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A258718 Value of DIS ("Decimal Integer Series") constant based on sequence of squares.

%I A258718 #54 Jun 28 2025 16:13:17
%S A258718 1,8,1,9,0,5,8,9,0,2,0,0,8,0,1,2,1,5,6,7,6,2,0,9,6,7,7,9,0,2,8,7,2,1,
%T A258718 2,3,4,0,4,7,9,5,5,0,2,6,4,8,5,2,1,1,5,2,1,7,5,8,8,5,4,2,1,4,3,2,1,8,
%U A258718 7,9,9,0,1,4,9,1,4,2,1,1,8,9,2,7,3
%N A258718 Value of DIS ("Decimal Integer Series") constant based on sequence of squares.
%C A258718 The following problem was proposed in Popular Computing in 1973. If m is a k-digit number, let c(m) = m/100^k. For example, if m=16, c(m) = .0016 = 16/100^2. For a sequence S = a(1), a(2), a(3), ..., the "DIS" constant based on S is defined to be f(S) = Sum_{n >= 1} c(a(n)).
%C A258718 If S = 1, 4, 9, 16, 25, 36, ... the nonzero squares then f(S) is the sum of the infinite series
%C A258718 .01
%C A258718 .04
%C A258718 .09
%C A258718 .0016
%C A258718 .0025
%C A258718 .0036
%C A258718 ...
%C A258718 Problem 22 in Popular Computing asks for the values of f(S) when S is respectively the squares (A000290), the cubes (A259929), the powers of 2 (A259838), powers of 3 (A259930), Fibonacci numbers (A000045), factorials (A259837), and subfactorials (A000166). To this list we might add the triangular numbers (A000217), the Catalan numbers (A000108), and the Motzkin numbers (A001006). But not the primes (A000040), for in that case the series would diverge.
%C A258718 Solution in the case of squares from _Alex Meiburg_, Jun 17 2015:
%C A258718 (Start)
%C A258718 If we group the sum (for squares) by number of digits -- that is,  A = (.01
%C A258718 + .04 + .09 + .0016 + .0025 + ...) + (.000100 + .000121 + .000144 )...,
%C A258718 Mathematica gives a closed form for each term. Specifically,
%C A258718 f[n_] := 1/3 2^(-5-4 n) 25^(-2-2 n) (10^(1+n)-Ceiling[10^(1/2+n)])
%C A258718 (1+2^(3+2 n) 5^(2+2 n)-3 10^(1+n)-3 Ceiling[10^(1/2+n)]+2^(2+n) 5^(1+n)
%C A258718 Ceiling[10^(1/2+n)]+2 Ceiling[10^(1/2+n)]^2)-1/3 2^(-3-4 n) 5^(-2-4 n)
%C A258718 (-1+10^n-Floor[10^(1/2+n)]) (2^(1+2 n) 5^(2
%C A258718 n)-10^n+Floor[10^(1/2+n)]+2^(1+n) 5^n Floor[10^(1/2+n)]+2
%C A258718 Floor[10^(1/2+n)]^2)
%C A258718 this is found by breaking into those less than 10^(n+1/2) and those more
%C A258718 than 10^(n+1/2), and each sum can be done exactly. The above expression
%C A258718 is then summed over n from 0 onwards. This allows it to converge very
%C A258718 rapidly, yielding (in 1 second on my computer)
%C A258718 A = 0.
%C A258718 1819058902008012156762096779028721234047955026485211521758854214321879901491421189273371395956634796
%C A258718 1904208362098771470319180456038832179432219577663082261873935853836211627222184657331459477127289143
%C A258718 6276893929762580722032999375515089312368984249626008647664116996102082800886283059357211021253063111
%C A258718 7152305529806492037632632133219059593583028182894120128297404653399814673445759857199248725452332778
%C A258718 2701081771512735871942465499976300270627184777090377542514576075322995673512370243399897097464883624
%C A258718 2192516987267080161614437137250709716131980302821936381137562051251267791554480374065719655469612472
%C A258718 1097775213404047734600505041085860200873512242914767206681950461568207388112576403619589771428626620
%C A258718 4125627257974347552135871228668804281367024808511880657288473617154807673305471279796586999731661629
%C A258718 7031493974952970572666731437316703731493871634266094228909721713279127991559139591293931720140406675
%C A258718 3533810115103080147864906643290214333852271864692146387518404231194138382947749162680518961898521092
%C A258718 ... The same technique should work for cubes, and perhaps some sequences that grow exponentially; but almost certainly not with factorials.
%C A258718 (End)
%D A258718 N. J. A. Sloane, Alex Meiburg, Olivier Gérard, "A Computational Challenge from 1973", Postings to Sequence Fans Mailing List, Jun 17 2015.
%H A258718 Robert G. Wilson v, <a href="/A258718/b258718.txt">Table of n, a(n) for n = 0..999</a>
%H A258718 Popular Computing (Calabasas CA), <a href="/A258718/a258718.jpg">Problem 22: Decimal Integer Series (DIS)</a>, Vol. 1 (No. 8, Nov 1973), page PC8-14.
%Y A258718 Cf. A000290, A000079, A000142, A000166, A000045, A000217, A000040.
%K A258718 nonn,cons,base
%O A258718 0,2
%A A258718 _N. J. A. Sloane_, Jun 17 2015