A165278 -id:A165278 - cp's OEIS frontend

A165276 Number of even-indexed Fibonacci numbers in the Zeckendorf representation of n.

1, 0, 1, 2, 0, 1, 0, 1, 2, 1, 2, 3, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 1, 2, 3, 1, 2, 1, 2, 3, 2, 3, 4, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 1, 2, 3, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 1, 2, 3, 1, 2, 1, 2, 3, 2, 3, 4, 1, 2, 1, 2, 3, 1, 2, 1, 2, 3, 2, 3, 4, 2, 3, 2, 3, 4, 3, 4, 5, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 1, 2, 3, 0, 1, 0, 1

Offset: 1

Clark Kimberling, Sep 12 2009

nonn

We begin the indexing at 2; that is, 1=F(2), 2=F(3), 3=F(4), 5=F(5), ...

For a count of odd-indexed Fibonacci summands, see A165277.

			6 = 5 + 1 = F(5) + F(2), so that a(6) = 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A014417, A165277, A165278, A165279.

fibEvenCount[n_] := Plus @@ (Reverse@IntegerDigits[n, 2])[[1 ;; -1 ;; 2]]; fibEvenCount /@ Select[Range[1000], BitAnd[#, 2 #] == 0 &] (* Amiram Eldar, Jan 20 2020 *)

A165277 Number of odd-indexed Fibonacci numbers in the Zeckendorf representation of n.

Original entry on oeis.org

0, 1, 0, 0, 1, 1, 2, 0, 0, 1, 0, 0, 1, 1, 2, 1, 1, 2, 2, 3, 0, 0, 1, 0, 0, 1, 1, 2, 0, 0, 1, 0, 0, 1, 1, 2, 1, 1, 2, 2, 3, 1, 1, 2, 1, 1, 2, 2, 3, 2, 2, 3, 3, 4, 0, 0, 1, 0, 0, 1, 1, 2, 0, 0, 1, 0, 0, 1, 1, 2, 1, 1, 2, 2, 3, 0, 0, 1, 0, 0, 1, 1, 2, 0, 0, 1, 0, 0, 1, 1, 2, 1, 1, 2, 2, 3, 1, 1, 2, 1, 1, 2, 2, 3, 2

Offset: 1

Clark Kimberling, Sep 12 2009

nonn

We begin the indexing at 2; that is, 1=F(2), 2=F(3), 3=F(4), 5=F(5), ...

For a count of even-indexed Fibonacci summands, see A165276.

			6 = 5 + 1 = F(5) + F(2), so that a(6) = 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A014417, A165276, A165278, A165279.

fibOddCount[n_] := Plus @@ (Reverse@IntegerDigits[n, 2])[[2 ;; -1 ;; 2]]; fibOddCount /@ Select[Range[1000], BitAnd[#, 2 #] == 0 &] (* Amiram Eldar, Jan 20 2020 *)

A165279 Table read by antidiagonals: T(n, k) is the k-th number with n-1 odd-indexed Fibonacci numbers in its Zeckendorf representation.

Original entry on oeis.org

1, 3, 2, 4, 5, 7, 8, 6, 15, 20, 9, 10, 18, 41, 54, 11, 13, 19, 49, 109, 143, 12, 14, 28, 52, 130, 287, 376, 21, 16, 36, 53, 138, 342, 753, 986, 22, 17, 39, 75, 141, 363, 897, 1973, 2583, 24, 23, 40, 96, 142, 371, 952, 2350, 5167, 6764, 25, 26, 44, 104, 198, 374

Offset: 1

Clark Kimberling, Sep 13 2009

For n>=0, row n is the monotonic sequence of positive integers m such that the number of odd-indexed Fibonacci numbers in the Zeckendorf representation of m is n.
We begin the indexing at 2; that is, 1=F(2), 2=F(3), 3=F(4), 5=F(5),...
Every positive integer occurs exactly once in the array, so that as a sequence it is a permutation of the positive integers.
For counts of even-indexed Fibonacci numbers, see A165278.
Essentially, (row 0)=A054204, (column 1)=A035508.

			Northwest corner:
1....3....4....8....9...11...12...21...22...
2....5....6...10...13...14...16...17...23...
7...15...18...19...28...36...39...40...44...
20..41...49...52...53...75...96..104..107...
Examples:
12=8+3+1=F(6)+F(4)+F(2), zero odds, so 12 is in row 0.
28=21+5+2=F(8)+F(5)+F(3), two odds, so 28 is in row 2.

Cf. A165276, A165277, A165278.

f[n_] := Module[{i = Ceiling[Log[GoldenRatio, Sqrt[5]*n]], v = {}, m = n}, While[i > 1, If[Fibonacci[i] <= m, AppendTo[v, 1]; m -= Fibonacci[i], If[v != {}, AppendTo[v, 0]]]; i--]; Total[Reverse[v][[1 ;; -1 ;; 2]]]]; T = GatherBy[SortBy[ Range[10^4], f], f]; Table[Table[T[[n - k + 1, k]], {k, n, 1, -1}], {n, 1, Length[T]}] // Flatten (* Amiram Eldar, Feb 04 2020 *)

More terms from Amiram Eldar, Feb 04 2020

A165274 Table read by antidiagonals: T(n, k) is the k-th number with n-1 even-power summands in its base 2 representation.

