A002024 -id:A002024 - cp's OEIS frontend

A271439 If n is a triangular number, a(n) = 0, otherwise a(n) = n - A002024(n) + 1.

0, 0, 1, 0, 2, 3, 0, 4, 5, 6, 0, 7, 8, 9, 10, 0, 11, 12, 13, 14, 15, 0, 16, 17, 18, 19, 20, 21, 0, 22, 23, 24, 25, 26, 27, 28, 0, 29, 30, 31, 32, 33, 34, 35, 36, 0, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 0, 56, 57, 58

Offset: 0

Peter Kagey, Apr 07 2016

a(n) gives the number above n when the natural numbers are represented as a square array, or 0 if n is in the top row.
   1 3 6 10
   2 5 9
   4 8
   7

			As a regular triangle, sequence begins:
  0;
  0, 1;
  0, 2, 3;
  0, 4, 5, 6;
  0, 7, 8, 9, 10;
  ...

Peter Kagey, Table of n, a(n) for n = 0..10000

Cf. A000217 (right diagonal), A006003 (row sums).

Riffle[#, 0] &@ Map[# (# - 1)/2 + Range@ # &, Range[0, 11]] /. {} -> {0} // Flatten (* Michael De Vlieger, Apr 07 2016 *)

from math import isqrt
def A271439(n): return n-m+(kChai Wah Wu, Nov 07 2024

A064520 a(n) = + 1 - 2 - 3 + 4 + 5 + 6 - 7 - 8 - 9 - 10 + 11 + 12 + 13 + 14 + 15 - ... + (+-1)*n, where there is one plus, two minuses, three pluses, etc. (see A002024).

Original entry on oeis.org

1, -1, -4, 0, 5, 11, 4, -4, -13, -23, -12, 0, 13, 27, 42, 26, 9, -9, -28, -48, -69, -47, -24, 0, 25, 51, 78, 106, 77, 47, 16, -16, -49, -83, -118, -154, -117, -79, -40, 0, 41, 83, 126, 170, 215, 169, 122, 74, 25, -25, -76, -128, -181, -235, -290, -234, -177, -119, -60, 0, 61, 123, 186, 250, 315, 381, 314, 246, 177

Offset: 1

Jonathan Ayres (jonathan.ayres(AT)btinternet.com), Oct 07 2001

|a(n)| takes its locally maximal values when n is a triangular number, the maximal values being given by A019298.

The maximal positive/negative values occur for n = 1, 3, 6, 10, 15, 21 ... the triangular numbers and are a(n) = 1, -4, 11, -23, 42, -69,106, 215, 381, 616 ... +- int(sqrt(n^3/2) + 0.22098 * sqrt(n)). a(n) = n for n = 5, 13, 25, 41, 61, 85, ... m*(m*2-2)+1 and the previous number is equal to 0. Positive numbers which do not occur in this sequence are 2, 3, 6, 7, 8, 10, 12, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 43, 44, 45, 46, 48, ...

			a(9) = -13 because 1 - 2 - 3 + 4 + 5 + 6 - 7 - 8 - 9 = -13.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A002024, A019298, A064528.

a := proc(n) option remember: if n=1 then RETURN(1) fi: a(n-1) + n*(-1)^( floor(1/2 + sqrt(2*n)+1)); end: for n from 1 to 150 do printf(`%d,`,a(n)) od:

Accumulate[Flatten[Table[(-1)^(n+1) Range[(n(n-1))/2+1,(n(n+1))/2], {n,15}]]] (* Harvey P. Dale, Apr 22 2015 *)

t(n) = floor(1/2+sqrt(2*n))
for(n=1,200,print1(sum(k=1,n,(-1)^(t(k)+1)*k)," "))

t(n)= { floor(sqrt(2*n) + 1/2) }
{ for (n=1, 1000, a=sum(k=1, n, (-1)^(t(k) + 1)*k); write("b064520.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 17 2009

from math import isqrt
def A064520(n): return sum(k if (isqrt(k<<3)+1>>1)&1 else -k for k in range(1,n+1)) # Chai Wah Wu, Oct 16 2022

a(n) = Sum_{k=1..n} (-1)^(A002024(k)+1)*k.

More terms from James Sellers, Jason Earls and Vladeta Jovovic, Oct 08 2001

A180272 a(n) = binomial(n, A002024(n+1)-1) where A002024 is "n appears n times".

