This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
seq = {0}; n = 0; s = 0; While[Length[seq] < 100,
s += Length[Length /@ Split[IntegerDigits[++n, 2]]]; If[IntegerQ@ Sqrt@ s, AppendTo[seq, s]]]; seq (* Giovanni Resta, Jul 27 2013 *)
(define (A227744 n) (A173318 (A227743 n)))
Select[Sqrt[#]&/@Accumulate[Join[{0},Table[Length[Split[IntegerDigits[n,2]]],{n,10000}]]],IntegerQ] (* Harvey P. Dale, Nov 20 2022 *)
(define (A227745 n) (sqrt (A227744 n)))
Considered as a triangle with 2^k terms per row, the first few rows are: 1 2, 1 2, 3, 2, 1 2, 3, 4, 3, 2, 3, 2, 1 ... The n-th row becomes right half of next row; left half is mirrored terms of n-th row increased by one. - _Gary W. Adamson_, Jun 20 2012
import Data.List (group) a005811 0 = 0 a005811 n = length $ group $ a030308_row n a005811_list = 0 : f [1] where f (x:xs) = x : f (xs ++ [x + x `mod` 2, x + 1 - x `mod` 2]) -- Reinhard Zumkeller, Feb 16 2013, Mar 07 2011
A005811 := proc(n) local i, b, ans; if n = 0 then return 0 ; end if; ans := 1; b := convert(n, base, 2); for i from nops(b)-1 to 1 by -1 do if b[ i+1 ]<>b[ i ] then ans := ans+1 fi od; return ans ; end proc: seq(A005811(i), i=1..50) ; # second Maple program: a:= n-> add(i, i=Bits[Split](Bits[Xor](n, iquo(n, 2)))): seq(a(n), n=0..100); # Alois P. Heinz, Feb 01 2023
Table[ Length[ Length/@Split[ IntegerDigits[ n, 2 ] ] ], {n, 1, 255} ]
a[n_] := DigitCount[BitXor[n, Floor[n/2]], 2, 1]; Array[a, 100, 0] (* Amiram Eldar, Jul 11 2024 *)
a(n)=sum(k=1,n,(-1)^((k/2^valuation(k,2)-1)/2))
a(n)=if(n<1,0,a(n\2)+(a(n\2)+n)%2) \\ Benoit Cloitre, Jan 20 2014
a(n) = hammingweight(bitxor(n, n>>1)); \\ Gheorghe Coserea, Sep 03 2015
def a(n): return bin(n^(n>>1))[2:].count("1") # Indranil Ghosh, Apr 29 2017
19 has binary expansion "10011", thus the maximal runs of identical bits (scanned from right to left) are [2,2,1]. We subtract one from each after the first one, to get [2,1,0] and then form their partial sums as [2,2+1,2+1+0], which thus maps to unordered partition {2+3+3} which adds to 8. Thus a(19)=8.
Table[Function[b, Total@ Accumulate@ Prepend[If[Length@ b > 1, Rest[b] - 1, {}], First@ b] - Boole[n == 0]]@ Map[Length, Split@ Reverse@ IntegerDigits[n, 2]], {n, 0, 79}] // Flatten (* Michael De Vlieger, May 09 2017 *)
def A227183(n): '''Sum of parts of the unique unordered partition encoded in the run lengths of the binary expansion of n.''' s = 0 b = n%2 i = 1 while (n != 0): n >>= 1 if ((n%2) == b): # Staying in the same run of bits? i += 1 else: # The run changes. b = n%2 s += i return(s)
1 has one run in its binary representation "1", thus 1 occurs once. 2 has two runs in its binary representation "10", thus 2 occurs twice. 3 has one run in its binary representation "11", thus 3 occurs only once. 4 has two runs in its binary representation "100", thus 4 occurs twice. 5 has three runs in its binary representation "101", thus 5 occurs three times. The sequence thus begins as 1, 2,2, 3, 4,4, 5,5,5, ...
Table[ConstantArray[n, Length@ Split@ IntegerDigits[n, 2]], {n, 26}] // Flatten (* Michael De Vlieger, May 09 2017 *)
Table[PadRight[{},Length[Split[IntegerDigits[n,2]]],n],{n,40}]//Flatten (* Harvey P. Dale, Jul 23 2021 *)
Table[Range[0, #] &@ Total@ Flatten@ Map[Abs@ Differences@ # &,
Partition[IntegerDigits[n, 2], 2, 1]], {n, 34}] // Flatten (* Michael De Vlieger, May 09 2017 *)
(define (A227740 n) (- n (+ 1 (A173318 (- (A227737 n) 1)))))
For 11, whose binary expansion is "1011", the run lengths, when starting scanning from the right, are: [2,1,1]. Their partial sums are [2,2+1,2+1+1] = [2,3,4] which sum to total 9, thus a(11)=9. Equivalently, the zero-based positions (counted from the right) where bits change from one to zero or vice versa in "...01011" are 2, 3, 4 and 2+3+4 = 9. For 13, whose binary expansion is "1101", the run lengths similarly scanned are [1,1,2]. Their partial sums are [1,1+1,1+1+2] = [1,2,4] which sum to total 7, thus a(13)=7. Equivalently, the positions where bits change in "...01101" are 1, 2, 4 and 1+2+4 = 7.
Table[Tr[FoldList[Plus,0,Length /@ Split[Reverse[IntegerDigits[n,2]]]] ],{n,71}] (* Wouter Meeussen, Aug 22 2013 *)
a(n)=local(b,s,t);b=binary(n);s=#b;t=b[#b];forstep(i=#b-1,1,-1,if(b[i]!=t,s=s+#b-i;t=!t));s /* Ralf Stephan, Sep 04 2013 */
def A227192(n): '''Sum of the partial sums of the run lengths of binary expansion of n, starting from the least significant end.''' s = 0 b = n%2 i = 0 while (n != 0): n >>= 1 i += 1 if((n%2) != b): b = n%2 s += i return(s)
def a(n)
k = n.to_s(2).scan(/((\d)\2*)/)
k.each_index.map { |i| (i + 1) * k[i][0].size }.reduce(:+)
end # Peter Kagey, Aug 06 2015
(define (A227192 n) (let loop ((i (- (A005811 n) 1)) (s 0)) (cond ((< i 0) s) (else (loop (- i 1) (+ s (A227188bi n i))))))) ;; This version sums the nonzero terms of the n-th row of table A227188. (define (A227192v2 n) (+ (A227183 n) (A000217 (- (A005811 n) 1)))) ;; Another variant. (define (A227192v3 n) (add A227738 (+ 1 (A173318 (- n 1))) (A173318 n))) ;; This sums terms of table A227738. ;; With the help of this function that implements Sum_{i=lowlim..uplim} intfun(i) (define (add intfun lowlim uplim) (let sumloop ((i lowlim) (res 0)) (cond ((> i uplim) res) (else (sumloop (1+ i) (+ res (intfun i)))))))
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