A002024 -id:A002024 - cp's OEIS frontend

A131950 A002024 + A131821 + A007318 - 2*A000012 as infinite lower triangular matrices.

1, 3, 3, 5, 4, 5, 7, 6, 6, 7, 9, 8, 10, 8, 9, 11, 10, 15, 15, 10, 11, 13, 12, 21, 26, 21, 12, 13, 15, 14, 28, 42, 42, 28, 14, 15, 17, 16, 36, 64, 78, 64, 36, 16, 17, 19, 18, 45, 93, 135, 135, 93, 45, 18, 19

Offset: 0

Gary W. Adamson, Jul 30 2007

Row sums = A131951: (1, 6, 14, 26, 44, 72, ...).

			First few rows of the triangle:
   1;
   3,  3;
   5,  4,  5;
   7,  6,  6,  7;
   9,  8, 10,  8,  9;
  11, 10, 15, 15, 10, 11;
  13, 12, 21, 26, 21, 12, 13;
  15, 14, 28, 42, 42, 28, 14, 15;
  ...

Cf. A002024, A131821, A007318, A000012, A131951.

A132071 A007318 + A002024 - A103451 as infinite lower triangular matrices.

Original entry on oeis.org

1, 2, 2, 3, 5, 3, 4, 7, 7, 4, 5, 9, 11, 9, 5, 6, 11, 16, 16, 11, 6, 7, 13, 22, 27, 22, 13, 7, 8, 15, 29, 43, 43, 29, 15, 8, 9, 17, 37, 65, 79, 65, 37, 17, 9, 10, 19, 46, 94, 136, 136, 94, 46, 19, 10

Offset: 0

Gary W. Adamson, Aug 09 2007

Row sums = A132072: (1, 4, 11, 22, 39, 66, 111, ...).

			First few rows of the triangle:
  1;
  2,  2;
  3,  5,  3;
  4,  7,  7,  4;
  5,  9, 11,  9,  5;
  6, 11, 16, 16, 11,  6;
  7, 13, 22, 27, 22, 13,  7;
  ...

Cf. A007318, A002024, A103451, A132072.

A135471 a(n) = Sum{i=1..n} ( i2^(i-1) ) - ( A002024(n)(A002024(n)+1)/2 - n ) * 2^(A002024(n)-1).

Original entry on oeis.org

1, 3, 17, 41, 125, 321, 745, 1777, 4089, 9217, 20417, 45009, 98273, 212977, 458753, 982881, 2097025, 4456353, 9437121, 19922913, 41943041, 88080001, 184549057, 385875713, 805306177, 1677721473, 3489660865, 7247757313, 15032384641, 31138512129, 64424508801

Offset: 1

N. J. A. Sloane, Feb 08 2008

nonn

At one time this expression was proposed (erroneously) as a formula for A007664.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A007664.

f[n_] := Floor[Sqrt[2*n] + 1/2]; a[n_] := Sum[ i*2^(i - 1), {i, 1, n}] - (f[n]*(f[n] + 1)/2 - n)*2^(f[n] - 1); Table[a[n], {n, 1, 25}] (* G. C. Greubel, Oct 14 2016 *)

A182245 Integers n such that either (a) A186053(n) does not equal A002024(n) + A182298(A025581(n)) or (b) A182298(n) does not equal 1 + A002024(n) + A186053(A025581(n)).

Original entry on oeis.org

0, 2, 4, 7, 8, 11, 16, 17, 29, 92, 125, 154, 155, 174, 361, 390, 441, 473, 529, 564, 601, 637, 704, 742, 743, 783, 837, 1003, 1147, 1184, 1340, 1341, 1380, 1394, 1548, 1549, 1606, 1665, 1771, 1772, 1833, 1896

Offset: 1

Patrick Devlin, Apr 23 2012

Comprehensive list of the 177 terms can be found in the paper. The link proves this list of counterexamples is comprehensive, but there is nothing currently known that explains the behavior of this list.

Michel Marcus, Table of n, a(n) for n = 1..177 (from Devlin papers).
Patrick Devlin, Sets with High Volume and Low Perimeter, arXiv:1107.2954 [math.CO], 2011.
Patrick Devlin, Integer Subsets with High Volume and Low Perimeter, arXiv:1202.1331 [math.CO], 2012.
Patrick Devlin, Integer Subsets with High Volume and Low Perimeter, INTEGERS, Vol. 12, #A32.

