A045926 -id:A045926 - cp's OEIS frontend

A084544 Alternate number system in base 4.

1, 2, 3, 4, 11, 12, 13, 14, 21, 22, 23, 24, 31, 32, 33, 34, 41, 42, 43, 44, 111, 112, 113, 114, 121, 122, 123, 124, 131, 132, 133, 134, 141, 142, 143, 144, 211, 212, 213, 214, 221, 222, 223, 224, 231, 232, 233, 234, 241, 242, 243, 244, 311, 312, 313, 314, 321

Offset: 1

Robert R. Forslund (forslund(AT)tbaytel.net), Jun 27 2003

			From _Hieronymus Fischer_, Jun 06 2012: (Start)
a(100)  = 1144.
a(10^3) = 33214.
a(10^4) = 2123434.
a(10^5) = 114122134.
a(10^6) = 3243414334.
a(10^7) = 211421121334.
a(10^8) = 11331131343334.
a(10^9) = 323212224213334. (End)

Hieronymus Fischer, Table of n, a(n) for n = 1..10000
EMIS, Mirror site for Southwest Journal of Pure and Applied Mathematics
R. R. Forslund, A logical alternative to the existing positional number system, Southwest Journal of Pure and Applied Mathematics, Vol. 1 1995 pp. 27-29.
R. R. Forslund, Positive Integer Pages [Broken link]
James E. Foster, A Number System without a Zero-Symbol, Mathematics Magazine, Vol. 21, No. 1. (1947), pp. 39-41.
Index entries for 10-automatic sequences.

Cf. A007931, A007932, A052382, A084545, A046034, A089581, A084984, A001742, A001743, A001744, A202267, A202268, A014261, A014263.

def A084544(n):
    m = (3*n+1).bit_length()-1>>1
    return int(''.join((str(((3*n+1-(1<<(m<<1)))//(3<<((m-1-j)<<1))&3)+1) for j in range(m)))) # Chai Wah Wu, Feb 08 2023

From Hieronymus Fischer, Jun 06 and Jun 08 2012: (Start)
The formulas are designed to calculate base-10 numbers only using the digits 1..4.
a(n) = Sum_{j=0..m-1} (1 + b(j) mod 4)*10^j,
where m = floor(log_4(3*n+1)), b(j) = floor((3*n+1-4^m)/(3*4^j)).
Special values:
a(k*(4^n-1)/3) = k*(10^n-1)/9, k = 1,2,3,4.
a((7*4^n-4)/3) = (13*10^n-4)/9 = 10^n + 4*(10^n-1)/9.
a((4^n-1)/3 - 1) = 4*(10^(n-1)-1)/9, n > 1.
Inequalities:
a(n) <= (10^log_4(3*n+1)-1)/9, equality holds for n=(4^k-1)/3, k>0.
a(n) > (4/10)*(10^log_4(3*n+1)-1)/9, n > 0.
Lower and upper limits:
lim inf a(n)/10^log_4(3*n) = 2/45, for n --> infinity.
lim sup a(n)/10^log_4(3*n) = 1/9, for n --> infinity.
G.f.: g(x) = (x^(1/3)*(1-x))^(-1) Sum_{j>=0} 10^j*z(j)^(4/3)*(1 - 5z(j)^4 + 4z(j)^5)/((1-z(j))(1-z(j)^4)), where z(j) = x^4^j.
Also: g(x) = (1/(1-x)) Sum_{j>=0} (1-5(x^4^j)^4 + 4(x^4^j)^5)*x^4^j*f_j(x)/(1-x^4^j), where f_j(x) = 10^j*x^((4^j-1)/3)/(1-(x^4^j)^4). The f_j obey the recurrence f_0(x) = 1/(1-x^4), f_(j+1)(x) = 10x*f_j(x^4).
Also: g(x) = (1/(1-x))* (h_(4,0)(x) + h_(4,1)(x) + h_(4,2)(x) + h_(4,3)(x) - 4*h_(4,4)(x)), where h_(4,k)(x) = Sum_{j>=0} 10^j*x^((4^(j+1)-1)/3) * (x^4^j)^k/(1-(x^4^j)^4).
(End)
a(n) = A045926(n) / 2. - Reinhard Zumkeller, Jan 01 2013

Offset set to 1 according to A007931, A007932 by Hieronymus Fischer, Jun 06 2012

A192370 Sum of all the n-digit numbers whose digits are all even and nonzero: 2,4,6,8.

Original entry on oeis.org

20, 880, 35520, 1422080, 56888320, 2275553280, 91022213120, 3640888852480, 145635555409920, 5825422221639680, 233016888886558720, 9320675555546234880, 372827022222184939520, 14913080888888739758080, 596523235555554959032320, 23860929422222219836129280

Offset: 1

Bernard Schott, Dec 31 2012

A192107 is the similar sequence when all the digits are odd.

A220094 is the similar sequence with the digits belonging to {1, 2, 3, 4, 5, 6, 7, 8, 9}.

			a(1) = 2 + 4 + 6 + 8 = 20.
a(2) = 22 + 24 + 26 + 28 + 42 + ... + 68 + 82 + 84 + 86 + 88 = 880.

