A098704 Decimal form of the binary numbers 10, 100010, 1000100010, 10001000100010, 100010001000100010,...

2, 34, 546, 8738, 139810, 2236962, 35791394, 572662306, 9162596898, 146601550370, 2345624805922, 37529996894754, 600479950316066, 9607679205057058, 153722867280912930, 2459565876494606882

Offset: 2

Author

Simone Severini, Oct 26 2004

Keywords

• nonn
• base
• easy

Comments

Decimal form of the hexadecimal numbers 2, 22, 222, 2222, 22222, 222222, ...; e.g., 2*16^0 + 2*16^1 = 2 + 32 = 34. - Zerinvary Lajos, Feb 01 2007

For n>0: A131852(a(n+1))=n and ABS(A131852(m))A131865(n-2). - Reinhard Zumkeller, Jul 22 2007

Third quadrisection of A115451. - Klaus Purath, Mar 14 2021

Links

• Harvey P. Dale, Table of n, a(n) for n = 2..832
• Index entries for linear recurrences with constant coefficients, signature (17,-16).

Programs

Mathematica

s=0;lst={};Do[s+=2^n;AppendTo[lst, s], {n, 1, 2*5!, 4}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 07 2008 *)
FromDigits[#,2]&/@Table[Join[PadRight[{},4n,{1,0,0,0}],{1,0}],{n,0,20}] (* Harvey P. Dale, Apr 06 2020 *)

PARI

for(n=0,20,print(2*sum(k=0,n,2^(4*k))))
for(k=0,20,print(2*(1-16^(k+1))/-15))

Formula

lim_{n -> infinity} a(n)/a(n-k) = 2^(4*(n-k)).

2*Sum_{k=0..n} 16^k = 2*(16^(n+1) - 1)/15.

From Klaus Purath, Mar 14 2021: (Start)

a(n) = (2^(4*n-3)-2)/15.

a(n) = 17*a(n-1) - 16*a(n-2).

a(n) = 16*a(n-1) + 2.

a(n) = 2*16^(n-2) + a(n-1).

a(n) = 2*A131865(n-2). (End)

Extensions

More terms from Ray Chandler, Nov 02 2004

More terms from Vladimir Joseph Stephan Orlovsky, Nov 07 2008