A000012 -id:A000012 - cp's OEIS frontend

A166752 Interleave A007583 and A000012.

1, 1, 3, 1, 11, 1, 43, 1, 171, 1, 683, 1, 2731, 1, 10923, 1, 43691, 1, 174763, 1, 699051, 1, 2796203, 1, 11184811, 1, 44739243, 1, 178956971, 1, 715827883, 1, 2863311531, 1, 11453246123, 1, 45812984491, 1, 183251937963, 1, 733007751851, 1

Offset: 0

Paul Barry, Oct 21 2009

Partial sums are A166753.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,5,0,-4).

[(4*4^Floor(n/2)-1)/3 - 2*Floor(2^n/3): n in [0..25]]; // G. C. Greubel, Oct 10 2017

LinearRecurrence[{0, 5, 0, -4}, {1, 1, 3, 1}, 100] (* G. C. Greubel, May 24 2016 *)

x='x+O('x^50); Vec((1+x-2*x^2-4*x^3)/(1-5*x^2+4*x^4)) \\ G. C. Greubel, Oct 10 2017

G.f.: (1+x-2*x^2-4*x^3)/(1-5*x^2+4*x^4).
G.f.: (1+x)/(1-5*x^2+4*x^4) - 2*x^2*(1+2*x)/(1-5*x^2+4*x^4).
a(n) = (4*4^floor(n/2)-1)/3 - 2*floor(2^n/3).
a(n) = 4*4^floor(n/2)/3 - 2*2^n/3 - (-1)^n/3 + 2/3.
a(n) = A002450(floor(n/2)+1) - 2*A000975(n-1).

A168534 Triangle read by rows, A168532 * A000012; as infinite lower triangular matrices.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 5, 2, 1, 1, 7, 1, 1, 1, 1, 11, 4, 2, 1, 1, 1, 15, 1, 1, 1, 1, 1, 1, 22, 5, 2, 2, 1, 1, 1, 1, 30, 3, 3, 1, 1, 1, 1, 1, 1, 42, 8, 2, 2, 2, 1, 1, 1, 1, 1, 56, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 77, 14, 7, 4, 2, 2, 1, 1, 1, 1, 1, 1

Offset: 1

Gary W. Adamson, Nov 28 2009

Row sums = A078392: (1, 3, 5, 9, 11, 20, 21,...).
Triangle A168533 = A000012 * A168532
Left border = the partition numbers, A000041 starting with offset 1.

			First few rows of the triangle =
1;
2, 1;
3, 1, 1;
5, 2, 1, 1;
7, 1, 1, 1, 1;
11, 4, 2, 1, 1, 1;
15, 1, 1, 1, 1, 1, 1;
22, 5, 2, 2, 1, 1, 1, 1;
30, 3, 3, 1, 1, 1, 1, 1, 1;
42, 8, 2, 2, 2, 1, 1, 1, 1, 1;
56, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
77, 14, 7, 4, 2, 2, 1, 1, 1, 1, 1, 1;
101, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
135, 16, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1;
176, 9, 9, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
231, 22, 5, 5, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1;
...

Cf. A168532, A168533, A000041, A078392

Triangle read by rows, A168532 * A000012; where A000012 = an infinite lower
triangular matrix with all 1's. The operation takes partial row sums
starting from the right of each row.

A246964 Limiting sequence of transformations when we start with the all 1's sequence a=A000012 and at step n>=0 replace a(n+a(n)) with Sum_{k=n..n+a(n)} a(k).

Original entry on oeis.org

1, 2, 1, 5, 1, 2, 1, 5, 10, 1, 2, 1, 23, 1, 2, 1, 5, 1, 39, 1, 2, 47, 50, 1, 2, 1, 5, 1, 2, 1, 5, 10, 1, 2, 1, 105, 1, 2, 1, 5, 1, 121, 1, 2, 129, 132, 1, 2, 1, 5, 1, 2, 1, 5, 10, 1, 2, 206, 432, 1, 2, 1, 5, 1, 449, 1, 2, 457, 889, 1, 2, 1, 820, 1, 2, 1, 5, 1

Offset: 0

Floor van Lamoen, Mar 02 2015

			Start . . . . . . . . . . . . . . . . .       : 1,1,1,1,1,...
Step 0: a(0+a(0)) = a(1)<- a(0)+a(1) = 2      : 1,2,1,1,1,...
Step 1: a(1+a(1)) = a(3)<- a(1)+a(2)+a(3) = 4 : 1,2,1,4,1,...
Step 2: a(2+a(2)) = a(3)<- a(2)+a(3) = 5      : 1,2,1,5,1,...

