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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.

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

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 1-2 of 2 results.

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