A386014 Distinct values occurring in first differences of A030655(n), listed in order of first appearance.

22, 832, 202, 89302, 2002, 8993002, 20002, 899930002, 200002, 89999300002, 2000002, 8999993000002, 20000002, 899999930000002, 200000002, 89999999300000002, 2000000002, 8999999993000000002, 20000000002, 899999999930000000002, 200000000002, 89999999999300000000002, 2000000000002

Offset: 1

Author

Carin Maria Sakin, Jul 14 2025

Keywords

• nonn
• easy
• base

Comments

A030655(t) = 2*t-1||2*t = (2*t-1)*10^k + 2*t, where || indicates digit concatenation and 2*t has digit length k.

Difference A030655(t+1) - A030655(t) = 2*(10^k+1) when 2*t and 2*t+2 are both digit length k and this is the odd n case in the formulas below.

2*t and 2*t+2 differ in length only at t = j(k) = 5*10^(k-1)-1 with 2*t+2 then having length k+1, and this is the even n case in the formulas below.

Examples

			At t = j(3) = 499, difference A030655(t+1) - A030655(t) = 9991000 - 997998 = 8993002 which is term a(6).
		

Links

• Index entries for linear recurrences with constant coefficients, signature (1,110,-110,-1000,1000).

Crossrefs

Cf. A007376, A030655, A057147, A061084.

Programs

Maple

a := proc(n)
    if type(n, even) then
        k := n/2;
        return 9*100^k - 7*10^k + 2;
    else
        k := (n + 1)/2;
        return 2*(10^k + 1);
    end if;
end proc:
seq(a(n), n=1..12);

Mathematica

a[n_] := Which[
  EvenQ[n], Module[{k = n/2}, 9*100^k - 7*10^k + 2],
  OddQ[n], Module[{k = (n + 1)/2}, 2*(10^k + 1)]
];
Table[a[n], {n, 1, 12}]

PARI

a(n) = if(n%2==1, 2*10^((n+1)/2)+2, 9*10^n - 7*10^(n/2) + 2) \\ David A. Corneth, Jul 17 2025

Python

def a(n):
    k = (n + 1)//2
    if n % 2 == 0:
        return 9 * 100**k - 7 * 10**k + 2
    else:
        return 2 * (10**k + 1)

Formula

a(n) = 2*(10^k + 1) for odd n, where k = (n+1)/2.

a(n) = 9*100^k - 7*10^k + 2 for even n, where k = n/2.

Extensions

More terms from Michel Marcus, Jul 16 2025

More terms from David A. Corneth, Jul 17 2025