A121544 Sum of all proper base 4 numbers with n digits (those not beginning with 0).

6, 114, 1896, 30624, 491136, 7862784, 125822976, 2013241344, 32212156416, 515395682304, 8246335635456, 131941389041664, 2111062300164096, 33776997104615424, 540431954881806336, 8646911282940739584, 138350580546379186176, 2213609288819376390144

Offset: 1

Author

Jonathan Vos Post, Sep 08 2006

Keywords

• easy
• nonn
• base

Comments

Sum of the first 3 * 4^(n-1) integers starting with 4^(n-1).

Sum of the integers from 4^(n-1) to 4^n -1.

First differences of A026337.

Examples

			a(1) = 6 = 1 + 2 + 3.
a(2) = 114 = 10_4 + 11_4 + 12_4 + 13_4 + 20_4 + 21_4 + 22_4 + 23_4 + 30_4 + 31_4 + 32_4 + 33_4 = (4+5+6+7+8+9+10+11+12+13+14+15)_10.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..820
• Index entries for linear recurrences with constant coefficients, signature (20,-64).

Crossrefs

Cf. A007090, A010036, A026121, A026337, A101291.

Programs

Magma

[3*Binomial(5*4^(n-1), 2)/5: n in [1..20]]; // G. C. Greubel, Nov 07 2024

Mathematica

Table[3*4^(n-1)*(5*4^(n-1) - 1)/2, {n,20}] (* James C. McMahon, Oct 19 2024 *)

SageMath

def A121544(n): return 3*binomial(5*4^(n-1), 2)//5
[A121544(n) for n in range(1,21)] # G. C. Greubel, Nov 07 2024

Formula

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

G.f.: 6*x*(1-x) / ((1-4*x)*(1-16*x)). - Colin Barker, Apr 30 2013

From G. C. Greubel, Nov 07 2024: (Start)

a(n) = (3/5)*binomial(5*4^(n-1), 2).

E.g.f.: (3/32)*(-1 - 4*exp(4*x) + 5*exp(16*x)). (End)

Extensions

More terms from Colin Barker, Apr 30 2013

Edited by Michel Marcus, Apr 15 2024