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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A261032 a(n) = (-1)^n*(n^8 + 4*n^7 - 14*n^5 + 28*n^3 - 17*n)/2.

Original entry on oeis.org

0, -1, 255, -6306, 59230, -331395, 1348221, -4416580, 12360636, -30686085, 69313915, -145044966, 284936730, -530793991, 944995065, -1617895560, 2677071736, -4298685705, 6721274871, -10262288170, 15337711830, -22485147531, 32390726005, -45920259276, 64155054900, -88432835725
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 18 2015

Keywords

Comments

Alternating sum of eighth powers (A001016).
For n>0, a(n) is divisible by A000217(n).

Examples

			a(0) = 0^8 = 0,
a(1) = 0^8 -1^8 = -1,
a(2) = 0^8 -1^8 + 2^8 = 255,
a(3) = 0^8 -1^8 + 2^8 - 3^8 = -6306,
a(4) = 0^8 -1^8 + 2^8 - 3^8 + 4^8 = 59230,
a(5) = 0^8 -1^8 + 2^8 - 3^8 + 4^8 - 5^8 = -331395, etc.

Links

Crossrefs

Cf. A000217, A001016, A000542, A089594, A232599, A062392, A062393, A152725, A152726.

Programs

  • Magma
    [(-1)^n*(n^8+4*n^7-14*n^5+28*n^3-17*n)/2: n in [0..30]]; // Vincenzo Librandi, Nov 20 2015
  • Maple
    seq((-1)^n*(n^8 + 4*n^7 - 14*n^5 + 28*n^3 - 17*n)/2, n = 0 .. 100); # Robert Israel, Nov 18 2015
  • Mathematica
    Table[(1/2) (-1)^n n (n + 1) (n^6 + 3 n^5 - 3 n^4 - 11 n^3 + 11 n^2 + 17 n - 17), {n, 0, 25}]
  • PARI
    vector(100, n, n--; (-1)^n*(n^8+4*n^7-14*n^5+28*n^3-17*n)/2) \\ Altug Alkan, Nov 18 2015
  • Sage
    [(-1)^n*(n^8 +4*n^7 -14*n^5 +28*n^3 -17*n)/2 for n in (0..40)] # G. C. Greubel, Apr 02 2021

Formula

G.f.: -x*(1 - 246*x + 4047*x^2 - 11572*x^3 + 4047*x^4 - 246*x^5 + x^6)/(1 + x)^9.
a(n) = Sum_{k = 0..n} (-1)^k*k^8.
a(n) = (-1)^n*n*(n + 1)*(n^6 + 3*n^5 - 3*n^4 - 11*n^3 + 11*n^2 + 17*n - 17)/2.
Sum_{n>0} 1/a(n) = -0.9962225712723456482...
Sum_{j=0..9} binomial(9,j)*a(n-j) = 0. - Robert Israel, Nov 18 2015
E.g.f.: (x/2)*(-2 +253*x -1848*x^2 +2961*x^3 -1596*x^4 +350*x^5 -32*x^6 +x^7)*exp(-x). - G. C. Greubel, Apr 02 2021

A270694 Alternating sum of centered heptagonal pyramidal numbers.

Original entry on oeis.org

0, -1, 8, -23, 51, -94, 157, -242, 354, -495, 670, -881, 1133, -1428, 1771, -2164, 2612, -3117, 3684, -4315, 5015, -5786, 6633, -7558, 8566, -9659, 10842, -12117, 13489, -14960, 16535, -18216, 20008, -21913, 23936, -26079, 28347, -30742, 33269
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 21 2016

Keywords

Comments

More generally, the ordinary generating function for the alternating sum of centered k-gonal pyramidal numbers is -x*(1 - (k - 2)*x + x^2)/((1 - x)*(1 + x)^4).

Links

Crossrefs

Cf. A004126 (centered heptagonal pyramidal numbers).
Cf. A000330, A006323 (partial sums of centered heptagonal pyramidal numbers), A019298, A232599.