Alois P. Heinz, Table of n, a(n) for n = 0..20000

mx:= 20000:  # maximal index needed
b:= proc() 1 end:
a:= proc(n) option remember; global mx; local t;
      if n<0 then 0 else a(n-1); t:= b(n);
        if n+t<= mx then b(n+t):= add(b(k), k=n..n+t) fi; t
      fi
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 04 2015

mx = 20000; (* Maximal index needed *)
b[_] = 1;
a[n_] := a[n] = Module[{t}, If[n<0, 0, t = b[n]; If[n+t <= mx, b[n+t] = Sum[b[k], {k, n, n+t}]]; t]];
a /@ Range[0, 100] (* Jean-François Alcover, Nov 13 2020, after Alois P. Heinz *)

A127108 Triangle read by rows, A127099 * A000012.

Original entry on oeis.org

1, 5, 2, 7, 3, 3, 17, 10, 4, 4, 11, 5, 5, 5, 5, 35, 23, 15, 6, 6, 6, 15, 7, 7, 7, 7, 7, 7, 49, 34, 20, 20, 8, 8, 8, 8, 34, 21, 21, 9, 9, 9, 9, 9, 9, 55, 37, 25, 25, 25, 10, 10, 10, 10, 10, 23, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 119, 91, 67, 46, 30, 30, 12, 12, 12, 12, 12, 12

Offset: 0

Gary W. Adamson, Jan 05 2007, Jul 27 2008

The operation A000012 * A127099 generates n-th row of the triangle by taking partial sums of n-th row of triangle A127099. Row 4 of A127099 (7, 6, 0, 4) becomes row 4 of A127108: (17, 10, 4, 4).
Row sums = A001001: (1, 7, 13, 35, 31, 91, ...).
Left column of the triangle = A060640: (1, 5, 7, 17, 11, 35, ...).

			First few rows of the triangle:
   1;
   5,  2;
   7,  3,  3;
  17, 10,  4,  4;
  11,  5,  5,  5,  5;
  35, 23, 15,  6,  6,  6;
  15,  7,  7,  7,  7,  7,  7;
  49, 34, 20, 20,  8,  8,  8,  8;
  34, 21, 21,  9,  9,  9,  9,  9,  9;
  55, 37, 25, 25, 25, 10, 10, 10, 10, 10;
  ...

Cf. A060640, A001001, A126988, A127093, A000203, A127094, A123229, A127096, A127098, A127099.

Triangle read by rows, A127099 * A000012.

Edited by N. J. A. Sloane, Aug 13 2008 at the suggestion of R. J. Mathar

A128138 A000012 * A128132.

Original entry on oeis.org

1, 0, 2, 0, 1, 3, 0, 1, 2, 4, 0, 1, 2, 3, 5, 0, 1, 2, 3, 4, 6, 0, 1, 2, 3, 4, 5, 7, 0, 1, 2, 3, 4, 5, 6, 8, 0, 1, 2, 3, 4, 5, 6, 7, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 0, 1, 2, 3

Offset: 1

Gary W. Adamson, Feb 16 2007

			First few rows of the triangle are:
1;
0, 2;
0, 1, 3;
0, 1, 2, 4;
0, 1, 2, 3, 5;
0, 1, 2, 3, 4, 6;
0, 1, 2, 3, 4, 5, 7;
...

Cf. A000012, A128132, A000124 (row sums).

Table[Delete[Range[0, n], -2], {n, 14}] // Flatten (* or *)
Table[If[k == n - 1, k + 1, k], {n, 14}, {k, 0, n - 1}] (* Michael De Vlieger, Apr 26 2016 *)

A000012 * A128132 as infinite lower triangular matrices.
T(n,n) = n.
T(n,k) = k-1, 0

A128219 A000012 * A127701. a(1) = 1, a(2) = 2, a(3) = 2; by rows, n-1 terms of 2, 3, 4, ... followed by "n".

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Feb 19 2007

			First few rows of the triangle:
  1;
  2, 2;
  2, 3, 3;
  2, 3, 4, 4;
  2, 3, 4, 5, 5;
  2, 3, 4, 5, 6, 6;
  2, 3, 4, 5, 6, 7, 7;
  ...

Cf. A000012, A127701, A034856 (row sums), A128220.

trm[i_]:=Join[Range[2,i],{i}]; Flatten[Table[trm[n],{n,13}]] (* Harvey P. Dale, Nov 14 2012 *)

A000012 * A127701 as infinite lower triangular matrices.

A128220 Triangle, A127701 * A000012.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Feb 19 2007

Row sums = A028387: (1, 5, 11, 19, 29, 41, 55, ...) A000012 * A127701 = A128219.