Cf. A186053.

Missing term 1548 added by Michel Marcus, May 27 2015

A182246 Integers n such that A186053(n) does not equal A002024(n) + A182298(A025581(n)).

Original entry on oeis.org

2, 4, 7, 8, 11, 16, 17, 29, 125, 154, 155, 174, 390, 473, 564, 601, 637, 704, 742, 743, 783, 1147, 1340, 1341, 1394, 1549, 1606, 1665, 1771, 1772, 1833, 1896

Offset: 1

Patrick Devlin, Apr 23 2012

A comprehensive list of the 114 counterexamples can be found in the Devlin paper.

Patrick Devlin, Sets with High Volume and Low Perimeter, arXiv:1107.2954 [math.CO], 2011.
Patrick Devlin, Integer Subsets with High Volume and Low Perimeter, arXiv:1202.1331 [math.CO], 2012.
Patrick Devlin, Integer Subsets with High Volume and Low Perimeter, INTEGERS, Vol. 12, #A32.

A218036 a(n) = (n+1) + (n+3/2)*H(n) - (H(n)^3)/2, where H(n) = A002024(n).

Original entry on oeis.org

4, 6, 9, 8, 12, 16, 10, 15, 20, 25, 12, 18, 24, 30, 36, 14, 21, 28, 35, 42, 49, 16, 24, 32, 40, 48, 56, 64, 18, 27, 36, 45, 54, 63, 72, 81, 20, 30, 40, 50, 60, 70, 80, 90, 100, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144

Offset: 1

Michel Marcus, Oct 19 2012

All terms are composite.

			Sequence can be seen as a triangle that begins:
   4;
   6,  9;
   8, 12, 16;
  10, 15, 20, 25;
  12, 18, 24, 30, 36;
  14, 21, 28, 35, 42, 49;
  16, 24, 32, 40, 48, 56, 64;
  ...

Vincenzo Librandi, Table of n, a(n) for n = 1..3655 (Rows n=1..85 of triangle, flattened).
Blake Ralston, Elemental Complete Composite Number Generators, The Fibonacci Quarterly, Volume 23, Number 2, May 1985, pp. 149-150.

Cf. A002024.

/* As triangle */ [[n*k +n + k+1: k in [1..n]]: n in [1.. 20]]; // Vincenzo Librandi, Jan 27 2025

Table[(k+1)*(n+1),{n,1,11},{k,1,n}]//Flatten (* Stefano Spezia, Nov 23 2019 *)

from math import isqrt
def A218036(n): return ((m:=isqrt((k:=n<<1)<<2)+1>>1)*(k+3-m**2)>>1)+n+1 # Chai Wah Wu, Jun 14 2025

a(n) = (A002024(n)+1)*(n+1-A002024(n)*(A002024(n)-1)/2).

As a triangle: T(n, k) = (k + 1)*(n + 1) with 1 <= k <= n. - Stefano Spezia, Nov 23 2019

A293670 Square array made of (W, N, S, E) quadruplets read by antidiagonals. Numeric structure of an anamorphosis of A002024 (see comments).

Original entry on oeis.org

1, -1, 0, 2, 1, 0, 2, -1, 1, 2, 0, 3, 1, 1, 2, 0, 3, -1, 2, 2, 1, 3, 0, 4, 1, 2, 2, 1, 3, 0, 4, -1, 3, 2, 2, 3, 1, 4, 0, 5, 1, 3, 2, 2, 3, 1, 4, 0, 5, -1, 4, 2, 3, 3, 2, 4, 1, 5, 0, 6, 1, 4, 2, 3, 3, 2, 4, 1, 5, 0, 6, -1, 5, 2, 4, 3, 3, 4, 2, 5, 1, 6, 0, 7, 1, 5, 2, 4, 3, 3, 4, 2, 5, 1, 6, 0, 7, -1, 6, 2, 5, 3, 4, 4, 3, 5, 2, 6, 1, 7, 0, 8, 1, 6, 2, 5, 3, 4, 4, 3, 5, 2, 6, 1, 7, 0, 8, -1, 7, 2, 6, 3, 5, 4, 4, 5, 3, 6, 2, 7, 1, 8, 0, 9, 1, 7, 2, 6, 3, 5, 4, 4, 5, 3, 6, 2, 7, 1, 8, 0, 9, -1