Programs

  • Magma
    [((-1)^n*(2*n + 1)*(14*n^2 + 14*n - 9) + 9)/48 : n in [0..40]]; // Wesley Ivan Hurt, Mar 21 2016
  • Maple
    A270694:= n-> ((-1)^n*(2*n+1)*(14*n^2+14*n-9) + 9)/48; seq(A270694(n), n=0..40); # G. C. Greubel, Apr 02 2021
  • Mathematica
    LinearRecurrence[{-3, -2, 2, 3, 1}, {0, -1, 8, -23, 51}, 39]
Table[((-1)^n (2 n + 1) (14 n^2 + 14 n - 9) + 9)/48, {n, 0, 38}]
  • PARI
    my(x='x+O('x^50)); concat(0, Vec(-x*(1-5*x+x^2)/((1-x)*(1+x)^4))) \\ Altug Alkan, Mar 21 2016
  • Sage
    [((-1)^n*(2*n+1)*(14*n^2+14*n-9) +9)/48 for n in (0..40)] # G. C. Greubel, Apr 02 2021

Formula

G.f.: -x*(1 - 5*x + x^2)/((1 - x)*(1 + x)^4).
a(n) = -3*a(n-1) - 2*a(n-2) + 2*a(n-3) + 3*a(n-4) + a(n-5).
a(n) = ((-1)^n*(2*n + 1)*(14*n^2 + 14*n - 9) + 9)/48.
E.g.f.: (1/48)*(9*exp(x) - (9 + 66*x - 126*x^2 + 28*x^3)*exp(-x)). - G. C. Greubel, Mar 28 2016

A270695 Alternating sum of centered octagonal pyramidal numbers.

Original entry on oeis.org

0, -1, 9, -26, 58, -107, 179, -276, 404, -565, 765, -1006, 1294, -1631, 2023, -2472, 2984, -3561, 4209, -4930, 5730, -6611, 7579, -8636, 9788, -11037, 12389, -13846, 15414, -17095, 18895, -20816, 22864, -25041, 27353, -29802, 32394, -35131, 38019
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 21 2016

Keywords

Links

Crossrefs

Cf. A000447 (centered octagonal pyramidal numbers).
Cf. A000330, A006324 (partial sums of centered octagonal pyramidal numbers), A019298, A232599.

Programs

  • Magma
    [((-1)^n*(4*n^2 - 1)*(2*n + 3) + 3)/12 : n in [0..40]]; // Wesley Ivan Hurt, Mar 21 2016
  • Maple
    A270695:= n-> ((-1)^n*(4*n^2 -1)*(2*n+3) +3)/12: seq(A270695(n), n=0..40); # G. C. Greubel, Apr 02 2021
  • Mathematica
    LinearRecurrence[{-3, -2, 2, 3, 1}, {0, -1, 9, -26, 58}, 39]
Table[((-1)^n (4 n^2 - 1) (2 n + 3) + 3)/12, {n, 0, 38}]
  • PARI
    x='x+O('x^100); concat(0, Vec(-x*(1-6*x+x^2)/((1-x)*(1+x)^4))) \\ Altug Alkan, Mar 21 2016
  • Sage
    [((-1)^n*(4*n^2 -1)*(2*n+3) +3)/12 for n in (0..40)] # G. C. Greubel, Apr 02 2021

Formula

G.f.: -x*(1 - 6*x + x^2)/((1 - x)*(1 + x)^4).
E.g.f.: (1/12)*(3*exp(x) - (3 + 18*x - 36*x^2 + 8*x^3)*exp(-x)).
a(n) = -3*a(n-1) - 2*a(n-2) + 2*a(n-3) + 3*a(n-4) + a(n-5).
a(n) = ((-1)^n*(4*n^2 - 1)*(2*n + 3) + 3)/12.

Extensions

a(6)=179 inserted by Georg Fischer, Apr 03 2019
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