A074279 -id:A074279 - cp's OEIS frontend

A345018 For each n, append to the sequence n^2 consecutive integers, starting from n.

1, 2, 3, 4, 5, 3, 4, 5, 6, 7, 8, 9, 10, 11, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41

Offset: 1

Paolo Xausa, Jun 05 2021

Irregular triangle read by rows T(n,k) in which row n lists the integers from n to n + n^2 - 1, with n >= 1.

			Written as an irregular triangle T(n,k) the sequence begins:
  n\k|  1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16 ...
  ---+---------------------------------------------------------------
   1 |  1;
   2 |  2,  3,  4,  5;
   3 |  3,  4,  5,  6,  7,  8,  9, 10, 11;
   4 |  4,  5,  6,  7,  8,  9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19;
  ...

Paolo Xausa, Table of n, a(n) for n = 1..10416 (rows 1..31 of the triangle, flattened)

Column 1: A000027.
Right border: A028387.
Row lengths: A000290.
Row sums: A255499.
Cf. A064866, A074279.

T:= n-> (t-> seq(n+i, i=0..t-1))(n^2):
seq(T(n), n=1..6);  # Alois P. Heinz, Nov 05 2024

Table[Range[n,n^2+n-1],{n,6}] (* Paolo Xausa, Sep 05 2023 *)

row(n) = vector(n^2, k, n+k-1); \\ Michel Marcus, Jun 08 2021

from sympy import integer_nthroot
def A345018(n): return n-1+(k:=(m:=integer_nthroot(3*n,3)[0])+(6*n>m*(m+1)*((m<<1)+1)))*(k*(3-(k<<1))+5)//6 # Chai Wah Wu, Nov 05 2024

T(n,k) = n + k - 1, with n >= 1 and 1 <= k <= n^2.

A376276 Table T(n, k) n > 0, k > 2 read by upward antidiagonals. The sequences in each column k is a triangle read by rows (blocks), where each row is a permutation of the numbers of its constituents. The length of the row number n in column k is equal to the n-th k-gonal number A086270.

Original entry on oeis.org

1, 3, 1, 4, 4, 1, 2, 3, 4, 1, 8, 5, 5, 5, 1, 7, 2, 3, 4, 5, 1, 9, 10, 6, 6, 6, 6, 1, 6, 11, 2, 3, 4, 5, 6, 1, 10, 9, 13, 7, 7, 7, 7, 7, 1, 5, 12, 12, 2, 3, 4, 5, 6, 7, 1, 16, 8, 14, 15, 8, 8, 8, 8, 8, 8, 1, 15, 13, 11, 16, 2, 3, 4, 5, 6, 7, 8, 1, 17, 7, 15, 14, 18, 9, 9, 9, 9, 9, 9, 9, 1, 14, 14, 10, 17, 17, 2, 3, 4, 5, 6, 7, 8, 9, 1, 18, 6, 16, 13, 19, 20, 10, 10, 10

Offset: 1

Boris Putievskiy, Sep 18 2024

A209278 presents an algorithm for generating permutations.

The sequence is an intra-block permutation of integer positive numbers.

			Table begins:
  k =      3   4   5   6   7   8
--------------------------------------
  n = 1:   1,  1,  1,  1,  1,  1, ...
  n = 2:   3,  4,  4,  5,  5,  6, ...
  n = 3:   4,  3,  5,  4,  6,  5, ...
  n = 4:   2,  5,  3,  6,  4,  7, ...
  n = 5:   8,  2,  6,  3,  7,  4, ...
  n = 6:   7, 10,  2,  7,  3,  8, ...
  n = 7:   9, 11, 13,  2,  8,  3, ...
  n = 8:   6,  9, 12, 15,  2,  9, ...
  n = 9:  10, 12, 14, 16, 18,  2, ...
  n =10:   5,  8, 11, 14, 17, 20, ...
  n =11:  16, 13, 15, 17, 19, 21, ...
  n =12:  15, 7,  10, 13, 16, 19, ...
  n =13:  17, 14, 16, 18, 20, 22, ...
  n =14:  14,  6,  9, 12, 15, 18, ...
  n =15:  18, 23, 17, 19, 21, 23, ...
  n =16:  13, 22,  8, 11, 14, 17, ...
  n =17:  19, 24, 18, 20, 22, 24, ...
  n =18:  12, 21,  7, 10, 13, 16, ...
  n =19:  20, 25, 30, 21, 23, 25, ...
  n =20:  11, 20, 29,  9, 12, 15, ...
          ... .
For k = 3 the first 4 blocks have lengths 1,3,6 and 10.
For k = 4 the first 3 blocks have lengths 1,4, and 9.
For k = 5 the first 3 blocks have lengths 1,5, and 12.
Each block is a permutation of the numbers of its constituents.
The first 6 antidiagonals are:
  1;
  3, 1;
  4, 4, 1;
  2, 3, 4, 1;
  8, 5, 5, 5, 1;
  7, 2, 3, 4, 5, 1;

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 45.