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 20, 35, 56, 84, 210, 330, 495, 715, 1001, 3003, 4368, 6188, 8568, 11628, 15504, 54264, 74613, 100947, 134596, 177100, 230230, 296010, 1184040, 1560780, 2035800, 2629575, 3365856, 4272048, 5379616, 6724520, 30260340, 38608020, 48903492

Offset: 0

Paul D. Hanna, Jan 17 2011

nonn

Number of subsets of [n] in which exactly half of the elements are triangular numbers: a(6) = 20: {}, {1,2}, {1,4}, {1,5}, {2,3}, {2,6}, {3,4}, {3,5}, {4,6}, {5,6}, {1,2,3,4}, {1,2,3,5}, {1,2,4,6}, {1,2,5,6}, {1,3,4,5}, {1,4,5,6}, {2,3,4,6}, {2,3,5,6}, {3,4,5,6}, {1,2,3,4,5,6}. - Alois P. Heinz, Oct 11 2022

			G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 6*x^4 + 10*x^5 + 20*x^6 +...
Terms are shown below in parenthesis as they appear in Pascals triangle:
(1);
1,(1);
1,(2),1;
1,3,(3),1;
1,4,(6),4,1;
1,5,(10),5,1;
1,6,15,(20),15,6,1;
1,7,21,(35),35,21,7,1;
1,8,28,(56),70,56,28,8,1;
1,9,36,(84),126,126,84,36,9,1;
1,10,45,120,(210),252,210,120,45,10,1; ...

Paul D. Hanna, Table of n, a(n) for n = 0..1000

Cf. A000217, A002024.

{a(n)=binomial(n,(sqrtint(8*n+1)-1)\2)}

from math import comb, isqrt
def A180272(n): return comb(n,(isqrt(n+1<<3)+1>>1)-1) # Chai Wah Wu, Oct 17 2022

A204169 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of (i+j-1), as in A002024.

Original entry on oeis.org

1, -1, -1, -4, 1, 0, 6, 9, -1, 0, 0, -20, -16, 1, 0, 0, 0, 50, 25, -1, 0, 0, 0, 0, -105, -36, 1, 0, 0, 0, 0, 0, 196, 49, -1, 0, 0, 0, 0, 0, 0, -336, -64, 1, 0, 0, 0, 0, 0, 0, 0, 540, 81, -1, 0, 0, 0, 0, 0, 0, 0, 0, -825, -100, 1

Offset: 1

Clark Kimberling, Jan 12 2012

Let p(n)=p(n,x) be the characteristic polynomial of the n-th principal submatrix. The zeros of p(n) are real, and they interlace the zeros of p(n+1). See A202605 and A204016 for guides to related sequences.

			Top of the array:
2....-1
-1....-4.....1
0.....6.....9....-1
0.....0....-20...-16...1

(For references regarding interlacing roots, see A202605.)

Cf. A002024, A202605, A204016.

f[i_, j_] := i + j - 1;
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[8]] (* 8x8 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
  {n, 1, 15}, {i, 1, n}]]  (* A002024 *)
p[n_] := CharacteristicPolynomial[m[n], x];
c[n_] := CoefficientList[p[n], x]
TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
Table[c[n], {n, 1, 12}]
Flatten[%]                (* A204169 *)
TableForm[Table[c[n], {n, 1, 10}]]

A004797 Convolution of A002024 with itself.

Original entry on oeis.org

1, 4, 8, 14, 22, 30, 41, 54, 67, 82, 99, 118, 138, 160, 182, 206, 234, 262, 292, 322, 353, 388, 425, 462, 501, 542, 583, 626, 671, 718, 766, 818, 870, 922, 976, 1030, 1088, 1148, 1210, 1274, 1338, 1402, 1469, 1538, 1607, 1678, 1753, 1828, 1905, 1984, 2063

Offset: 0

N. J. A. Sloane

nonn

Robert Israel, Table of n, a(n) for n = 0..10000
Raphael Schumacher, The self-counting identity, Fib. Quart., 55 (No. 2 2017), 157-167.

Cf. A002024.

a002024:= [seq(i$i,i=1..10)]:
g002024:= add(a002024[i]*x^(i-1),i=1..nops(a002024)):
g:= expand(g002024^2):
seq(coeff(g,x,i),i=0..degree(g002024)); # Robert Israel, May 30 2017

nn(n)=(sqrtint(n*8)+1)\2;
a(n) = sum(k=1, n, nn(k)*nn(n-k+1)); \\ Michel Marcus, May 30 2017

G.f.: (1/(1 - x)^2)*Product_{k>=1} (1 - x^(2*k))^2/(1 - x^(2*k-1))^2. - Ilya Gutkovskiy, May 30 2017

A088673 a(n) = n mod A002024(n), where A002024 is "n appears n times": 1, 2, 2, 3, 3, 3, ...

Original entry on oeis.org

0, 0, 1, 1, 2, 0, 3, 0, 1, 2, 1, 2, 3, 4, 0, 4, 5, 0, 1, 2, 3, 1, 2, 3, 4, 5, 6, 0, 5, 6, 7, 0, 1, 2, 3, 4, 1, 2, 3, 4, 5, 6, 7, 8, 0, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 8, 9, 10, 11, 12, 13, 0, 1, 2

Offset: 1

Gary W. Adamson, Oct 04 2003

			a(7) = 3: A002024(7) = 4 and 7 mod 4 = 3.

Cf. A002024.