Original entry on oeis.org

2, 8, 1, 10, 3, 5, 32, 4, 7, 21, 34, 6, 13, 23, 85, 40, 9, 15, 29, 87, 341, 42, 11, 17, 31, 93, 343, 1365, 128, 12, 19, 53, 95, 349, 1367, 5461, 130, 14, 20, 55, 117, 351, 1373, 5463, 21845, 136, 16, 22, 61, 119, 373, 1375, 5469, 21847, 87381, 138, 18, 25, 63

Offset: 1

Clark Kimberling, Sep 12 2009

For n>=0, row n is the ordered sequence of positive integers m such that the number of even powers of 2 in the base 2 representation of m is n.
Every positive integer occurs exactly once in the array, so that as a sequence it is a permutation of the positive integers.
For odd powers, see A165275.
For the number of even powers of 2 in the base 2 representation of n, see A139351; for odd, see A139352.
Essentially, (Row 0)=A062880, (Row 1)=A158705, (Column 1)=A002450, also possibly (Column 2)=A163832.

			Northwest corner:
2....8...10...32...34...40...42...129
1....3....4....6....9...11...12...14
5....7...13...15...17...19...20...22
21..23...29...31...53...55...61...63
Examples:
40 = 32 + 8 = 2^5 + 2^3, so that 40 is in row 0.
13 = 8 + 4 + 1 = 2^3 + 2^2 + 2^0, so that 13 is in row 2.

Cf. A139351, A139352, A165275, A165276, A165277, A165278, A165279.

f[n_] := Total[(Reverse@IntegerDigits[n, 2])[[1 ;; -1 ;; 2]]]; T = GatherBy[ SortBy[Range[10^5], f], f]; Table[Table[T[[n - k + 1, k]], {k, n, 1, -1}], {n, 1, Length[T]}] // Flatten (* Amiram Eldar, Feb 04 2020*)

More terms from Amiram Eldar, Feb 04 2020

A165275 Table read by antidiagonals: T(n, k) is the k-th number with n-1 odd-power summands in its base 2 representation.

Original entry on oeis.org

1, 4, 2, 5, 3, 10, 16, 6, 11, 42, 17, 7, 14, 43, 170, 20, 8, 15, 46, 171, 682, 21, 9, 26, 47, 174, 683, 2730, 64, 12, 27, 58, 175, 686, 2731, 10922, 65, 13, 30, 59, 186, 687, 2734, 10923, 43690, 68, 18, 31, 62, 187, 698, 2735, 10926, 43691, 174762, 69, 19, 34

Offset: 1

Clark Kimberling, Sep 12 2009

For n>=0, row n is the ordered sequence of positive integers m such that the number of odd powers of 2 in the base 2 representation of m is n.
Every positive integer occurs exactly once in the array, so that as a sequence it is a permutation of the positive integers.
For even powers, see A165274. For the number of even powers of 2 in the base 2 representation of n, see A139351; for odd, see A139352.
Essentially, (Row 0)=A000695, (Column 1)=A020988, also possibly (Column 2)=A007583.
It appears that, for n>=3, a(t(n)) = 4*a(t(n-1))+2, where t(n) is the n-th triangular number t(n)=n(n+1)/2 (A000217). [John W. Layman, Sep 15 2009]

			Northwest corner:
  1....4....5...16...17...20...21...64
  2....3....6....7....8....9...12...13
  10..11...14...26...27...30...31...34
  42..43...46...47...58...59...62...63
Examples:
20 = 16 + 4 = 2^4 + 2^2, so that 20 is in row 0.
13 = 8 + 4 + 1 = 2^3 + 2^2 + 2^0, so that 13 is in row 1.

Cf. A139351, A139352, A165274, A165276, A165277, A165278, A165279.

Cf. A000217.

f[n_] := Total[(Reverse@IntegerDigits[n, 2])[[2 ;; -1 ;; 2]]]; T = GatherBy[ SortBy[Range[10^5], f], f]; Table[Table[T[[n - k + 1, k]], {k, n, 1, -1}], {n, 1, Length[T]}] // Flatten (* Amiram Eldar, Feb 04 2020*)

a(27) corrected and a(28)-a(54) added by John W. Layman, Sep 15 2009

More terms from Amiram Eldar, Feb 04 2020

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A165276 Number of even-indexed Fibonacci numbers in the Zeckendorf representation of n.

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A165277 Number of odd-indexed Fibonacci numbers in the Zeckendorf representation of n.

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A165279 Table read by antidiagonals: T(n, k) is the k-th number with n-1 odd-indexed Fibonacci numbers in its Zeckendorf representation.

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A165274 Table read by antidiagonals: T(n, k) is the k-th number with n-1 even-power summands in its base 2 representation.

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A165275 Table read by antidiagonals: T(n, k) is the k-th number with n-1 odd-power summands in its base 2 representation.

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