A269025 a(n) = Sum_{k = 0..n} 60^k.

1, 61, 3661, 219661, 13179661, 790779661, 47446779661, 2846806779661, 170808406779661, 10248504406779661, 614910264406779661, 36894615864406779661, 2213676951864406779661, 132820617111864406779661, 7969237026711864406779661, 478154221602711864406779661

Offset: 0

Author

Ilya Gutkovskiy, Feb 18 2016

Keywords

• nonn
• easy

Comments

Partial sums of powers of 60 (A159991).

Converges in a 10-adic sense to ...762711864406779661.

More generally, the ordinary generating function for the Sum_{k = 0..n} m^k is 1/((1 - m*x)*(1 - x)). Also, Sum_{k = 0..n} m^k = (m^(n + 1) - 1)/(m - 1).

Links

• Index entries related to partial sums
• Index entries for linear recurrences with constant coefficients, signature (61,-60).

Crossrefs

Cf. A159991.

Cf. similar sequences of the form (k^n-1)/(k-1): A000225 (k=2), A003462 (k=3), A002450 (k=4), A003463 (k=5), A003464 (k=6), A023000 (k=7), A023001 (k=8), A002452 (k=9), A002275 (k=10), A016123 (k=11), A016125 (k=12), A091030 (k=13), A135519 (k=14), A135518 (k=15), A131865 (k=16), A091045 (k=17), A218721 (k=18), A218722 (k=19), A064108 (k=20), A218724-A218734 (k=21..31), A132469 (k=32), A218736-A218753 (k=33..50), this sequence (k=60), A133853 (k=64), A094028 (k=100), A218723 (k=256), A261544 (k=1000).

Programs

Mathematica

Table[Sum[60^k, {k, 0, n}], {n, 0, 15}]
Table[(60^(n + 1) - 1)/59, {n, 0, 15}]
LinearRecurrence[{61, -60}, {1, 61}, 15]

PARI

a(n)=60^n + 60^n\59 \\ Charles R Greathouse IV, Jul 26 2016

Formula

G.f.: 1/((1 - 60*x)*(1 - x)).

a(n) = (60^(n + 1) - 1)/59 = 60^n + floor(60^n/59).

a(n+1) = 60*a(n) + 1, a(0)=1.

a(n) = Sum_{k = 0..n} A159991(k).

Sum_{n>=0} 1/a(n) = 1.016671221665660580331...

E.g.f.: exp(x)*(60*exp(59*x) - 1)/59. - Stefano Spezia, Mar 23 2023