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

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

Offset: 0

Author

N. J. A. Sloane, Jun 17 2015

Keywords

• nonn
• cons
• base

Comments

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)).

If S = 1, 4, 9, 16, 25, 36, ... the nonzero squares then f(S) is the sum of the infinite series

.01

.04

.09

.0016

.0025

.0036

...

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.

Solution in the case of squares from Alex Meiburg, Jun 17 2015:

(Start)

If we group the sum (for squares) by number of digits -- that is, A = (.01

+ .04 + .09 + .0016 + .0025 + ...) + (.000100 + .000121 + .000144 )...,

Mathematica gives a closed form for each term. Specifically,

f[n_] := 1/3 2^(-5-4 n) 25^(-2-2 n) (10^(1+n)-Ceiling[10^(1/2+n)])

(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)

Ceiling[10^(1/2+n)]+2 Ceiling[10^(1/2+n)]^2)-1/3 2^(-3-4 n) 5^(-2-4 n)

(-1+10^n-Floor[10^(1/2+n)]) (2^(1+2 n) 5^(2

n)-10^n+Floor[10^(1/2+n)]+2^(1+n) 5^n Floor[10^(1/2+n)]+2

Floor[10^(1/2+n)]^2)

this is found by breaking into those less than 10^(n+1/2) and those more

than 10^(n+1/2), and each sum can be done exactly. The above expression

is then summed over n from 0 onwards. This allows it to converge very

rapidly, yielding (in 1 second on my computer)

A = 0.

1819058902008012156762096779028721234047955026485211521758854214321879901491421189273371395956634796

1904208362098771470319180456038832179432219577663082261873935853836211627222184657331459477127289143

6276893929762580722032999375515089312368984249626008647664116996102082800886283059357211021253063111

7152305529806492037632632133219059593583028182894120128297404653399814673445759857199248725452332778

2701081771512735871942465499976300270627184777090377542514576075322995673512370243399897097464883624

2192516987267080161614437137250709716131980302821936381137562051251267791554480374065719655469612472

1097775213404047734600505041085860200873512242914767206681950461568207388112576403619589771428626620

4125627257974347552135871228668804281367024808511880657288473617154807673305471279796586999731661629

7031493974952970572666731437316703731493871634266094228909721713279127991559139591293931720140406675

3533810115103080147864906643290214333852271864692146387518404231194138382947749162680518961898521092

... The same technique should work for cubes, and perhaps some sequences that grow exponentially; but almost certainly not with factorials.

(End)

References

• N. J. A. Sloane, Alex Meiburg, Olivier Gérard, "A Computational Challenge from 1973", Postings to Sequence Fans Mailing List, Jun 17 2015.

Links

• Robert G. Wilson v, Table of n, a(n) for n = 0..999
• Popular Computing (Calabasas CA), Problem 22: Decimal Integer Series (DIS), Vol. 1 (No. 8, Nov 1973), page PC8-14.

Crossrefs

Cf. A000290, A000079, A000142, A000166, A000045, A000217, A000040.