Offset: 1

Luc Rousseau, Oct 14 2017

Numeric characterization:
Row n is the value of a list after n iterations of the following algorithm:
- start with an empty list (assimilable to row number 0)
- Iteration n consists of
-- if n is odd, appending 1 to the left of the list and -1 to the right;
-- if n is even, replacing each value in the list by its complement to n/2.
Underlying definition and interest: this sequence represents a square array in which each cell is a structure made of 4 values arranged in W/N/S/E fashion. These values are twice the areas of elementary right triangles that enter the composition of quadrilaterals delimited by two families of lines, with the following equations:
- for m = 1, 2, 3, ...: y = mx - (m-1)^2 {x <= m-1}
- for n = -1, 0, 1, ...: y = -nx - (n+1)^2 {x >= 1-n}
Globally these quadrilaterals form an anamorphosis of A002024. See provided link for explanations and illustrations.

			Array begins (characterization)(x stands for -1):
              1 x
              0 2
            1 0 2 x
            1 2 0 3
          1 1 2 0 3 x
          2 2 1 3 0 4
        1 2 2 1 3 0 4 x
        3 2 2 3 1 4 0 5
      1 3 2 2 3 1 4 0 5 x
      4 2 3 3 2 4 1 5 0 6
    1 4 2 3 3 2 4 1 5 0 6 x
    5 2 4 3 3 4 2 5 1 6 0 7
  1 5 2 4 3 3 4 2 5 1 6 0 7 x
Or (definition)(to be read by antidiagonals):
    x     x     x     x
  1   2 2   3 3   4 4   5 ...
    0     0     0     0
    0     0     0     0
  1   2 2   3 3   4 4   5 ...
    1     1     1     1
    1     1     1     1
  1   2 2   3 3   4 4   5 ...
    2     2     2     2
    2     2     2     2
  1   2 2   3 3   4 4   5 ...
    3     3     3     3
    3     3     3     3
  1   2 2   3 3   4 4   5 ...
    4     4     4     4
  ...

Luc Rousseau, Relation between A293670 and A002024 - Numeric structure of an anamorphosis

Cf. A293578, A002024.

evolve(L,n)=if(n%2==1,listinsert(L,1,1);listinsert(L,-1,#L+1),L=apply(v->n/2-v,L));L
N=30;L=List();for(n=1,N,L=evolve(L,n);for(i=1,#L,print1(L[i],", "));print())

A379777 Array A(n, k), n, k >= 0, read by upward antidiagonals; for any v >= 0, the value appears twice in the array: in row A002262(v) and in row A002024(v+1); values in each row are given in strictly increasing order.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 3, 2, 4, 6, 6, 4, 5, 7, 10, 10, 7, 5, 8, 11, 15, 15, 11, 8, 9, 12, 16, 21, 21, 16, 12, 9, 13, 17, 22, 28, 28, 22, 17, 13, 14, 18, 23, 29, 36, 36, 29, 23, 18, 14, 19, 24, 30, 37, 45, 45, 37, 30, 24, 19, 20, 25, 31, 38, 46, 55, 55, 46, 38, 31, 25, 20, 26, 32, 39, 47, 56, 66

Offset: 0

Rémy Sigrist, Jan 02 2025

This sequence was inspired by the game Dobble: this game is based on cards with symbols such that two distinct cards always have exactly one common symbol. Here, two distinct rows have exactly one common term.
This square array combines two symetrical copies of the triangular view of A001477 (the nonnegative integers):
     0 1 3 6 .
       2 4 7 .      0 1 3 6 .
         5 8 .      0 2 4 7 .
     0     9 .  ->  1 2 5 8 .
     1 2     .      3 4 5 9 .
     3 4 5          6 7 8 9 .
     6 7 8 9        . . . . .
     . . . . .

