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.

Previous Showing 11-15 of 15 results.

A269429 Alternating sum of octagonal pyramidal numbers.

Original entry on oeis.org

0, -1, 8, -22, 48, -87, 144, -220, 320, -445, 600, -786, 1008, -1267, 1568, -1912, 2304, -2745, 3240, -3790, 4400, -5071, 5808, -6612, 7488, -8437, 9464, -10570, 11760, -13035, 14400, -15856, 17408, -19057, 20808, -22662, 24624, -26695, 28880, -31180, 33600
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 26 2016

Keywords

Links

Crossrefs

Cf. A000292, A002414, A002717, A173196, A266677.

Programs

  • Mathematica
    Table[((2 n^3 + 4 n^2 - 1) (-1)^n + 1)/4, {n, 0, 40}]
LinearRecurrence[{-3, -2, 2, 3, 1}, {0, -1, 8, -22, 48}, 41]
  • PARI
    a(n)=(2*n^3 + 4*n^2 - 1)*(-1)^n\/4 \\ Charles R Greathouse IV, Jul 26 2016

Formula

G.f.: x*(1 - 5*x)/((x - 1)*(x + 1)^4).
a(n) = ((2*n^3 + 4*n^2 - 1)*(-1)^n + 1)/4.
a(n) = Sum_{k = 0..n} (-1)^k*A002414(k).
Sum_{n>=1} 1/a(n) = -0.906890389180715042293808708467278316660747358... . - Vaclav Kotesovec, Feb 26 2016

A371972 a(n) is the number of distinct areas of triangles with integer sides whose largest side is n.

Original entry on oeis.org

1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36, 42, 49, 56, 64, 72, 81, 90, 100, 110, 120, 131, 144, 156, 168, 182, 196, 210, 225, 239, 256, 270, 288, 306, 321, 342, 361, 380, 399, 420, 441, 460, 484, 505, 527, 552, 576, 599, 623, 649, 673, 702, 729, 752, 781, 808, 840, 870, 900
Offset: 1

Views

Author

Hugo Pfoertner, Apr 16 2024

Keywords

Crossrefs

Cf. A316841, A371973, A371345.
See the formula section for the relationships with A002620, A173196, A316843, A316853.

Programs

  • PARI
    A2(a,b,c) = {my(s=(a+b+c)/2);s*(s-a)*(s-b)*(s-c)};
a371972(n) = {my (A=List()); for(s2=1,n, for(s3=1,s2, if(s2+s3>n, listput(A, A2(n,s2,s3))))); #Set(A)};

Formula

a(n) <= A002620(n+1), with equality for n <= 20.
a(n) = |{A316853(k) : A316843(k) = n}| = |{A316853(k) : A173196(n) < k <= A173196(n+1)}|. - Peter Munn, Jul 30 2025

A384063 Partial sums of A172471.

Original entry on oeis.org

0, 1, 3, 5, 7, 10, 13, 16, 20, 24, 28, 32, 36, 41, 46, 51, 56, 61, 67, 73, 79, 85, 91, 97, 103, 110, 117, 124, 131, 138, 145, 152, 160, 168, 176, 184, 192, 200, 208, 216, 224, 233, 242, 251, 260, 269, 278, 287, 296, 305, 315, 325, 335, 345, 355, 365, 375, 385, 395, 405, 415, 426, 437
Offset: 0

Views

Author

Hoang Xuan Thanh, May 18 2025

Keywords

Links

Crossrefs

Cf. A172471, A173196, A000217.

Programs

  • Mathematica
    Accumulate[Floor[Sqrt[2*Range[0, 100]]]] (* Paolo Xausa, Jun 04 2025 *)
  • PARI
    a(n) = sum(k=1, n, sqrtint(2*k)); \\ Michel Marcus, May 23 2025

Formula

a(n) = m*n - floor((m-1)*(m+3)*(2m-1)/12), where m = A172471(n).
a(n) = m*n - A000217(m-1) - 2*A173196(m-1), where m = A172471(n).

A269440 Alternating sum of 9-gonal (or decagonal) pyramidal numbers.

Original entry on oeis.org

0, -1, 9, -25, 55, -100, 166, -254, 370, -515, 695, -911, 1169, -1470, 1820, -2220, 2676, -3189, 3765, -4405, 5115, -5896, 6754, -7690, 8710, -9815, 11011, -12299, 13685, -15170, 16760, -18456, 20264, -22185, 24225, -26385, 28671, -31084, 33630, -36310, 39130
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 26 2016

Keywords

Links

Crossrefs

Cf. A000292, A002717, A007584, A173196, A266677.

Programs

  • Mathematica
    Table[(-1)^n (2 n - 1) ((14 n^2 + 34 n + 15)/48) + 5/16, {n, 0, 40}]
LinearRecurrence[{-3, -2, 2, 3, 1}, {0, -1, 9, -25, 55}, 41]

Formula

G.f.: x*(1 - 6*x)/((x - 1)*(x + 1)^4).
a(n) = (-1)^n*(2*n - 1)*(14*n^2 + 34*n + 15)/48 + 5/16.
a(n) = Sum_{k = 0..n} (-1)^k*A007584(k).

A269441 Alternating sum of 10-gonal (or decagonal) pyramidal numbers.

Original entry on oeis.org

0, -1, 10, -28, 62, -113, 188, -288, 420, -585, 790, -1036, 1330, -1673, 2072, -2528, 3048, -3633, 4290, -5020, 5830, -6721, 7700, -8768, 9932, -11193, 12558, -14028, 15610, -17305, 19120, -21056, 23120, -25313, 27642, -30108, 32718, -35473, 38380, -41440, 44660
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 26 2016

Keywords

Links

Crossrefs

Cf. A000292, A002717, A007585, A173196, A266677, A269428, A269429.

Programs

  • Magma
    [((-1)^n*(16*n^3+30*n^2-4*n-9)+9)/24: n in [0..40]]; // Vincenzo Librandi, Feb 27 2016
  • Mathematica
    Table[((-1)^n (16 n^3 + 30 n^2 - 4 n - 9) + 9)/24, {n, 0, 40}]
LinearRecurrence[{-3, -2, 2, 3, 1}, {0, -1, 10, -28, 62}, 41]

Formula

G.f.: x*(1 - 7*x)/((x - 1)*(x + 1)^4).
a(n) = ((-1)^n*(16*n^3 + 30*n^2 - 4*n - 9) + 9) /24.
a(n) = Sum_{k = 0..n} (-1)^k*A007585(k).
Sum_{n>=1} 1/a(n) = -0.9251958836055717745244669... . - Vaclav Kotesovec, Feb 26 2016
Previous Showing 11-15 of 15 results.

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