Boris Putievskiy, Table of n, a(n) for n = 1..9870
Boris Putievskiy, Integer Sequences: Irregular Arrays and Intra-Block Permutations, arXiv:2310.18466 [math.CO], 2023.
Eric Weisstein's World of Mathematics, Polygonal Number.
Index entries for sequences that are permutations of the natural numbers.
Index to sequences related to polygonal numbers

Cf. A000012, A004526, A028242, A074279, A084964, A086270, A209278, A360010, A375725, A375797

T[n_,k_]:=Module[{L,R,P,Res,result},L=Ceiling[Max[x/.NSolve[x^3*(k-2)+3*x^2-x*(k-5)-6*n==0,x,Reals]]];
R=n-(((L-1)^3)*(k-2)+3*(L-1)^2-(L-1)*(k-5))/6;P=Which[OddQ[R]&&OddQ[k*L*(L-1)/2-L^2+2*L],((k*L*(L-1)/2-L^2+2*L+1-R)+1)/2,OddQ[R]&&EvenQ[k*L*(L-1)/2-L^2+2*L],(R+k*L*(L-1)/2-L^2+2*L+1)/2,EvenQ[R]&&OddQ[k*L*(L-1)/2-L^2+2*L],Ceiling[(k*L*(L-1)/2-L^2+2*L+1)/2]+R/2,EvenQ[R]&&EvenQ[k*L*(L-1)/2-L^2+2*L],Ceiling[(k*L*(L-1)/2-L^2+2*L+1)/2]-R/2];
Res=P+((L-1)^3*(k-2)+3*(L-1)^2-(L-1)*(k-5))/6;result=Res;result]
Nmax=6;Table[T[n,k],{n,1,Nmax},{k,3,Nmax+2}]

T(n,k) = P(n,k) + ((L(n,k)-1)^3*(k-2)+3*(L(n,k)-1)^2-(L(n,k)-1)*(k-5))/6, where L(n,k) = ceiling(x(n,k)), x(n,k) is largest real root of the equation x^3*(k - 2) + 3*x^2 - x*(k - 5) - 6*n = 0. R(n,k) = n - ((L(n,k) - 1)^3*(k-2)+3*(L(n,k)-1)^2-(L(n,k)-1)*(k-5))/6. P(n,k) = ((k * L(n,k) * (L(n,k) - 1) / 2) - L(n,k)^2 + 2 * L(n,k) + 2 - R(n,k)) / 2 if R is odd and (k * L(n,k) * (L(n,k) - 1) / 2) - L(n,k)^2 + 2 * L(n,k) is odd, P(n,k) = (R(n,k) + (k * L(n,k) * (L(n,k) - 1) / 2) - L(n,k)^2 + 2 * L(n,k) + 1) / 2 if R is odd and (k * L(n,k) * (L(n,k) - 1) / 2) - L(n,k)^2 + 2 * L(n,k) is even, P(n,k) = ceiling(((k * L(n,k) * (L(n,k) - 1) / 2) - L(n,k)^2 + 2 * L(n,k) + 1) / 2) + (R(n,k) / 2) if R is even and (k * L(n,k) * (L(n,k) - 1) / 2) - L(n,k)^2 + 2 * L(n,k) is odd, P(n,k) = ceiling(((k * L(n,k) * (L(n,k) - 1) / 2) - L(n,k)^2 + 2 * L(n,k) + 1) / 2) - (R(n,k) / 2) if R is even and (k * L(n,k) * (L(n,k) - 1) / 2) - L(n,k)^2 + 2 * L(n,k) is even.

T(1,n) = A000012(n). T(2,n) = A004526(n+7). T(3,n) = A028242(n+6). T(4,n) = A084964(n+5). T(n-2,n) = A000027(n) for n > 3. L(n,3) = A360010(n). L(n,4) = A074279(n).

A377721 Numbers that are not of the form k(k+1)(2k+1)/6 + k.

Original entry on oeis.org

1, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72

Offset: 1

Chai Wah Wu, Nov 04 2024

nonn

Complement of A145066. Numbers that are not of the form A000330(k)+k.

Cf. A000330, A074279, A145066.

from sympy import integer_nthroot
def A377721(n): return n+(m:=integer_nthroot(3*n,3)[0])-(6*n<=m*(m+1)*((m<<1)+1))

a(n) = n+m if n > m(m+1)(2m+1)/6 and a(n) = n+m-1 otherwise where m = floor((3n)^(1/3)).

a(n) = A074279(n)+n-1.

A377722 n appears n^4 times.

Original entry on oeis.org

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

Offset: 1

Chai Wah Wu, Nov 04 2024

nonn

Paolo Xausa, Table of n, a(n) for n = 1..15333

Cf. A002024, A074279, A108582.

A377722[n_] := # + Boole[n > #*(# + 1)*(2*# + 1)*(3*#^2 + 3*# - 1)/30] & [Floor[(5*n)^(1/5)]];
Array[A377722, 354] (* or *)
Flatten[Table[k, {k, 4}, {k^4}]] (* Paolo Xausa, Nov 05 2024 *)

from sympy import integer_nthroot
def A377722(n): return (m:=integer_nthroot(5*n,5)[0])+(30*n>m*(m+1)*((m<<1)+1)*(3*m*(m+1)-1))

a(n) = m+1 if n>m(m+1)(2m+1)(3m^2+3m-1)/30 and a(n) = m otherwise where m = floor((5n)^(1/5)).

For a sequence a_k(n) where n appears n^(k-1) times, a_k(n) = m+1 if n > Sum_{i=1..m} i^(k-1) and a_k(n) = m otherwise where m = floor((kn)^(1/k)).

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A345018 For each n, append to the sequence n^2 consecutive integers, starting from n.

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A376276 Table T(n, k) n > 0, k > 2 read by upward antidiagonals. The sequences in each column k is a triangle read by rows (blocks), where each row is a permutation of the numbers of its constituents. The length of the row number n in column k is equal to the n-th k-gonal number A086270.

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A377721 Numbers that are not of the form k(k+1)(2k+1)/6 + k.

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A377722 n appears n^4 times.

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