A329719 - OEIS

%I #41 Dec 14 2019 08:08:04

%S 112,123,134,145,156,167,178,189,213,224,235,246,257,268,279,314,325,

%T 336,347,358,369,415,426,437,448,459,516,527,538,549,617,628,639,718,

%U 729,819,1123,1235,1347,1459,1910,2134,2246,2358,2810,2911,3145,3257,3369

%N Numbers whose digits can be partitioned into at least 3 segments (not beginning with 0) where each segment is the sum of the previous two segments.

%H Bi Cheng Wu, <a href="/A329719/b329719.txt">Table of n, a(n) for n = 1..4958 (a(n) < 1000000)</a>

%e 112 -> 1+1=2;

%e 1235 -> 1+2=3 and 2+3=5;

%e 224610 -> 2+2=4 and 2+4=6 and 4+6=10.

%t part[n_] := part[n] = Select[Flatten[ Permutations /@ Reverse /@ IntegerPartitions[n, {3,n}], 1], 0 <= #[[3]] - Max[#[[1]], #[[2]]] <= 1 && AllTrue[Rest@ Rest@ Differences@ #, 0 <= # <= 1 &] &]; spl[x_, L_] := Map[ FromDigits@ Take[x, #] &, Transpose[{Most@ #, Rest[#]-1}& [ FoldList[ Plus, 1, L]]]]; sumQ[w_] := AllTrue[Range[3, Length@w], w[[#]] == w[[#-1]] + w[[#-2]] &]; ok[n_] := Block[{m = IntegerLength@ n}, AnyTrue[ spl[ IntegerDigits[n], #] & /@ part[m], sumQ[#] && Total[ IntegerLength /@ #] == m &]]; Select[ Range[100, 6000], ok] (* _Giovanni Resta_, Dec 03 2019 *)

%o (Python)

%o def isSegmentSum(digits,segment1=None,segment2=None):

%o     digits = str(digits)

%o     N = len(digits)

%o     if N == 0:

%o         return True

%o     else:

%o         if (segment1 is None) and (segment2 is None):

%o             for i in range(N):

%o                 try:

%o                     slice1 = digits[:i+1]

%o                     for j in range(N-(i+1)):

%o                         slice2 = digits[i+1:i+1+j+1]

%o                         slice3 = digits[i+1+j+1:]

%o                         if (isSegmentSum(slice3,slice1,slice2) and

%o                          len(slice3)>0 and not (slice1.startswith("0") or

%o                          slice2.startswith("0") or

%o                          (slice3.startswith("0")))):

%o                             return True

%o                 except:

%o                     return False

%o         else:

%o             sumOfDigits = str(int(segment1)+int(segment2))

%o             nS = len(sumOfDigits)

%o             try:

%o                 if digits[:nS] == sumOfDigits:

%o                     return isSegmentSum(digits[nS:],segment2,digits[:nS])

%o                 else:

%o                     return False

%o             except:

%o                 return False

%o     return False

%o def findSegmentSum(lower,upper):

%o     for i in range(lower,upper+1):

%o         if isSegmentSum(i):

%o             print(str(i))

%o findSegmentSum(1, 5200)

%Y Shares subsequences with A108203, A308104, A214527.

%Y Cf. A019523.

%K nonn,base

%O 1,1

%A _Bi Cheng Wu_, Nov 19 2019