A261720 -id:A261720 - cp's OEIS frontend

A376353 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-pyramidal number A261720.

1, 4, 1, 3, 4, 1, 5, 5, 5, 1, 2, 3, 4, 5, 1, 11, 6, 6, 6, 6, 1, 10, 2, 3, 4, 5, 6, 1, 12, 14, 7, 7, 7, 7, 7, 1, 9, 13, 2, 3, 4, 5, 6, 7, 1, 13, 15, 17, 8, 8, 8, 8, 8, 8, 1, 8, 12, 16, 2, 3, 4, 5, 6, 7, 8, 1, 14, 16, 18, 20, 9, 9, 9, 9, 9, 9, 9, 1, 7, 11, 15, 19, 2, 3, 4, 5, 6, 7, 8, 9, 1, 15, 17, 19, 21, 23, 10, 10, 10, 10, 10, 10, 10, 10, 1, 6, 10, 14, 18, 22, 2, 3

Offset: 1

Boris Putievskiy, Sep 21 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:   4,  4,  5,  5,  6,  6, ...
  n = 3:   3,  5,  4,  6,  5,  7, ...
  n = 4:   5,  3,  6,  4,  7,  5, ...
  n = 5:   2,  6,  3,  7,  4,  8, ...
  n = 6:  11,  2,  7,  3,  8,  4, ...
  n = 7:  10, 14,  2,  8,  3,  9, ...
  n = 8:  12, 13, 17,  2,  9,  3, ...
  n = 9:   9, 15, 16, 20,  2, 10, ...
  n = 10: 13, 12, 18, 19, 23,  2, ...
  n = 11:  8, 16, 15, 21, 22, 26, ...
  n = 12: 14, 11, 19, 18, 24, 25, ...
  n = 12:  7, 17, 14, 22, 21, 27, ...
  n = 14: 15, 10, 20, 17, 25, 24, ...
  n = 15:  6, 18, 13, 23, 20, 28, ...
          ... .
For k = 3 the first 3 blocks have lengths 1,4 and 10.
For k = 4 the first 2 blocks have lengths 1 and 5.
For k = 5 the first 2 blocks have lengths 1 and 6.
Each block is a permutation of the numbers of its constituents.
The first 6 antidiagonals are:
   1;
   4, 1;
   3, 4, 1;
   5, 5, 5, 1;
   2, 3, 4, 5, 1;
  11, 6, 6, 6, 6, 1;

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93.

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, Pyramidal Number.
Index entries for sequences that are permutations of the natural numbers.
Index to sequences related to pyramidal numbers.

Cf. A000012, A000027, A004526, A028242, A084964, A168230, A209278, A261720, A375725, A375797.

T[n_,k_]:=Module[{L,R,result},L=Ceiling[Max[x/.NSolve[(k-2)*x^4+2*k*x^3+(14-k)*x^2+(12-2*k)*x-24*n==0,x,Reals]]]; R=n-((k-2)*(L-1)^4+2*k*(L-1)^3+(14-k)*(L-1)^2+(12-2*k)*(L-1))/24; P=Which[OddQ[R]&&OddQ[(L^3*(k-2)+3*L^2-L*(k-5))/6],((L^3*(k-2)+3*L^2-L*(k-5))/6+2-R)/2,OddQ[R]&&EvenQ[(L^3*(k-2)+3*L^2-L*(k-5))/6],(R+(L^3*(k-2)+3*L^2-L*(k-5))/6+1)/2,EvenQ[R]&&OddQ[(L^3*(k-2)+3*L^2-L*(k-5))/6],Ceiling[((L^3*(k-2)+3*L^2-L*(k-5))/6+1)/2]+R/2,EvenQ[R]&&EvenQ[(L^3*(k-2)+3*L^2-L*(k-5))/6],Ceiling[((L^3*(k-2)+3*L^2-L*(k-5))/6+1)/2]-R/2]; Res= P +((k-2)*(L-1)^4+2*k*(L-1)^3+(14-k)*(L-1)^2+(12-2*k)*(L-1))/24; result=Res] Nmax=6; Table[T[n,k],{n,1,Nmax},{k,3,Nmax+2}]

T(n,k) = P(n,k) + ((k-2)*(L(n,k)-1)^4+2*k*(L(n,k)-1)^3+(14-k)*(L(n,k)-1)^2+(12-2*k)*(L(n,k)-1))/24, where L(n,k) = ceiling(x(n,k)), x(n,k) is largest real root of the equation (k-2)*x^4+2*k*x^3+(14-k)*x^2+(12-2*k)*x-24*n = 0. R(n,k) = n - ((k-2)*(L(n,k)-1)^4+2*k*(L(n,k)-1)^3+(14-k)*(L(n,k)-1)^2+(12-2*k)*(L(n,k)-1))/24. P(n,k) = ((L(n,k)^3*(k-2)+3*L(n,k)^2-L(n,k)*(k-5))/6+2-R(n,k))/2 if R(n,k) is odd and (L(n,k)^3*(k-2)+3*L(n,k)^2-L(n,k)*(k-5))/6 is odd, P(n,k) = ((L(n,k)^3*(k-2)+3*L(n,k)^2-L(n,k)*(k-5))/6+1)+R(n,k))/2 if R(n,k) is odd and (L(n,k)^3*(k-2)+3*L(n,k)^2-L(n,k)*(k-5))/6 is even, P = ceiling(((L(n,k)^3*(k-2)+3*L(n,k)^2-L(n,k)*(k-5))/6+1)/2)+R(n,k)/2) if R(n,k) is even and (L(n,k)^3*(k-2)+3*L(n,k)^2-L(n,k)*(k-5))/6 is odd, P = ceiling(((L(n,k)^3*(k-2)+3*L(n,k)^2-L(n,k)*(k-5))/6+1)/2)-R(n,k)/2) if R(n,k) is even and (L(n,k)^3*(k-2)+3*L(n,k)^2-L(n,k)*(k-5))/6 is even.

