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 = 4, the only blocks that occur are {0101, 0110, 0111, 1010, 1011, 1101, 1110, 1111}, so a(4) = 8.
a010061 n = a010061_list !! (n-1) a010061_list = filter ((== 0) . a228085) [1..] -- Reinhard Zumkeller, Oct 13 2013
# For Maple code see A230091. - N. J. A. Sloane, Oct 10 2013
Table[n + Total[IntegerDigits[n, 2]], {n, 0, 300}] // Complement[Range[Last[#]], #]& (* Jean-François Alcover, Sep 03 2013 *)
/* Gen(n, b) returns a list of the generators of n in base b. Written by Max Alekseyev (see Alekseyev et al., 2021). For example, Gen(101, 10) returns [91, 101]. - N. J. A. Sloane, Jan 02 2022 */ { Gen(u, b=10) = my(d, m, k); if(u<0 || u==1, return([]); ); if(u==0, return([0]); ); d = #digits(u, b)-1; m = u\b^d; while( sumdigits(m, b) > u - m*b^d, m--; if(m==0, m=b-1; d--; ); ); k = u - m*b^d - sumdigits(m, b); vecsort( concat( apply(x->x+m*b^d, Gen(k, b)), apply(x->m*b^d-1-x, Gen((b-1)*d-k-2, b)) ) ); }
a092391 n = n + a000120 n -- Reinhard Zumkeller, May 13 2012
Table[n + Total[IntegerDigits[n, 2]], {n, 0, 100}] (* Jean-François Alcover, Sep 03 2013 *)
A092391(n)=n+hammingweight(n) \\ M. F. Hasler, Oct 05 2013
def a(n): return n + bin(n).count("1")
print([a(n) for n in range(72)]) # Michael S. Branicky, May 26 2022
a228085 n = length $ filter ((== n) . a092391) [n - a070939 n .. n] -- Reinhard Zumkeller, Oct 13 2013
For Maple code see A230091. - N. J. A. Sloane, Oct 10 2013 # Find all inverses of m under x -> x + wt(x) - N. J. A. Sloane, Oct 19 2013 A000120 := proc(n) local w, m, i; w := 0; m := n; while m > 0 do i := m mod 2; w := w+i; m := (m-i)/2; od; w; end: wt := A000120; F:=proc(m) local ans,lb,n,i; lb:=m-ceil(log(m+1)/log(2)); ans:=[]; for n from max(1,lb) to m do if (n+wt(n)) = m then ans:=[op(ans),n]; fi; od: [seq(ans[i],i=1..nops(ans))]; end;
nmax = 8191; Clear[a]; a[_] = 0;
Scan[Set[a[#[[1]]], #[[2]]]&, Tally[Table[n + DigitCount[n, 2, 1], {n, 0, nmax}]]];
a /@ Range[0, nmax] (* Jean-François Alcover, Oct 29 2019 *)
a[n_] := Module[{k, cnt = 0}, For[k = n - Floor[Log[2, n]] - 1, k < n, k++, If[n == k + DigitCount[k, 2, 1], cnt++]]; cnt];
a /@ Range[0, 100] (* Jean-François Alcover, Nov 28 2020 *)
0 is in this sequence because there is a unique k such that k+A000120(k)=0, in this case k=0. 1 is not in this sequence because there is no such k that k+A000120(k) would be 1. (Instead 1 is in A010061). 2 is in this sequence because there is exactly one k that satisfies k+A000120(k)=2, namely k=1. 3 is in this sequence because there is exactly one k that satisfies k+A000120(k)=3, namely k=2. 4 is not in this sequence because there is no such k that k+A000120(k) would be 4. (Instead 4 is in A010061.) 5 is not in this sequence because there is more than one k that satisfies k+A000120(k)=5, namely k=3 and k=4.
a228088 n = a228088_list !! (n-1) a228088_list = 0 : filter ((== 1) . a228085) [1..] -- Reinhard Zumkeller, Oct 13 2013
For Maple code see A230091. - N. J. A. Sloane, Oct 10 2013
0 has no children distinct from itself (we only have A092391(0)=0), so we define a(0) = (0+1) = 1,
1 has no children (it is one of the terms of A010061), so a(1) = (0+1) = 1,
4 and 6 are also members of A010061, so both a(4) and a(6) = (0+1) = 1,
7 has 1,2,3,4 and 5 among its descendants (as A092391(5)=7, A092391(3)=A092391(4)=5, A092391(2)=3, A092391(1)=2), so a(7) = (5+1) = 6,
8 has 6 as a child value, so a(8) = (1+1) = 2,
9 has 6 and 8 as descendants, so a(9) = (2+1) = 3,
10 has {1,2,3,4,5,7} so a(10) = (6+1) = 7.
;; A deficient definition which works only up to n=128: (definec (A227643deficient n) (cond ((zero? n) 1) ((zero? (A228085 n)) 1) ((= 1 (A228085 n)) (+ 1 (A227643deficient (A228086 n)))) ((= 2 (A228085 n)) (+ 1 (A227643deficient (A228086 n)) (A227643deficient (A228087 n)))) (else (error "Not yet implemented for cases where n has more than two immediate children!")))) ;; Another definition that works for all n, but is somewhat slower: (definec (A227643full n) (cond ((zero? n) 1) (else (+ 1 (add (lambda (i) (if (= (A092391 i) n) (A227643full i) 0)) (A228086 n) (A228087 n)))))) ;; Auxiliary function add 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))))))) ;; by Antti Karttunen, Aug 16 2013, macro definec can be found in his IntSeq-library.
The top-left corner of the square array: 1, 2, 3, 5, 7, 10, 12, 14, ... 4, 5, 7, 10, 12, 14, 17, 19, ... 6, 8, 9, 11, 14, 17, 19, 22, ... 13, 16, 17, 19, 22, 25, 28, 31, ... 15, 19, 22, 25, 28, 31, 36, 38, ... 18, 20, 22, 25, 28, 31, 36, 38, ... 21, 24, 26, 29, 33, 35, 38, 41, ... 23, 27, 31, 36, 38, 41, 44, 47, ... ... The non-initial terms on each row are obtained by adding to the preceding term the number of 1-bits in its binary representation (A000120).
nmax0 = 100;
nmax := Length[col[1]];
col[1] = Table[n + DigitCount[n, 2, 1], {n, 0, nmax0}] // Complement[Range[Last[#]], #]&;
col[k_] := col[k] = col[k - 1] + DigitCount[col[k-1], 2, 1];
T[n_, k_] := col[k][[n]];
Table[T[n-k+1, k], {n, 1, nmax}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Nov 28 2020 *)
0.2526602590088829221550627143278941418252...
m0 = 100; dm = 100; digits = 100; Clear[lambda]; lambda[m_] := lambda[m] = Total[1/2^Union[Table[n + Total[IntegerDigits[n, 2]], {n, 0, m}]]]^2/8 // N[#, 2*digits]& // RealDigits[#, 10, 2*digits]& // First; lambda[m0]; lambda[m = m0 + dm]; While[lambda[m] != lambda[m - dm], Print["m = ", m]; m = m + dm]; lambda[m][[1 ;; digits]]
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