A022122 -id:A022122 - cp's OEIS frontend

A127830 a(n) = Sum_{k=0..n} (binomial(floor(k/2),n-k) mod 2).

1, 1, 1, 2, 2, 1, 2, 3, 3, 3, 2, 2, 3, 2, 3, 5, 5, 4, 4, 5, 4, 3, 3, 3, 4, 4, 3, 4, 5, 3, 5, 8, 8, 7, 6, 7, 7, 5, 6, 8, 7, 6, 5, 5, 5, 4, 4, 5, 6, 5, 5, 7, 6, 4, 5, 6, 7, 7, 5, 6, 8, 5, 8, 13, 13, 11, 10, 12, 11, 8, 9, 11, 11, 10, 8, 9, 10, 7, 9, 13, 12

Offset: 0

Paul Barry, Feb 01 2007

Row sums of number triangle A127829.
From Johannes W. Meijer, Jun 05 2011: (Start)
The Ze3 and Ze4 triangle sums, see A180662 for their definitions, of Sierpinski's triangle A047999 equal this sequence.
The sequences A127830(2^n-p), p>=0, are apparently all Fibonacci like sequences, i.e., the next term is the sum of the two nonzero terms that precede it; see the crossrefs. (End)

Cf.: A000045 (p=0), A000204 (p=7), A001060 (p=13), A000285 (p=14), A022095 (p=16), A022120 (p=24), A022121 (p=25), A022113 (p=28), A022096 (p=30), A022097 (p=31), A022098 (p=32), A022130 (p=44), A022137 (p=48), A022138 (p=49), A022122 (p=52), A022114 (p=53), A022123 (p=56), A022115 (p=60), A022100 (p=62), A022101 (p=63), A022103 (p=64), A022136 (p=79), A022388 (p=80), A022389 (p=88). - Johannes W. Meijer, Jun 05 2011

A127830 := proc(n) local k: option remember: add(binomial(floor(k/2), n-k) mod 2, k=0..n) end: seq(A127830(n), n=0..80); # Johannes W. Meijer, Jun 05 2011

Table[Sum[Mod[Binomial[Floor[k/2],n-k],2],{k,0,n}],{n,0,80}] (* James C. McMahon, Jan 04 2025 *)

def A127830(n): return sum(not ~(k>>1)&n-k for k in range(n+1)) # Chai Wah Wu, Jul 29 2025

a(2^n) = F(n); a(2^(n+1)+1) = L(n).

a(n) mod 2 = A000931(n+5) mod 2 = A011656(n+4).

A199535 Clark Kimberling's even first column Stolarsky array read by antidiagonals.

Original entry on oeis.org

1, 2, 4, 3, 7, 6, 5, 11, 9, 10, 8, 18, 15, 17, 12, 13, 29, 24, 27, 19, 14, 21, 47, 39, 44, 31, 23, 16, 34, 76, 63, 71, 50, 37, 25, 20, 55, 123, 102, 115, 81, 60, 41, 33, 22, 89, 199, 165, 186, 131, 97, 66, 53, 35, 26, 144, 322, 267, 301, 212, 157, 107, 86, 57, 43, 28

Offset: 1

Casey Mongoven, Nov 07 2011

The rows of the array can be seen to have the form A(n, k) = p(n)*Fibonacci(k) + q(n)*Fibonacci(k+1) where p(n) is the sequence {0, 1, 3, 3, 3, 5, 7, 7, 9, 9, 11, 11, 13, 13, 15, 15, 17, ...}{n >= 1} and q(n) is the sequence {1, 3, 3, 7, 2, 9, 9, 13, 13, 17, 17, 19, 19, 23, 23, 25, ...}{n >= 1}. - G. C. Greubel, Jun 23 2022

			The even first column stolarsky array (EFC array), northwest corner:
  1......2.....3.....5.....8....13....21....34....55....89...144 ... A000045;
  4......7....11....18....29....47....76...123...199...322...521 ... A000032;
  6......9....15....24....39....63...102...165...267...432...699 ... A022086;
  10....17....27....44....71...115...186...301...487...788..1275 ... A022120;
  12....19....31....50....81...131...212...343...555...898..1453 ... A013655;
  14....23....37....60....97...157...254...411...665..1076..1741 ... A000285;
  16....25....41....66...107...173...280...453...733..1186..1919 ... A022113;
  20....33....53....86...139...225...364...589...953..1542..2495 ... A022096;
  22....35....57....92...149...241...390...631..1021..1652..2673 ... A022130;
Antidiagonal rows (T(n, k)):
   1;
   2,   4;
   3,   7,   6;
   5,  11,   9,  10;
   8,  18,  15,  17, 12;
  13,  29,  24,  27, 19, 14;
  21,  47,  39,  44, 31, 23, 16;
  34,  76,  63,  71, 50, 37, 25, 20;
  55, 123, 102, 115, 81, 60, 41, 33, 22;

Clark Kimberling, The first column of an interspersion, Fibonacci Quarterly 32 (1994), pp. 301-314.