Table[Mod[n,Floor[1/2+Sqrt[2n]]],{n,100}] (* Harvey P. Dale, Feb 09 2013 *)

from math import isqrt
def A088673(n): return n%((m:=isqrt(k:=n<<1))+(k>m*(m+1))) # Chai Wah Wu, Nov 07 2024

Corrected and extended by Ray Chandler, Oct 04 2003

Corrected offset by Chai Wah Wu, Nov 07 2024

A131413 Triangle read by rows: A002024 + A128076 - A000012 as infinite lower triangular matrices.

Original entry on oeis.org

1, 4, 3, 7, 6, 5, 10, 9, 8, 7, 13, 12, 11, 10, 9, 16, 15, 14, 13, 12, 11, 19, 18, 17, 16, 15, 14, 13, 22, 21, 20, 19, 18, 17, 16, 15, 25, 24, 23, 22, 21, 20, 19, 18, 17, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23

Offset: 0

Gary W. Adamson, Jul 08 2007

Row sums = A000566, the heptagonal numbers: (1, 7, 18, 34, 55, ...).

			First few rows of the triangle:
   1;
   4,  3;
   7,  6,  5;
  10,  9,  8,  7;
  13, 12, 11, 10,  9;
  16, 15, 14, 13, 12, 11;
  19, 18, 17, 16, 15, 14, 13;
  ...

Cf. A000566, A002024, A128076.

By rows, (n+1) terms of 3n+1, 3n, 3n-1, ...

a(5) = 5, a(30) = 20 corrected and more terms from Georg Fischer, Jun 07 2023

A131901 2*A002024 - A131821.

Original entry on oeis.org

1, 2, 2, 3, 5, 3, 4, 7, 7, 4, 5, 9, 9, 9, 5, 6, 11, 11, 11, 11, 6, 7, 13, 13, 13, 13, 13, 7, 8, 15, 15, 15, 15, 15, 15, 8, 9, 17, 17, 17, 17, 17, 17, 17, 9, 10, 19, 19, 19, 19, 19, 19, 19, 19, 10

Offset: 1

Gary W. Adamson, Jul 26 2007

Row sums = A084849: (1, 4, 11, 22, 37, ...).

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

Cf. A002024, A131821, A084849.

2*A002024 - A131821 as infinite lower triangular matrices.

A131925 2*A002024 - A000012(signed).

Original entry on oeis.org

1, 5, 3, 5, 7, 5, 9, 7, 9, 7, 9, 11, 9, 11, 9, 13, 11, 13, 11, 13, 11, 13, 15, 13, 15, 13, 15, 13, 17, 15, 17, 15, 17, 15, 17, 15, 17, 19, 17, 19, 17, 19, 17, 19, 17

Offset: 0

Gary W. Adamson, Jul 29 2007

Row sums = A077221: (1, 8, 17, 32, 49, 72, 97, ...).

			First few rows of the triangle:
   1;
   5,  3;
   5,  7,  5;
   9,  7,  9,  7;
   9, 11,  9, 11,  9;
  13, 11, 13, 11, 13, 11;
  13, 15, 13, 15, 13, 15, 13;
  ....

Cf. A002024, A000012, A077221.

2*A002024 - A000012 (signed + - + -, ... by columns).

A002024 = (1; 2,2; 3,3,3; ...); A000012(signed) = (1; -1,1; 1,-1,1; ...).

A131948 Triangle T(n,k) = 2*A002024(n+1,k+1) + A007318(n,k) - 2, read by rows.

Original entry on oeis.org

1, 3, 3, 5, 6, 5, 7, 9, 9, 7, 9, 12, 14, 12, 9, 11, 15, 20, 20, 15, 11, 13, 18, 27, 32, 27, 18, 13, 15, 21, 35, 49, 49, 35, 21, 15, 17, 24, 44, 72, 86, 72, 44, 24, 17, 19, 27, 54, 102, 144, 144, 102, 54, 27, 19

Offset: 0

Gary W. Adamson, Jul 30 2007

Row sums = A131949.

			First few rows of the triangle are:
1;
3, 3;
5, 6, 5;
7, 9, 9, 7;
9, 12, 14, 12, 9;
11, 15, 20, 20, 15, 11;
13, 18, 27, 32, 27, 18, 13;
15, 21, 35, 49, 49, 35, 21, 15;
...

Cf. A002024, A007318, A131949.

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

cp's OEIS Frontend

A271439 If n is a triangular number, a(n) = 0, otherwise a(n) = n - A002024(n) + 1.

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A064520 a(n) = + 1 - 2 - 3 + 4 + 5 + 6 - 7 - 8 - 9 - 10 + 11 + 12 + 13 + 14 + 15 - ... + (+-1)*n, where there is one plus, two minuses, three pluses, etc. (see A002024).

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A180272 a(n) = binomial(n, A002024(n+1)-1) where A002024 is "n appears n times".

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A204169 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of (i+j-1), as in A002024.

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A004797 Convolution of A002024 with itself.

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A088673 a(n) = n mod A002024(n), where A002024 is "n appears n times": 1, 2, 2, 3, 3, 3, ...

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A131413 Triangle read by rows: A002024 + A128076 - A000012 as infinite lower triangular matrices.

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A131901 2*A002024 - A131821.

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