			Array A(n, k) begins:
  n\k |  0   1   2   3   4   5   6   7   8   9
  ----+---------------------------------------
    0 |  0   1   3   6  10  15  21  28  36  45
    1 |  0   2   4   7  11  16  22  29  37  46
    2 |  1   2   5   8  12  17  23  30  38  47
    3 |  3   4   5   9  13  18  24  31  39  48
    4 |  6   7   8   9  14  19  25  32  40  49
    5 | 10  11  12  13  14  20  26  33  41  50
    6 | 15  16  17  18  19  20  27  34  42  51
    7 | 21  22  23  24  25  26  27  35  43  52
    8 | 28  29  30  31  32  33  34  35  44  53
    9 | 36  37  38  39  40  41  42  43  44  54
   10 | 45  46  47  48  49  50  51  52  53  54

Rémy Sigrist, Table of n, a(n) for n = 0..10010
Wikipedia, Dobble

Cf. A000096, A000217, A001477, A002024, A002262.

A(n, k) = { my (x, y); if (n > k, x = n-1; y = k, x = k; y = n;); x*(x+1)/2 + y }

A(0, k) = A000217(k).
A(n, k) = A(k+1, n) = A000217(k) + n for any n in 0..k.
A(n, n) = A000096(n).

A065979 Binomial transform of A002024.

Original entry on oeis.org

1, 3, 7, 16, 36, 79, 170, 362, 767, 1619, 3402, 7112, 14797, 30673, 63427, 130951, 270031, 556111, 1143537, 2347476, 4810758, 9843932, 20118655, 41081297, 83832648, 170987463, 348581862, 710242310, 1446198858

Offset: 1

Robert A. Stump (bee_ess107(AT)yahoo.com), Dec 09 2001

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 97.

F. Ellermann, Illustration of binomial transforms
N. J. A. Sloane, Transforms

Cf. A001792, A002024.

rep(n) = floor(1/2 + sqrt(2*n)); /* A002024 */
a(n) = sum(k = 0, n-1, binomial(n-1, k) * rep(k+1)); \\ Michel Marcus, Aug 31 2013

A076922 Smallest number greater than the previous term such that the highest common factor of successive pairs follows the pattern 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, ... (n repeated n times, A002024).

Original entry on oeis.org

1, 2, 4, 6, 9, 15, 24, 28, 32, 36, 40, 45, 50, 55, 65, 90, 96, 102, 108, 114, 120, 126, 133, 140, 147, 154, 161, 175, 224, 232, 240, 248, 256, 264, 272, 280, 288, 297, 306, 315, 324, 333, 342, 351, 369, 450, 460, 470, 480, 490, 500, 510, 520, 530

Offset: 1

Amarnath Murthy, Oct 17 2002

nonn

Cf. A002024.

More terms from Diana L. Mecum, Jun 04 2007

cp's OEIS Frontend

A131950 A002024 + A131821 + A007318 - 2*A000012 as infinite lower triangular matrices.

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A132071 A007318 + A002024 - A103451 as infinite lower triangular matrices.

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A135471 a(n) = Sum{i=1..n} ( i*2^(i-1) ) - ( A002024(n)*(A002024(n)+1)/2 - n ) * 2^(A002024(n)-1).

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A182245 Integers n such that either (a) A186053(n) does not equal A002024(n) + A182298(A025581(n)) or (b) A182298(n) does not equal 1 + A002024(n) + A186053(A025581(n)).

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A182246 Integers n such that A186053(n) does not equal A002024(n) + A182298(A025581(n)).

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A218036 a(n) = (n+1) + (n+3/2)*H(n) - (H(n)^3)/2, where H(n) = A002024(n).

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A293670 Square array made of (W, N, S, E) quadruplets read by antidiagonals. Numeric structure of an anamorphosis of A002024 (see comments).

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A379777 Array A(n, k), n, k >= 0, read by upward antidiagonals; for any v >= 0, the value appears twice in the array: in row A002262(v) and in row A002024(v+1); values in each row are given in strictly increasing order.

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A065979 Binomial transform of A002024.

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A135471 a(n) = Sum{i=1..n} ( i2^(i-1) ) - ( A002024(n)(A002024(n)+1)/2 - n ) * 2^(A002024(n)-1).