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.
For n = 42: - A063037(1+42) = 86, - the binary expansion of 86 is "1010110", - reversing run lengths yields "1001010", - this corresponds to 74 = A063037(1+34), - hence a(42) = 34.
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For n = 42: - A166535(42) = 50, - the binary expansion of 50 is "110010", - reversing run lengths yields "101100", - this corresponds to 44 = A166535(38), - hence a(42) = 38.
See Links section.
By definition a(0)=0 and a(1)=1. a(2)=2, as 2 = 10 in binary, thus its run counts are (1 1), which reversed yields the same (1 1), from which we collect a binary number 10, i.e. 2 in decimal. a(4)=6, as 4 = 100 in binary, thus its run counts are (1 2), which reversed and each element mapped through a itself yields (2 1), which are run counts of binary number 110, i.e. 6 in decimal. a(16)=126, as 16 = 10000 in binary, thus its run counts are (1 4), which reversed yields (4 1) and a(4)=6 and a(1)=1, thus we find that the run counts (6 1) form a binary 1111110 i.e. 126 in decimal.
(define (A105726 n) (cond ((< n 2) n) (else (runcount1list->binexp (map A105726 (reverse! (binexp->runcount1list n))))))) (define (binexp->runcount1list n) (if (zero? n) (list) (let loop ((n n) (rc (list)) (count 0) (prev-bit (modulo n 2))) (if (zero? n) (cons count rc) (if (eq? (modulo n 2) prev-bit) (loop (floor->exact (/ n 2)) rc (1+ count) (modulo n 2)) (loop (floor->exact (/ n 2)) (cons count rc) 1 (modulo n 2))))))) (define (runcount1list->binexp lista) (let loop ((lista lista) (s 0) (state 1)) (cond ((null? lista) s) (else (loop (cdr lista) (+ (* s (expt 2 (car lista))) (* state (- (expt 2 (car lista)) 1))) (- 1 state))))))
For n = 42: - A044813(42) = 159, - the binary expansion of 159 is "10011111", - reversing run lengths yields "11111001", - this corresponds to 249 = A044813(52), - hence a(42) = 52.
See Links section.
For n = 42: - A031443(42) = 210, - the binary expansion of 210 is "11010010", - reversing run lengths yields "10110100", - this corresponds to 180 = A031443(33), - hence a(42) = 33.
See Links section.
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
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