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 1-3 of 3 results.

A247840 a(n) = Sum_{k=2..n} 6^k.

Original entry on oeis.org

0, 36, 252, 1548, 9324, 55980, 335916, 2015532, 12093228, 72559404, 435356460, 2612138796, 15672832812, 94036996908, 564221981484, 3385331888940, 20311991333676, 121871948002092, 731231688012588, 4387390128075564, 26324340768453420
Offset: 1

Views

Author

Vincenzo Librandi, Sep 25 2014

Keywords

Links

Crossrefs

Cf. similar sequences listed in A247817.
Cf. A000400, A003464, A105281.

Programs

  • Magma
    [0] cat [&+[6^k: k in [2..n]]: n in [2..30]];
  • Magma
    [(6^(n+1)-36)/5: n in [1..30]];
  • Mathematica
    RecurrenceTable[{a[1] == 0, a[n] == a[n-1] + 6^n}, a, {n, 30}] (* or *) CoefficientList[Series[36 x / ((1 - x) (1 - 6 x)), {x, 0, 30}], x]
Join[{0},Accumulate[6^Range[2,30]]] (* or *) LinearRecurrence[{7,-6},{0,36},30] (* Harvey P. Dale, Jun 11 2016 *)
  • PARI
    a(n) = sum(k=2, n, 6^k); \\ Michel Marcus, Sep 25 2014

Formula

G.f.: 36*x^2/((1-x)*(1-6*x)).
a(n) = a(n-1) + 6^n = (6^(n+1) - 36)/5 = 7*a(n-1) - 6*a(n-2).
a(n) = A105281(n) - 6. - Michel Marcus, Sep 25 2014
a(n) = 36 * A003464(n-1). - Alois P. Heinz, Jan 14 2025

A247841 a(n) = Sum_{k=2..n} 8^k.

Original entry on oeis.org

0, 64, 576, 4672, 37440, 299584, 2396736, 19173952, 153391680, 1227133504, 9817068096, 78536544832, 628292358720, 5026338869824, 40210710958656, 321685687669312, 2573485501354560, 20587884010836544, 164703072086692416, 1317624576693539392
Offset: 1

Views

Author

Vincenzo Librandi, Sep 25 2014

Keywords

Links

Crossrefs

Cf. similar sequences listed in A247817.
Cf. A052379.

Programs

  • Magma
    [0] cat [&+[8^k: k in [2..n]]: n in [2..30]];
  • Magma
    [(8^(n+1)-64)/7: n in [1..30]];
  • Mathematica
    RecurrenceTable[{a[1] == 0, a[n] == a[n-1] + 8^n}, a, {n, 30}] (* or *) CoefficientList[Series[64 x / ((1 - x) (1 - 8 x)), {x, 0, 30}], x]
LinearRecurrence[{9,-8},{0,64},30] (* Harvey P. Dale, May 01 2018 *)

Formula

G.f.: 64*x^2/((1-x)*(1-8*x)).
a(n) = a(n-1) + 8^n.
a(n) = (8^(n+1) - 64)/7.
a(n) = 9*a(n-1) - 8*a(n-2).
a(n) = A052379(n) - 8. - Michel Marcus, Sep 25 2014

A247842 a(n) = Sum_{k=2..n} 9^k.

Original entry on oeis.org

0, 81, 810, 7371, 66420, 597861, 5380830, 48427551, 435848040, 3922632441, 35303692050, 317733228531, 2859599056860, 25736391511821, 231627523606470, 2084647712458311, 18761829412124880, 168856464709124001, 1519708182382116090, 13677373641439044891
Offset: 1

Views

Author

Vincenzo Librandi, Sep 25 2014

Keywords

Links

Crossrefs

Cf. similar sequences listed in A247817.
Cf. A052386.

Programs

  • Magma
    [0] cat [&+[9^k: k in [2..n]]: n in [2..30]];
  • Magma
    [(9^(n+1)-81)/8: n in [1..30]];
  • Maple
    A247842:=n->add(9^k, k=2..n): seq(A247842(n), n=1..30); # Wesley Ivan Hurt, Sep 26 2014
  • Mathematica
    RecurrenceTable[{a[1] == 0, a[n] == a[n-1] + 9^n}, a, {n, 30}] (* or *) CoefficientList[Series[81 x / ((1 - x) (1 - 9 x)), {x, 0, 30}], x]

Formula

G.f.: 81*x^2/((1-x)*(1-9*x)).
a(n) = a(n-1) + 9^n = (9^(n+1) - 81)/8 = 10*a(n-1) - 9*a(n-2).
a(n) = A052386(n) - 9. - Michel Marcus, Sep 25 2014
Showing 1-3 of 3 results.

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