Cf. A000032, A000045, A000285, A013655, A022086, A022088, A022095.
Cf. A022096, A022097, A022113, A022120, A022121, A022122, A022130.
Cf. A022388, A022390, A035506, A035513, A097135, A199536, A199537.

From G. C. Greubel, Jun 23 2022: (Start)
T(n, 1) = A000045(n+1).
T(n, 2) = A000032(n+1), n >= 2.
T(n, 3) = A022086(n) = A097135(n), n >= 3.
T(n, 4) = A022120(n-2), n >= 4.
T(n, 5) = A013655(n-1), n >= 5.
T(n, 6) = A000285(n-2), n >= 6.
T(n, 7) = A022113(n-4), n >= 7.
T(n, 8) = A022096(n-4), n >= 8.
T(n, 9) = A022130(n-6), n >= 9.
T(n, 10) = A022098(n-5), n >= 10.
T(n, 11) = A022095(n-7), n >= 11.
T(n, 12) = A022121(n-8), n >= 12.
T(n, 13) = A022388(n-10), n >= 13.
T(n, 14) = A022122(n-10), n >= 14.
T(n, 15) = A022097(n-10), n >= 15.
T(n, 16) = A022088(n-10), n >= 16.
T(n, 17) = A022390(n-14), n >= 17.
T(n, n) = A199536(n).
T(n, n-1) = A199537(n-1), n >= 2. (End)

More terms added by G. C. Greubel, Jun 23 2022

A217762 Square array T, read by antidiagonals: T(n,k) = F(n) + 2*F(k) where F(n) is the n-th Fibonacci number.

Original entry on oeis.org

0, 2, 1, 2, 3, 1, 4, 3, 3, 2, 6, 5, 3, 4, 3, 10, 7, 5, 4, 5, 5, 16, 11, 7, 6, 5, 7, 8, 26, 17, 11, 8, 7, 7, 10, 13, 42, 27, 17, 12, 9, 9, 10, 15, 21, 68, 43, 27, 18, 13, 11, 12, 15, 23, 34, 110, 69, 43, 28, 19, 15, 14, 17, 23, 36, 55, 178, 111, 69, 44, 29, 21

Offset: 0

Philippe Deléham, Apr 07 2013

			Square array begins:
...0....2....2....4....6...10...16...26...42...
...1....3....3....5....7...11...17...27...43...
...1....3....3....5....7...11...17...27...43...
...2....4....4....6....8...12...18...28...44...
...3....5....5....7....9...13...19...29...45...
...5....7....7....9...11...15...21...31...47...
...8...10...10...12...14...18...24...34...50...
..13...15...15...17...19...23...29...39...55...
..21...23...23...25...27...31...37...47...63...
..34...36...36...38...40...44...50...60...76...
..55...57...57...59...61...65...71...81...97...
..89...91...91...93...95...99..105..115..131...
.144..146..146..148..150..154..160..170..186...
...

Cf. A000045

T(n,0) = A000045(n).
T(1,k) = A001588(k).
T(n,1) = T(n,2) = A157725(n).
T(n,3) = A157727(n).
T(n,n)= A022086(n) = 3*A000045(n).
T(n+1,n) = A000032(n+1) = A000204(n+1).
T(n+2,n) = A000285(n).
T(n+3,n) = A013655(n+1) = A001060(n+1).
T(n+4,n) = A021120(n).
T(n+5,n) = A022088(n+2) = 5*A000045(n+2).
T(n+6,n) = A022097(n+2).
T(n+7,n) = A022122(n+2).
T(n+8,n) = 3*A013655(n+2).
T(n+9,n) = A097657(n+2).
T(n+10,n) = A022118(n+4).
T(n,n+1) = A000045(n+3).
T(n,n+2) = A013655(n+1) = A001060(n+1).
T(n,n+3) = A000032(n+3).
T(n,n+4) = A022095(n+2).
T(n,n+5) = A022120(n+2).
T(n,n+6) = A022136(n+2).
T(n,n+7) = A022098(n+4).
T(n,n+8) = A022380(n+4).
T(n,n+9) = A206419(n+6).
Sum(T(n-k,k), 0<=k<=n) = 3*A000071(n+2).

cp's OEIS Frontend

A127830 a(n) = Sum_{k=0..n} (binomial(floor(k/2),n-k) mod 2).

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A199535 Clark Kimberling's even first column Stolarsky array read by antidiagonals.

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A217762 Square array T, read by antidiagonals: T(n,k) = F(n) + 2*F(k) where F(n) is the n-th Fibonacci number.

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