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 1-6 of 6 results.

A319373 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, 2, -1, -10, -5, 20, 13, -36, -27, 54, 43, -78, -65, 104, 89, -136, -119, 170, 151, -210, -189, 252, 229, -300, -275, 350, 323, -406, -377, 464, 433, -528, -495, 594, 559, -666, -629, 740, 701, -820, -779, 902, 859, -990, -945, 1080, 1033, -1176, -1127
Offset: 1

Views

Author

Wesley Ivan Hurt, Sep 17 2018

Keywords

Comments

In general, for alternating sequences that multiply the first k natural numbers, and subtract/add the products of the next k natural numbers (preserving the order of operations up to n), we have a(n) = (-1)^floor(n/k) * Sum_{i=1..k-1} (1-sign((n-i) mod k)) * (Product_{j=1..i} (n-j+1)) + Sum_{i=1..n} (-1)^(floor(i/k)+1) * (1-sign(i mod k)) * (Product_{j=1..k} (i-j+1)). Here k=2.
An alternating version of A228958.

Examples

			a(1) = 1;
a(2) = 1*2 = 2;
a(3) = 1*2 - 3 = -1;
a(4) = 1*2 - 3*4 = -10;
a(5) = 1*2 - 3*4 + 5 = -5;
a(6) = 1*2 - 3*4 + 5*6 = 20;
a(7) = 1*2 - 3*4 + 5*6 - 7 = 13;
a(8) = 1*2 - 3*4 + 5*6 - 7*8 = -36;
a(9) = 1*2 - 3*4 + 5*6 - 7*8 + 9 = -27;
a(10) = 1*2 - 3*4 + 5*6 - 7*8 + 9*10 = 54;
a(11) = 1*2 - 3*4 + 5*6 - 7*8 + 9*10 - 11 = 43;
a(12) = 1*2 - 3*4 + 5*6 - 7*8 + 9*10 - 11*12 = -78;
a(13) = 1*2 - 3*4 + 5*6 - 7*8 + 9*10 - 11*12 + 13 = -65;
a(14) = 1*2 - 3*4 + 5*6 - 7*8 + 9*10 - 11*12 + 13*14 = 104;
a(15) = 1*2 - 3*4 + 5*6 - 7*8 + 9*10 - 11*12 + 13*14 - 15 = 89; etc.

Links

Crossrefs

Cf. A093361, A228958, A305189.
For similar sequences, see: A001057 (k=1), this sequence (k=2), A319543 (k=3), A319544 (k=4), A319545 (k=5), A319546 (k=6), A319547 (k=7), A319549 (k=8), A319550 (k=9), A319551 (k=10).

Programs

  • Mathematica
    Table[(Cos[n Pi/2] (1 - n - n^2) + Sin[n Pi/2] (1 + 3 n - n^2) - 1)/2, {n, 50}]
a[n_] := (-1)^Floor[n/2] Sum[(1 - Sign[Mod[n - i, 2]]) Product[n - j + 1, {j, 1, i}], {i, 1, 1}] + Sum[(-1)^(Floor[i/2] + 1) (1 - Sign[Mod[i, 2]]) Product[i - j + 1, {j, 1, 2}], {i, 1, n}]; Array[a, 30] (* Stefano Spezia, Sep 23 2018 *)
  • PARI
    Vec(x*(1 + x - 6*x^3 - x^4 + x^5) / ((1 - x)*(1 + x^2)^3) + O(x^50)) \\ Colin Barker, Sep 18 2018

Formula

a(n) = (cos(n*Pi/2)*(1-n-n^2) + sin(n*Pi/2)*(1+3*n-n^2) - 1)/2.
From Colin Barker, Sep 18 2018: (Start)
G.f.: x*(1 + x - 6*x^3 - x^4 + x^5) / ((1 - x)*(1 + x^2)^3).
a(n) = a(n-1) - 3*a(n-2) + 3*a(n-3) - 3*a(n-4) + 3*a(n-5) - a(n-6) + a(n-7) for n>7. (End)
a(n) = (-1 + (-1)^((n-1)*n/2))/2 + (-2 + (-1)^n)*(-1)^(n*(n+1)/2)*n/2 - (-1)^((n-1)*n/2)*n^2/2. - Bruno Berselli, Sep 25 2018

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)

A319391 a(n) = (1 + 2)^3 + (4 + 5)^6 + (7 + 8)^9 + (10 + 11)^12 + ... + (up to n).

Original entry on oeis.org

1, 3, 27, 31, 36, 531468, 531475, 531483, 38443890843, 38443890853, 38443890864, 7355865955277484, 7355865955277497, 7355865955277511, 2954320062416788976127, 2954320062416788976143, 2954320062416788976160, 2154028838712789034859190336
Offset: 1

Views

Author

Wesley Ivan Hurt, Sep 18 2018

Keywords

Examples

			a(1) = 1;
a(2) = 1 + 2 = 3;
a(3) = (1 + 2)^3 = 27;
a(4) = (1 + 2)^3 + 4 = 31;
a(5) = (1 + 2)^3 + 4 + 5 = 36;
a(6) = (1 + 2)^3 + (4 + 5)^6 = 531468;
a(7) = (1 + 2)^3 + (4 + 5)^6 + 7 = 531475;
a(8) = (1 + 2)^3 + (4 + 5)^6 + 7 + 8 = 531483;
a(9) = (1 + 2)^3 + (4 + 5)^6 + (7 + 8)^9 = 38443890843;
a(10) = (1 + 2)^3 + (4 + 5)^6 + (7 + 8)^9 + 10 = 38443890853; etc.

