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 A084545 Alternate number system in base 5. 12
 1, 2, 3, 4, 5, 11, 12, 13, 14, 15, 21, 22, 23, 24, 25, 31, 32, 33, 34, 35, 41, 42, 43, 44, 45, 51, 52, 53, 54, 55, 111, 112, 113, 114, 115, 121, 122, 123, 124, 125, 131, 132, 133, 134, 135, 141, 142, 143, 144, 145, 151, 152, 153, 154, 155, 211, 212, 213, 214, 215, 221, 222 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS 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 [Wayback Machine 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. FORMULA From Hieronymus Fischer, Jun 06 and Jun 08 2012: (Start) The formulas are designed to calculate base-10 numbers only using the digits 1..5. a(n) = Sum_{j=0..m-1} (1 + b(j) mod 5)*10^j, where m = floor(log_5(4*n+1)), b(j) = floor((4*n+1-5^m)/(4*5^j)). a(k*(5^n-1)/4) = k*(10^n-1)/9, for k = 1,2,3,4,5. a((9*5^n-5)/4) = (14*10^n-5)/9 = 10^n + 5*(10^n-1)/9. a((5^n-1)/4 - 1) = 5*(10^(n-1)-1)/9, n>1. a(n) <= (10^log_5(4*n+1)-1)/9, equality holds for n=(5^k-1)/4, k>0. a(n) > (5/10)*(10^log_5(4*n+1)-1)/9, n>0. lim inf a(n)/10^log_5(4*n) = 1/18, for n --> infinity. lim sup a(n)/10^log_5(4*n) = 1/9, for n --> infinity. G.f.: g(x) = (x^(1/4)*(1-x))^(-1) sum_{j>=0} 10^j*z(j)^(5/4)*(1 - 6z(j)^5 + 5z(j)^6)/((1-z(j))(1-z(j)^5)), where z(j) = x^5^j. Also: g(x) = (1/(1-x)) sum_{j>=0} (1-6(x^5^j)^5+5(x^5^j)^6)*x^5^j*f_j(x)/(1-x^5^j), where f_j(x) = 10^j*x^((5^j-1)/4)/(1-(x^5^j)^5). The f_j obey the recurrence f_0(x) = 1/(1-x^5), f_(j+1)(x) = 10x*f_j(x^5). Also: g(x) = 1/(1-x))*(h_(5,0)(x) + h_(5,1)(x) + h_(5,2)(x) + h_(4,1)(x) + h_(5,4)(x) - 5*h_(5,5)(x)), where h_(5,k)(x) = sum_{j>=0} 10^j*x^((5^(j+1)-1)/4) * (x^5^j)^k/(1-(x^5^j)^5). (End) EXAMPLE From Hieronymus Fischer, Jun 06 2012: (Start) a(100) = 345. a(10^3) = 12445. a(10^4) = 254445. a(10^5) = 11144445. a(10^6) = 223444445. a(10^7) = 4524444445. a(10^8) = 145544444445. a(10^9) = 3521444444445. (End) PROG (PARI) a(n) = my (w=5); while (n>w, n -= w; w *= 5); my (d=digits(w+n-1, 5)); d[1] = 0; fromdigits(d) + (10^(#d-1)-1)/9 \\ Rémy Sigrist, Dec 04 2019 CROSSREFS Cf. A007931, A007932, A052382, A084544, A046034, A089581, A084984, A001742, A001743, A001744, A202267, A202268, A014261, A014263. Sequence in context: A359999 A316538 A283206 * A069908 A130640 A214653 Adjacent sequences: A084542 A084543 A084544 * A084546 A084547 A084548 KEYWORD nonn,base,easy AUTHOR Robert R. Forslund (forslund(AT)tbaytel.net), Jun 27 2003 EXTENSIONS Offset set to 1 according to A007931, A007932 and more terms added by Hieronymus Fischer, Jun 06 2012 STATUS approved

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