A247817 Sum(4^k, k=2..n).

0, 16, 80, 336, 1360, 5456, 21840, 87376, 349520, 1398096, 5592400, 22369616, 89478480, 357913936, 1431655760, 5726623056, 22906492240, 91625968976, 366503875920, 1466015503696, 5864062014800, 23456248059216, 93824992236880, 375299968947536, 1501199875790160

Offset: 1

Author

Vincenzo Librandi, Sep 25 2014

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..200
• Index entries for linear recurrences with constant coefficients, signature (5,-4).

Crossrefs

Cf. Sum(h^k,k=2..n): A028329 (h=2), A168569 (h=3), this sequence (h=4), A168571 (h=5), A247840 (h=6), A168572 (h=7), A247841 (h=8), A247842 (h=9), A124166 (h=10).

Programs

Magma

[0] cat [&+[4^k: k in [2..n]]: n in [2..30]];

Magma

[(4^(n+1)-16)/3: n in [1..30]];

Mathematica

RecurrenceTable[{a[1] == 0, a[n] == a[n-1] + 4^n}, a, {n, 30}] (* or *) CoefficientList[ Series[16 x / ((1 - x) (1 - 4 x)),{x, 0, 40}], x]
LinearRecurrence[{5,-4},{0,16},30] (* Harvey P. Dale, Feb 19 2023 *)

PARI

a(n) = sum(k=2, n, 4^k); \\ Michel Marcus, Sep 25 2014

Formula

G.f.: 16*x^2/((1-x)*(1-4*x)).

a(n) = a(n-1) + 4^n = (4^(n+1) - 16)/3 = 5*a(n-1) - 4*a(n-2).

a(n) = A080674(n) - 4. - Michel Marcus, Sep 25 2014