Links

Crossrefs

Cf. A305189, A318868, A319014, A319258.

Programs

  • Maple
    f:= proc(n) option remember;
  if n mod 3 = 0 then procname(n-3)+(2*n-3)^n
  else procname(n-1)+n
  fi
end proc:
f(0):= 0:
map(f, [$1..20]); # Robert Israel, Oct 05 2018
  • Mathematica
    Table[Sum[(Floor[i/3] - Floor[(i - 1)/3])*(6*Floor[(i + 2)/3] - 3)^(3*Floor[(i + 2)/3]) + i*(Floor[(i - 1)/3] - Floor[(i - 2)/3]) + i*(Floor[(i + 1)/3] - Floor[i/3]) - (6*Floor[(i + 2)/3] - 3)*(Floor[i/3] - Floor[(i - 1)/3]), {i, n}], {n, 20}]

Formula

a(n) = Sum_{i=1..n} (floor(i/3)-floor((i-1)/3))*(6*floor((i+2)/3)-3)^(3*floor((i+2)/3)) + i*(floor((i-1)/3)-floor((i-2)/3))+i*(floor((i+1)/3)-floor(i/3))-(6*floor((i+2)/3)-3)*(floor(i/3)-floor((i-1)/3)).
If 3|n then a(n) = a(n-3)+(2*n-3)^n, otherwise a(n) = a(n-1)+n. - Robert Israel, Oct 05 2018

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).

A319493 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, 2, -1, 3, 19, 13, 20, 69, 60, 70, 170, 158, 171, 340, 325, 341, 597, 579, 598, 959, 938, 960, 1444, 1420, 1445, 2070, 2043, 2071, 2855, 2825, 2856, 3817, 3784, 3818, 4974, 4938, 4975, 6344, 6305, 6345, 7945, 7903, 7946, 9795, 9750, 9796, 11912, 11864, 11913
Offset: 1

Views

Author

Wesley Ivan Hurt, Sep 20 2018

Keywords

Examples

			a(1) = 1;
a(2) = 1*2 = 2;
a(3) = 1*2 - 3 = -1;
a(4) = 1*2 - 3 + 4 = 3;
a(5) = 1*2 - 3 + 4*5 = 19;
a(6) = 1*2 - 3 + 4*5 - 6 = 13;
a(7) = 1*2 - 3 + 4*5 - 6 + 7 = 20;
a(8) = 1*2 - 3 + 4*5 - 6 + 7*8 = 69;
a(9) = 1*2 - 3 + 4*5 - 6 + 7*8 - 9 = 60;
a(10) = 1*2 - 3 + 4*5 - 6 + 7*8 - 9 + 10 = 70;
a(11) = 1*2 - 3 + 4*5 - 6 + 7*8 - 9 + 10*11 = 170;
a(12) = 1*2 - 3 + 4*5 - 6 + 7*8 - 9 + 10*11 - 12 = 158;
a(13) = 1*2 - 3 + 4*5 - 6 + 7*8 - 9 + 10*11 - 12 + 13 = 171;
a(14) = 1*2 - 3 + 4*5 - 6 + 7*8 - 9 + 10*11 - 12 + 13*14 = 340; etc.

Links

Crossrefs

Cf. A305189, A319258.

Programs

  • Mathematica
    Table[Floor[(n + 1)/3]*(3*Floor[(n + 1)/3]^2 - 1) + n*(Floor[(n - 1)/3] - Floor[(n - 2)/3]) - 3*Floor[n/3]*(Floor[n/3] + 1)/2, {n, 50}]
CoefficientList[Series[(1 + x)*(1 - 3*x^2 + 4*x^3 + 9*x^4 - 6*x^5 + 4*x^6)/((1 - x)^4*(1 + x + x^2)^3), {x, 0, 50}], x] (* Stefano Spezia, Sep 23 2018 *)
  • PARI
    Vec(x*(1 + x)*(1 - 3*x^2 + 4*x^3 + 9*x^4 - 6*x^5 + 4*x^6) / ((1 - x)^4*(1 + x + x^2)^3) + O(x^50)) \\ Colin Barker, Sep 20 2018

Formula

a(n) = floor((n + 1)/3)*(3*floor((n + 1)/3)^2 - 1) + n*(floor((n - 1)/3) - floor((n - 2)/3)) - 3*floor(n/3)*(floor(n/3) + 1)/2.
From Colin Barker, Sep 20 2018: (Start)
G.f.: x*(1 + x)*(1 - 3*x^2 + 4*x^3 + 9*x^4 - 6*x^5 + 4*x^6) / ((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)
Showing 1-6 of 6 results.

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