A033922 -id:A033922 - cp's OEIS frontend

A206774 First differences of A033922.

0, 0, 1, -1, 1, 0, 1, -1, 1, 0, 1, -1, 1, 0, 1, -4, 1, 0, 1, -1, 1, 0, 1, -1, 1, 0, 1, -1, 1, 0, 1, -4, 1, 0, 1, -1, 1, 0, 1, -1, 1, 0, 1, -1, 1, 0, 1, -4, 1, 0, 1, -1, 1, 0, 1, -1, 1, 0, 1, -1, 1, 0, 1, -6, 1, 0, 1, -1, 1, 0, 1, -1, 1, 0, 1, -1, 1, 0, 1, -4, 1, 0, 1, -1, 1, 0, 1, -1, 1, 0, 1, -1, 1, 0, 1, -4, 1, 0, 1, -1, 1, 0, 1, -1, 1

Offset: 1

Benoit Cloitre, Jan 10 2013

Antti Karttunen, Table of n, a(n) for n = 1..16383

Cf. A033922.

up_to = 16383;
A033922list(n) = { my(v, k, e); v=vector(n+1); v[1]=1; for(m=1, n, k=m; e=0; while(k>0, if(k%2, v[m+1]+=v[e+1]); e++; k\=2)); (v); }; \\ From A033922
v033922 = A033922list(16383);
A033922(n) = v033922[1+n];
A206774(n) = (A033922(n)-A033922(n-1)); \\ Antti Karttunen, Nov 06 2018

a(n) = A033922(n) - A033922(n-1).

For n >= 1, a(2n+1)=1, a(4n+2)=0, a(4*A042968(n))=-1, a(16*A042968(n))=-4, a(64*(2n+1))=-6. The values < 0 taken by the sequence are -1,-4,-6,-7, ... see A206775.

More terms from Antti Karttunen, Nov 06 2018

A033639 Base-2 digital convolution sequence.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 6, 1, 3, 4, 13, 4, 11, 21, 49, 13, 17, 24, 62, 66, 103, 145, 338, 128, 297, 376, 1156, 763, 1564, 2592, 6451, 376, 1532, 1139, 4235, 4124, 11714, 8735, 26105, 5263, 21212, 18122, 77153, 35210, 100649, 135748, 369972, 95275, 207638

Offset: 0

David W. Wilson

			Suppose base = 3 and a(0)..a(13) are 1, 1, 2, 1, 3, 7, 6, 20, 52, 6, 26, 104, 32, 162. In base 3, 14 = 112, so we convolve the last three terms with 1, 1, 2 to obtain 104*1 + 32*1 + 162*2 = 460.

Rémy Sigrist, Table of n, a(n) for n = 0..5000
Rémy Sigrist, PARI program for A033639.

Cf. A033640, A033641, A033642, A033643, A033644, A033645, A033646, A033647, A033666, A033922, A260956.

a:= proc(n) option remember; `if`(n=0, 1, (l->
      add(l[i]*a(n-i), i=1..nops(l)))(Bits[Split](n)))
    end:
seq(a(n), n=0..49);  # Alois P. Heinz, Apr 14 2021

a[0] = 1; a[n_] := a[n] = Total[(d = IntegerDigits[n, 2]) * Table[a[n - Length[d] + k - 1], {k, 1, Length[d]}]]; Array[a, 50, 0] (* Amiram Eldar, Jul 25 2023 *)

PARI
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See Links section.
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A273004 Sum of coefficients in the hereditary representation of n in base 2.

Original entry on oeis.org

0, 1, 2, 3, 3, 4, 5, 6, 4, 5, 6, 7, 7, 8, 9, 10, 4, 5, 6, 7, 7, 8, 9, 10, 8, 9, 10, 11, 11, 12, 13, 14, 5, 6, 7, 8, 8, 9, 10, 11, 9, 10, 11, 12, 12, 13, 14, 15, 9, 10, 11, 12, 12, 13, 14, 15, 13, 14, 15, 16, 16, 17, 18, 19, 6, 7, 8, 9, 9, 10, 11, 12, 10, 11, 12, 13, 13, 14, 15, 16, 10, 11, 12, 13, 13, 14, 15, 16, 14, 15, 16, 17, 17, 18, 19, 20, 11, 12, 13, 14, 14

Offset: 0

M. F. Hasler, May 12 2016

The hereditary representation of a number n in base b is a [possibly empty] sum (possibly represented as a list) of monomials of the form m*b^e (possibly represented as a list [m,e] or as a single number m if e = 0) with coefficients 0 < m < b, and the (strictly increasing) exponents e > 0 recursively again expressed in the same form. Thus 0 = [], 1 = 1*b^0 = [1], b = 1*b^1 = [[1, [1]]] etc.

			266 = 1*2^1 + 1*2^(1+1*2^1) + 1*2^(1*2^(1+1*2^1)) which can be represented as [[1, [1]], [1, [1, [1, [1]]]], [1, [[1, [1, [1, [1]]]]]]], and there are 11 "1"s, therefore a(266) = 11.

