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Perl, 76 bytes

Includes +3 for -p (normally +1, but +3 to play fair with the other perl solution)

Run with the input on STDIN:

sum15.pl <<< "1 2 3"

sum15.pl:

#!/usr/bin/perl -p
s/$/ 15/;s/\d+/1x$&/eg;s/( *1){15}\K ?/
/g;s/
1*
*$//;s/1+|\B/length$&/eg

Look ma, no calculations whatsoever...

Perl, 76 bytes

Includes +3 for -p (normally +1, but +3 to play fair with the other perl solution)

Run with the input on STDIN (final newline on input is optional, but MUST be absent for the empty input)

sum15.pl <<< "1 2 3"

sum15.pl:

#!/usr/bin/perl -p
s/$/ 15/;s/\d+/1x$&/eg;s/( *1){15}\K ?/
/g;s/
1*
*$//;s/1+|\B/length$&/eg

Look ma, no calculations whatsoever...

Perl, 76 bytes

Includes +3 for -p (normally +1, but +3 to play fair with the other perl solution)

Run with the input on STDIN:

sum15.pl <<< "1 2 3"

sum15.pl:

#!/usr/bin/perl -p
s/$/ 15/;s/\d+/1x$&/eg;s/( *1){15}\K ?/
/g;s/
.*
*$//;s/1+|\B/length$&/eg

Look ma, no calculations whatsoever...

Perl, 76 bytes

Includes +3 for -p (normally +1, but +3 to play fair with the other perl solution)

Run with the input on STDIN:

sum15.pl <<< "1 2 3"

sum15.pl:

#!/usr/bin/perl -p
s/$/ 15/;s/\d+/1x$&/eg;s/( *1){15}\K ?/
/g;s/
1*
*$//;s/1+|\B/length$&/eg

Look ma, no calculations whatsoever...

Perl, 76 bytes

Doesn't properly handle final 0

Includes +3 for -p (normally +1, but +3 to play fair with the other perl solution)

Run with the input on STDIN (a terminating newline is OK except for the empty input)

sum15.pl <<< "1 2 3"

sum15.pl:

#!/usr/bin/perl -p
s/$/ 15/;s/\d+/1x$&/eg;s/( *1){15}\K ?/
/g;s/
.*
*$//;s/1+|\B/length$&/eg

Look ma, no calculations whatsoever...

Perl, 76 bytes

Includes +3 for -p (normally +1, but +3 to play fair with the other perl solution)

Run with the input on STDIN:

sum15.pl <<< "1 2 3"

sum15.pl:

#!/usr/bin/perl -p
s/$/ 15/;s/\d+/1x$&/eg;s/( *1){15}\K ?/
/g;s/
.*
*$//;s/1+|\B/length$&/eg

Look ma, no calculations whatsoever...