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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, 1958794348735327250973600, 181693537570273520779480800
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.
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
KEYWORD
base,nonn
AUTHOR
M. F. Hasler, Nov 23 2008
EXTENSIONS
a(13)-a(14) from Kevin P. Thompson, Dec 05 2021
STATUS
approved