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An explicit expression for binary digital sums. (English) Zbl 0162.06303


Keywords:

number theory
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Online Encyclopedia of Integer Sequences:

1’s-counting sequence: number of 1’s in binary expansion of n (or the binary weight of n).
a(n) = a(n-1) + sum of digits of n.
a(n) = a(n-1) + square of the sum of digits of n.
Sum of all digits in ternary expansions of 0, ..., n.
a(n) = Sum_{i=0..n} wt(i)^2, where wt(i) = A000120(i).
a(n) = Sum_{i=0..n} wt(i)^3, where wt() = A000120().
a(n) = Sum_{i=0..n} wt(i)^4, where wt() = A000120().
a(n) = Sum_{i=0..n} digsum_3(i)^2, where digsum_3(i) = A053735(i).
a(n) = Sum_{i=0..n} digsum_3(i)^3, where digsum_3(i) = A053735(i).
a(n) = Sum_{i=0..n} digsum_3(i)^4, where digsum_3(i) = A053735(i).
a(n) = Sum_{i=0..n} digsum_4(i), where digsum_4(i) = A053737(i).
a(n) = Sum_{i=0..n} digsum_4(i)^2, where digsum_4(i) = A053737(i).
a(n) = Sum_{i=0..n} digsum_4(i)^3, where digsum_4(i) = A053737(i).
a(n) = Sum_{i=0..n} digsum_4(i)^4, where digsum_4(i) = A053737(i).
a(n) = Sum_{i=0..n} digsum_5(i), where digsum_5(i) = A053824(i).
a(n) = Sum_{i=0..n} digsum_5(i)^2, where digsum_5(i) = A053824(i).
a(n) = Sum_{i=0..n} digsum_5(i)^3, where digsum_5(i) = A053824(i).
a(n) = Sum_{i=0..n} digsum_5(i)^4, where digsum_5(i) = A053824(i).
a(n) = Sum_{i=0..n} digsum_6(i), where digsum_6(i) = A053827(i).
a(n) = Sum_{i=0..n} digsum_6(i)^2, where digsum_6(i) = A053827(i).
a(n) = Sum_{i=0..n} digsum_6(i)^3, where digsum_6(i) = A053827(i).
a(n) = Sum_{i=0..n} digsum_6(i)^4, where digsum_6(i) = A053827(i).
a(n) = Sum_{i=0..n} digsum_7(i), where digsum_7(i) = A053828(i).
a(n) = Sum_{i=0..n} digsum_7(i)^2, where digsum_7(i) = A053828(i).
a(n) = Sum_{i=0..n} digsum_7(i)^3, where digsum_7(i) = A053828(i).
a(n) = Sum_{i=0..n} digsum_7(i)^4, where digsum_7(i) = A053828(i).
a(n) = Sum_{i=0..n} digsum_8(i), where digsum_8(i) = A053829(i).
a(n) = Sum_{i=0..n} digsum_8(i)^2, where digsum_8(i) = A053829(i).
a(n) = Sum_{i=0..n} digsum_8(i)^3, where digsum_8(i) = A053829(i).
a(n) = Sum_{i=0..n} digsum_8(i)^4, where digsum_8(i) = A053829(i).
a(n) = Sum_{i=0..n} digsum_9(i), where digsum_9(i) = A053830(i).
a(n) = Sum_{i=0..n} digsum_9(i)^2, where digsum_9(i) = A053830(i).
a(n) = Sum_{i=0..n} digsum_9(i)^3, where digsum_9(i) = A053830(i).
a(n) = Sum_{i=0..n} digsum_9(i)^4, where digsum_9(i) = A053830(i).
a(n) = Sum_{i=0..n} digsum(i)^3, where digsum(i) = A007953(i).
a(n) = Sum_{i=0..n} digsum(i)^4, where digsum(i) = A007953(i).