Index entries for linear recurrences with constant coefficients, signature (44,-160).

Cf. A192107, A220094, A045926.

Maple
```
A:=seq(5 * 4^n *(10^n-1)/9,n=1..20);
```

Table[(5*4^n*(10^n - 1))/9, {n, 20}] (* T. D. Noe, Dec 31 2012 *)

a(n)=(5*4^n*(10^n-1))/9 \\ Charles R Greathouse IV, Jul 06 2017

a(n) = (5 * 4^n * (10^n-1))/9.
From Colin Barker, Jan 04 2013: (Start)
a(n) = 44*a(n-1) - 160*a(n-2).
G.f.: 20*x/((4*x-1)*(40*x-1)). (End)

A045927 Digits even, nonzero and nondecreasing.

Original entry on oeis.org

2, 4, 6, 8, 22, 24, 26, 28, 44, 46, 48, 66, 68, 88, 222, 224, 226, 228, 244, 246, 248, 266, 268, 288, 444, 446, 448, 466, 468, 488, 666, 668, 688, 888, 2222, 2224, 2226, 2228, 2244, 2246, 2248, 2266, 2268, 2288, 2444, 2446, 2448, 2466, 2468, 2488, 2666, 2668

Offset: 1

Felice Russo

Index entries for 10-automatic sequences.

Cf. A014261, A014263, A045926.

More terms from Patrick De Geest, Jun 15 1999

A137103 Numbers k such that k and k^2 use only the digits 2, 4, 6 and 8.

Original entry on oeis.org

2, 8, 22, 68, 262, 668, 6668, 66668, 666668, 6666668, 66666668, 666666668, 6666666668, 66666666668, 666666666668, 6666666666668, 66666666666668, 666666666666668, 6666666666666668, 66666666666666668, 666666666666666668, 6666666666666666668, 66666666666666666668

Offset: 1

Jonathan Wellons (wellons(AT)gmail.com), Jan 22 2008

Generated with DrScheme.
From Bernard Schott, May 04 2022: (Start)
All terms end with 2 or 8, because when k ends with 4 or 6, the tens digit of k^2 is always odd.
Squares are a subsequence of A103751.
This sequence is infinite because terms of the form 8, 68, 668, 6668, ..., have respectively squares equal to 64, 4624, 446224, 44462224, ... In fact, if m = (10^k+20)/15 and k >= 2, then m^2 has successively (k-2) 4's, one 6, (k-2) 2's, and one 4 in its decimal representation; hence, A073555 \ {1} is a subsequence. (End)

			262^2 = 68644.

Michael S. Branicky, Table of n, a(n) for n = 1..43
Jonathan Wellons, Tables of Shared Digits

Subsequence of A045926.

Cf. A073555, A103751.

a(19) and beyond from Michael S. Branicky, May 04 2022

A258271 Decimal expansion of the sum of the reciprocal of the squares of the numbers whose digits are all even.

Original entry on oeis.org

0, 3, 6, 6, 3, 6, 0, 0, 3, 9, 7, 1, 9, 5, 2, 3, 2, 9, 5, 1, 7, 1, 8, 8, 2, 5, 0, 8, 9, 6, 7, 4, 1, 2, 4, 2, 6, 6, 2, 5, 1, 7, 3, 9, 5, 0, 3, 4, 2, 1, 1, 8, 7, 6, 0, 0, 2, 0, 0, 7, 1, 1, 3, 5, 0, 8, 5, 2, 8, 3, 3, 3, 2, 9, 3, 4, 9, 5, 1, 5, 7, 5, 8, 4, 4, 6, 5

Offset: 1

Paolo P. Lava, May 25 2015

A rational approximation (correct up to the 9th decimal digit) is 22781/62182.

Continued fraction: [0, 2, 1, 2, 1, 2, 3, 3, 1, 8, 5, 2, 1, 14,...].

			Decimal expansion of Sum_{k=1..oo}{1/A045926(k)^2} = 1/2^2 + 1/4^2 + 1/6^2 + 1/8^2 + 1/22^2 + 1/24^2 + 1/26^2 + ... = 0.3663600397195232951718825089674124266251739503421187600...

Cf. A045926, A194181, A194182.

P:=proc(q) local a,b,k,ok,n; a:=0; for n from 2 by 2 to q do ok:=1; b:=n;
for k from 1 to ilog10(n)+1 do if (b mod 10)=0 or ((b mod 10) mod 2)=1 then ok:=0;
break; else b:=trunc(b/10); fi; od; if ok=1 then a:=a+(1/n)^2; fi; od;
print(evalf(a,200)); end: P(10^9);

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A084544 Alternate number system in base 4.

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A192370 Sum of all the n-digit numbers whose digits are all even and nonzero: 2,4,6,8.

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A045927 Digits even, nonzero and nondecreasing.

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A137103 Numbers k such that k and k^2 use only the digits 2, 4, 6 and 8.

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A258271 Decimal expansion of the sum of the reciprocal of the squares of the numbers whose digits are all even.

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Maple