Andrei Zabolotskii, Table of n, a(n) for n = 0..8191
Eric Weisstein's World of Mathematics, Hereditary Representation.

Cf. A056004, A222112, A273005 (base 10 analog), A033922, A361838.

a[n_] := a[n] = Total[1 + a /@ Log2[DeleteCases[NumberExpand[n, 2], 0]]]; (* Vladimir Reshetnikov, Dec 21 2023 *)

(hr(n,b=2)=if(1<#n=digits(n,b),my(v=if(n[#n],[n[#n]],[]));forstep(i=#n-1,1,-1,n[i]&&v=concat(v,[[n[i],hr(#n-i,b)]]));v,n));(cc(v)=if(type(v)=="t_VEC",sum(i=1,#v,cc(v[i])),v)); a(n)=cc(hr(n))

def A273004(n):
  s=format(n,'b')[::-1]
  return sum(1+A273004(i) for i in range(len(s)) if s[i]=='1') # Pontus von Brömssen, Sep 17 2020

If n = Sum_{j=1..k} 2^e_j where 0 <= e_1 < ... < e_k, then a(n) = k + Sum_{j=1..k} a(e_j). - Pontus von Brömssen, Sep 17 2020

a(n) = A033922(n) + A361838(floor(n/2)) for n > 1. - Andrei Zabolotskii, Aug 24 2025

A033926 Base 6 digital convolution sequence.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 2, 3, 4, 5, 6, 7, 3, 4, 5, 6, 7, 8, 4, 5, 6, 7, 8, 9, 5, 6, 7, 8, 9, 10, 2, 3, 4, 5, 6, 7, 3, 4, 5, 6, 7, 8, 4, 5, 6, 7, 8, 9, 5, 6, 7, 8, 9, 10, 6, 7, 8, 9, 10, 11, 7, 8, 9, 10, 11, 12, 4, 5, 6, 7, 8, 9, 5, 6, 7, 8, 9, 10, 6, 7, 8, 9, 10, 11, 7, 8, 9, 10, 11, 12, 8, 9

Offset: 0

David W. Wilson

			181 = 501 in base 6, so a(181) = 5*a(2)+0*a(1)+1*a(0) = 5*2+0+1 = 11.

Michael De Vlieger, Table of n, a(n) for n = 0..7776

Cf. A007092.

Cf. A033922, A033923, A033924, A033925, A033927, A033928, A033929, A033930.

nn = 90; a[0] = 1; Do[k = Total@ MapIndexed[#1 a[First[#2] - 1] &, Reverse@ IntegerDigits[n, 6]]; a[n] = k, {n, nn}]; Array[a, nn + 1, 0] (* Michael De Vlieger, Nov 03 2022 *)

a(n) = if (n, my(d=digits(n, 6)); sum(k=1, #d, d[k]*a(#d-k)), 1); \\ Michel Marcus, Nov 03 2022

Offset 0 from Michel Marcus, Nov 03 2022

A381957 If n = Sum 2^e(k), then a(n) = Sum 2^a(e(k)), with a(0) = 1.

Original entry on oeis.org

1, 2, 4, 6, 16, 18, 20, 22, 64, 66, 68, 70, 80, 82, 84, 86, 65536, 65538, 65540, 65542, 65552, 65554, 65556, 65558, 65600, 65602, 65604, 65606, 65616, 65618, 65620, 65622, 262144, 262146, 262148, 262150, 262160, 262162, 262164, 262166, 262208, 262210, 262212, 262214, 262224, 262226

Offset: 0

Ilya Gutkovskiy, Mar 11 2025

Replace 2^k in the binary representation of n with 2^a(k).

			25 = 2^4 + 2^3 + 2^0, hence a(25) = 2^a(4) + 2^a(3) + 2^a(0) = 2^16 + 2^6 + 2^1 = 65602.

Cf. A029931, A033922, A073642.

e[n_] := -1 + Position[Reverse[IntegerDigits[n, 2]], 1] // Flatten; a[0] = 1; a[n_] := a[n] = Total[2^a /@ e[n]]; Array[a, 50, 0] (* Amiram Eldar, Mar 11 2025 *)

a(n) = if (n==0, 1, my(v=Vecrev(binary(n))); sum(k=1, #v, if (v[k], 2^a(k-1)))); \\ Michel Marcus, Mar 11 2025

G.f.: 1 + (1/(1 - x)) * Sum_{k>=0} 2^a(k) * x^(2^k) / (1 + x^(2^k)).

cp's OEIS Frontend

A206774 First differences of A033922.

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A033639 Base-2 digital convolution sequence.

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A273004 Sum of coefficients in the hereditary representation of n in base 2.

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A033926 Base 6 digital convolution sequence.

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A381957 If n = Sum 2^e(k), then a(n) = Sum 2^a(e(k)), with a(0) = 1.

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