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A147808 Sum of n-digit numbers which are balanced: the first [n/2] digits have the same sum as the last [n/2] digits. 4
45, 495, 49500, 3314850, 331431000, 27336542310, 2733612983100, 238305122029260, 23830484311542600, 2140037814262627400, 214003761418373774000, 19587943639318412097360 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Numbers such that the first half of digits have the same sum than the last half of digits are called balanced in the linked "Problem 217". (Note that here the meaning of "balanced" is neither that of A020492, nor that of A031443.)

Up to n=3 digits, the only balanced numbers are the palindromes, from n=4 on, there are non-palindromic balanced numbers, cf. A145808.

LINKS

Table of n, a(n) for n=1..12.

Project Euler, Problem 217.

FORMULA

lim a(2n+1)/a(2n) = 100, lim a(2n)/a(2n-1) = 90 (as n -> oo).

EXAMPLE

a(1) = 1+2+...+9; a(2) = 11+22+...+99 = 11 a(1); a(3) = 101+111+121+....+191+202+...+989+999 = (101*10 + 10*9)*a(1); a(4) = 1001+1010+1102+1111+1120+1203+...+9889+9898+9999.

MATHEMATICA

balQ[n_]:=Module[{idn=IntegerDigits[n], len=Floor[IntegerLength[n]/2]}, Total[ Take[ idn, len]] == Total[Take[idn, -len]]]; Table[Total[ Select[ Range[ 10^n, 10^(n+1)-1], balQ]], {n, 0, 5}] (* This will generate the first six terms of the sequence.  To generate more, (1) change the range of the Table from (0, 5) to (0, 6) or (0, 7), etc., but the program will take increasingly long to run. *) (* Harvey P. Dale, Apr 07 2013 *)

PROG

(PARI) A147808(n)={ local( t, c ); if( n==1, 45, /* global variable SC[sd] (used for n=2k and n=2k+1) stores [sum, count] of numbers with <= n\2 digits and digit sum = sd */ if( #SC != n\2*9, SC=vector( n\2*9, digsum, c=0; [sum( i=0, 10^(n\2)-1, if((i-digsum)%9==0 && digsum==sum(j=1, #t=Vecsmall(Str(i)), t[j])-48*#t, c++; i )), c] )); if( n%2==0, sum( i=10^((n\=2)-1), 10^n-1, SC[A007953(i)]*[1, i*10^n]~ ), t=10^(n\=2)*[100, 45]~; sum( i=10^(n-1), 10^n-1, SC[A007953(i)]*[10, [i, 1]*t]~ )))}

CROSSREFS

Sequence in context: A193434 A086576 A295319 * A190417 A093529 A197501

Adjacent sequences:  A147805 A147806 A147807 * A147809 A147810 A147811

KEYWORD

base,nonn

AUTHOR

M. F. Hasler, Nov 23 2008

STATUS

approved

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Last modified June 25 12:01 EDT 2019. Contains 324352 sequences. (Running on oeis4.)