			First few rows of the triangle:
  1;
  3, 2;
  4, 4, 3;
  5, 5, 5, 4;
  6, 6, 6, 6, 5;
  7, 7, 7, 7, 7, 6;
  ...

Cf. A128219, A000012, A127701, A028387.

A127701 * A000012 as infinite lower triangular matrices. Triangle read by rows: a(1) = 1; n-th row = (n-1) terms of (n+1) followed by "n".

A128409 Triangle read by rows: A000012 * A128408 as infinite lower triangular matrices.

Original entry on oeis.org

1, 0, -1, -1, -1, -1, -1, -1, -1, 0, -2, -1, -1, 0, -1, -1, 0, 0, 0, -1, 1, -2, 0, 0, 0, -1, 1, -1, -2, 0, 0, 0, -1, 1, -1, 0, -2, 0, 0, 0, -1, 1, -1, 0, 0, -1, 1, 0, 0, 0, 1, -1, 0, 0, 1, -2, 1, 0, 0, 0, 1, -1, 0, 0, 1, -1, -2, 1, 0, 0, 0, 1, -1, 0, 0, 1, -1, 0, -3, 1, 0, 0, 0, 1, -1, 0, 0, 1, -1, 0, -1

Offset: 1

Gary W. Adamson, Mar 01 2007

Left border = the Mertens sequence: A002321: (1, 0, -1, -1, -2, ...).
Right border = mu(n), A008683: (1, -1, -1, 0, -1, 1, -1, ...).
Row sums = A062563: (1, -1, -3, -3, -5, -1, -3, ...).

			First few rows of the triangle:
   1;
   0, -1;
  -1, -1, -1;
  -1, -1, -1,  0;
  -2, -1, -1,  0, -1;
  -1,  0,  0,  0, -1,  1;
  -2,  0,  0,  0, -1,  1, -1;
  ...

Cf. A002321, A008683, A062563, A128407, A128408.

Previous a(51) = 0 removed and more terms from Georg Fischer, Jun 08 2023

A128521 A128174 * A054525 * A000012.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 2, 2, 1, 1, 0, 0, 1, 2, 1, 1, 1, 3, 3, 2, 2, 1, 1, 0, 0, 1, 2, 2, 2, 1, 1, 1, 3, 3, 3, 3, 2, 2, 1, 1, 0, -1, 1, 2, 2, 3, 2, 2, 1, 1

Offset: 1

Gary W. Adamson, Mar 07 2007

Row sums = A106477: (1, 1, 3, 3, 7, 5, 13, 9, 19, 13, ...). A128522 = A054525 * A128174 * A000012.

			First few rows of the triangle:
  1;
  0, 1;
  1, 1, 1;
  0, 1, 1, 1;
  1, 2, 2, 1, 1;
  0, 0, 1, 2, 1, 1;
  1, 3, 3, 2, 2, 1, 1;
  0, 0, 1, 2, 2, 2, 1, 1;
  ...

Cf. A128174, A054525, A000012, A106477, A128522.

A128174 * A054525 * A000012 as infinite lower triangular matrices.

A128522 A054525 * A128174 * A000012.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Mar 07 2007

Row sums = A123323: (1, 1, 3, 4, 8,7, 15, 14, ...). Left column = A083290: (1, 0, 1, 1, 2, 1, 3, 2, 3, 2, ...) A128521 = A128174 * A054525 * A000012.

			First few rows of the triangle:
  1;
  0, 1;
  1, 1, 1;
  1, 1, 1, 1;
  2, 2, 2, 1, 1;
  1, 1, 1, 2, 1, 1;
  3, 3, 3, 2, 2, 1, 1;
  2, 2, 2, 2, 2, 2, 1, 1;
  3, 3, 3, 3, 3, 2, 2, 1, 1;
  ...

Cf. A054525, A128174, A000012, A083290, A123323, A128521.

A054525 * A128174 * A000012 as infinite lower triangular matrices.

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A166752 Interleave A007583 and A000012.

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A168534 Triangle read by rows, A168532 * A000012; as infinite lower triangular matrices.

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A246964 Limiting sequence of transformations when we start with the all 1's sequence a=A000012 and at step n>=0 replace a(n+a(n)) with Sum_{k=n..n+a(n)} a(k).

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A127108 Triangle read by rows, A127099 * A000012.

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A128138 A000012 * A128132.

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A128219 A000012 * A127701. a(1) = 1, a(2) = 2, a(3) = 2; by rows, n-1 terms of 2, 3, 4, ... followed by "n".

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A128220 Triangle, A127701 * A000012.

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A128409 Triangle read by rows: A000012 * A128408 as infinite lower triangular matrices.

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A128521 A128174 * A054525 * A000012.

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