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 A084544 Alternate number system in base 4. 13
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS a(n) = A045926(n) / 2. - Reinhard Zumkeller, Jan 01 2013 LINKS Hieronymus Fischer, Table of n, a(n) for n = 1..10000 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. 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..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) EXAMPLE 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) CROSSREFS Cf. A007931, A007932, A052382, A084545, A046034, A089581, A084984, A001742, A001743, A001744, A202267, A202268, A014261, A014263. Sequence in context: A108467 A265565 A265549 * A268236 A039023 A110918 Adjacent sequences:  A084541 A084542 A084543 * A084545 A084546 A084547 KEYWORD nonn,base AUTHOR Robert R. Forslund (forslund(AT)tbaytel.net), Jun 27 2003 EXTENSIONS Offset set to 1 according to A007931, A007932 by Hieronymus Fischer, Jun 06 2012 STATUS approved

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Last modified January 23 22:36 EST 2020. Contains 331177 sequences. (Running on oeis4.)