - 3*a(n-4) - 3*a(n-6) + 3*a(n-7) + a(n-9) - a(n-10) for n>10.
(End)

A318868 a(n) = 1^2 + 3^4 + 5^6 + 7^8 + 9^10 + 11^12 + 13^14 + ... + (up to n).

Original entry on oeis.org

1, 1, 4, 82, 87, 15707, 15714, 5780508, 5780517, 3492564909, 3492564920, 3141920941630, 3141920941643, 3940518306640919, 3940518306640934, 6572348874019531544, 6572348874019531561, 14069656800941744522553, 14069656800941744522572, 37604043114346899937878154
Offset: 1

Views

Author

Wesley Ivan Hurt, Sep 16 2018

Keywords

Examples

			a(1) = 1;
a(2) = 1^2 = 1;
a(3) = 1^2 + 3 = 4;
a(4) = 1^2 + 3^4 = 82;
a(5) = 1^2 + 3^4 + 5 = 87;
a(6) = 1^2 + 3^4 + 5^6 = 15707;
a(7) = 1^2 + 3^4 + 5^6 + 7 = 15714;
a(8) = 1^2 + 3^4 + 5^6 + 7^8 = 5780508;
a(9) = 1^2 + 3^4 + 5^6 + 7^8 + 9 = 5780517;
a(10) = 1^2 + 3^4 + 5^6 + 7^8 + 9^10 = 3492564909;
a(11) = 1^2 + 3^4 + 5^6 + 7^8 + 9^10 + 11 = 3492564920;
a(12) = 1^2 + 3^4 + 5^6 + 7^8 + 9^10 + 11^12 = 3141920941630, etc.

Links

Crossrefs

Cf. A093361, A228958, A305189.

Programs

  • Mathematica
    Table[(2*Floor[(n - 1)/2] + 1)*Mod[n, 2] + Sum[(2*i - 1)^(2*i), {i, Floor[n/2]}], {n, 25}]
  • PARI
    a(n) = (2*((n-1)\2) + 1)*(n % 2) + sum(i=1, n\2, (2*i - 1)^(2*i)); \\ Michel Marcus, Sep 18 2018

Formula

a(n) = (2*floor((n-1)/2) + 1)*(n mod 2) + Sum_{i=1..floor(n/2)} (2*i - 1)^(2*i).

A319258 a(n) = 1 + 2*3 + 4 + 5*6 + 7 + 8*9 + 10 + 11*12 + ... + (up to n).

Original entry on oeis.org

1, 3, 7, 11, 16, 41, 48, 56, 120, 130, 141, 262, 275, 289, 485, 501, 518, 807, 826, 846, 1246, 1268, 1291, 1820, 1845, 1871, 2547, 2575, 2604, 3445, 3476, 3508, 4532, 4566, 4601, 5826, 5863, 5901, 7345, 7385, 7426, 9107, 9150, 9194, 11130, 11176, 11223
Offset: 1

Views

Author

Wesley Ivan Hurt, Sep 16 2018

Keywords

Examples

			a(1) = 1;
a(2) = 1 + 2 = 3;
a(3) = 1 + 2*3 = 7;
a(4) = 1 + 2*3 + 4 = 11;
a(5) = 1 + 2*3 + 4 + 5 = 16;
a(6) = 1 + 2*3 + 4 + 5*6 = 41;
a(7) = 1 + 2*3 + 4 + 5*6 + 7 = 48;
a(8) = 1 + 2*3 + 4 + 5*6 + 7 + 8 = 56;
a(9) = 1 + 2*3 + 4 + 5*6 + 7 + 8*9 = 120;
a(10) = 1 + 2*3 + 4 + 5*6 + 7 + 8*9 + 10 = 130;
a(11) = 1 + 2*3 + 4 + 5*6 + 7 + 8*9 + 10 + 11 = 141;
a(12) = 1 + 2*3 + 4 + 5*6 + 7 + 8*9 + 10 + 11*12 = 262; etc.

Links

Crossrefs

Cf. A093361, A228958, A305189, A319014.

Programs

  • Mathematica
    Table[n (1 + Floor[(n - 2)/3] - Floor[n/3]) + 3 Floor[n/3]^2 (1 + Floor[n/3]) + Floor[(n + 2)/3] (3 Floor[(n + 2)/3] - 1)/2, {n, 50}]
  • PARI
    Vec(x*(1 + 2*x + 4*x^2 + x^3 - x^4 + 13*x^5 - 2*x^6 - x^7 + x^8) / ((1 - x)^4*(1 + x + x^2)^3) + O(x^40)) \\ Colin Barker, Sep 16 2018

Formula

a(n) = n*(1 + floor((n-2)/3) - floor(n/3)) + 3*floor(n/3)^2*(1 + floor(n/3)) + floor((n+2)/3)*(3*floor((n+2)/3) - 1)/2.
From Colin Barker, Sep 16 2018: (Start)
G.f.: x*(1 + 2*x + 4*x^2 + x^3 - x^4 + 13*x^5 - 2*x^6 - x^7 + x^8) / ((1 - x)^4*(1 + x + x^2)^3).
a(n) = a(n-1) + 3*a(n-3) - 3*a(n-4) - 3*a(n-6) + 3*a(n-7) + a(n-9) - a(n-10) for n>10.
(End)

A319438 a(n) = 1^2 - 3^4 + 5^6 - 7^8 + 9^10 - 11^12 + 13^14 - ... + (up to n).

Original entry on oeis.org

1, 1, -2, -80, -75, 15545, 15538, -5749256, -5749247, 3481035145, 3481035134, -3134947341576, -3134947341563, 3934241438357713, 3934241438357698, -6564474114274532912, -6564474114274532895, 14056519977953450458097, 14056519977953450458078
Offset: 1

Views

Author

Wesley Ivan Hurt, Sep 18 2018

Keywords

Comments

An alternating version of A318868.

Examples

			   a(1) = 1;
   a(2) = 1^2 = 1;
   a(3) = 1^2 - 3 = -2;
   a(4) = 1^2 - 3^4 = -80;
   a(5) = 1^2 - 3^4 + 5 = -75;
   a(6) = 1^2 - 3^4 + 5^6 = 15545;
   a(7) = 1^2 - 3^4 + 5^6 - 7 = 15538;
   a(8) = 1^2 - 3^4 + 5^6 - 7^8 = -5749256;
   a(9) = 1^2 - 3^4 + 5^6 - 7^8 + 9 = -5749247;
  a(10) = 1^2 - 3^4 + 5^6 - 7^8 + 9^10 = 3481035145;
  a(11) = 1^2 - 3^4 + 5^6 - 7^8 + 9^10 - 11 = 3481035134;
  a(12) = 1^2 - 3^4 + 5^6 - 7^8 + 9^10 - 11^12 = -3134947341576; etc .

Links

Crossrefs

Cf. A093361, A228958, A305189, A318868.

Programs

  • Mathematica
    Table[n*Mod[n, 2]*(-1)^(Floor[n/2]) + Sum[(2*i - 1)^(2*i)*(-1)^(i - 1), {i, Floor[n/2]}], {n, 30}]

Formula

a(n) = n*(n mod 2)*(-1)^floor(n/2) + Sum_{i=1..floor(n/2)} (2*i - 1)^(2*i)*(-1)^(i - 1).
Previous Showing 11-14 of 14 results.

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