T(1,n) = A000012(n). T(2,n) = A004526(n+8). T(3,n) = A028242(n+7). T(4,n) = A084964(n+6). T(5,n) = A168230(n+5). T(n-2,n) = 4*A000012(n) for n > 3. T(n-1,n) = A000027(n) for n > 2.

A257199 a(n) = n(n+1)(n+2)(n^2+2n+17)/120.

Original entry on oeis.org

1, 5, 16, 41, 91, 182, 336, 582, 957, 1507, 2288, 3367, 4823, 6748, 9248, 12444, 16473, 21489, 27664, 35189, 44275, 55154, 68080, 83330, 101205, 122031, 146160, 173971, 205871, 242296, 283712, 330616, 383537, 443037, 509712, 584193, 667147, 759278, 861328, 974078

Offset: 1

Luciano Ancora, Apr 18 2015

Antidiagonal sums of the array of pyramidal numbers shown in Table 2 of Sardelis and Valahas paper (see A261720).
This is the case j = 3 of (n^2 + (j-1)*n + (j+1)^2 + 1)*binomial(n+j-1, j)/((j+1)*(j+2)), where j is the space dimension: a(n) = (n^2+2*n+17)*binomial(n+2,3)/20.
The sequence is the binomial transform of (1, 4, 7, 7, 4, 1, 0, 0, 0, ...). - Gary W. Adamson, Aug 26 2015

D. A. Sardelis and T. M. Valahas, On Multidimensional Pythagorean Numbers, arXiv:0805.4070 [math.GM], 2008.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A257200, A257201.

For another version of the array, see A080851.

[n*(n+1)*(n+2)*(n^2+2*n+17)/120: n in [1..40]]; // Vincenzo Librandi, Apr 18 2015

Table[n (n + 1) (n + 2) (n^2 + 2n + 17)/120, {n, 40}]
LinearRecurrence[{6,-15,20,-15,6,-1},{1,5,16,41,91,182},40] (* Harvey P. Dale, Mar 18 2018 *)

G.f.: x*(1 - x + x^2)/(1 - x)^6.

A261721 Fourth-dimensional figurate numbers.

Original entry on oeis.org

1, 1, 5, 1, 6, 15, 1, 7, 20, 35, 1, 8, 25, 50, 70, 1, 9, 30, 65, 105, 126, 1, 10, 35, 80, 140, 196, 210, 1, 11, 40, 95, 175, 266, 336, 330, 1, 12, 45, 110, 210, 336, 462, 540, 495, 1, 13, 50, 125, 245, 406, 588, 750, 825, 715, 1, 14, 55, 140, 280, 476, 714, 960, 1155, 1210, 1001, 1

Offset: 1

Gary W. Adamson, Aug 30 2015

Generating polygons for the sequences are: Triangle, Square, Pentagon, Hexagon, Heptagon, Octagon, ... .
n-th row sequence is the binomial transform of the fourth row of Pascal's triangle (1,n) followed by zeros; and the fourth partial sum of (1, n, n, n, ...).
n-th row sequence is the binomial transform of:
((n-1) * (0, 1, 3, 3, 1, 0, 0, 0) + (1, 4, 6, 4, 1, 0, 0, 0)).
Given the n-th row of the array (1, b, c, d, ...), the next row of the array is (1, b, c, d, ...) + (0, 1, 5, 15, 35, ...)

			The array as shown in A257200:
  1,  5, 15,  35,  70, 126, 210,  330, ... A000332
  1,  6, 20,  50, 105, 196, 336,  540, ... A002415
  1,  7, 25,  65, 140, 266, 462,  750, ... A001296
  1,  8, 30,  80, 175, 336, 588,  960, ... A002417
  1,  9, 35,  95, 210, 406, 714, 1170, ... A002418
  1, 10, 40, 110, 245, 476, 840, 1380, ... A002419
  ...
(1, 7, 25, 65, 140, ...) is the third row of the array and is the binomial transform of the fourth row of Pascal's triangle (1,3) followed by zeros: (1, 6, 12, 10, 3, 0, 0, 0, ...); and the fourth partial sum of (1, 3, 3, 3, 0, 0, 0).
(1, 7, 25, 65, 140, ...) is the third row of the array and is the binomial transform of: (2 * (0, 1, 3, 3, 1, 0, 0, 0, ...) + (1, 4, 6, 4, 1, 0, 0, 0, ...)); that is, the binomial transform of (1, 6, 12, 10, 3, 0, 0, 0, ...).
Row 2 of the array is (1, 5, 15, 35, 70, ...) + (0, 1, 5, 15, 35, ...), = (1, 6, 20, 50, 105, ...).

Albert H. Beiler, "Recreations in the Theory of Numbers"; Dover, 1966, p. 195 (Table 80).

Alois P. Heinz, Antidiagonals n = 1..141, flattened
Index to sequences related to pyramidal numbers

Cf. A257200, A261720 (pyramidal numbers), A000332, A002415, A001296, A002417, A002418, A002419.
Similar to A080852 but without row n=0.
Main diagonal gives A256859.

A:= (n, k)-> binomial(k+3, 3) + n*binomial(k+3, 4):
seq(seq(A(d-k, k), k=0..d-1), d=1..13);  # Alois P. Heinz, Aug 31 2015

row[1] = LinearRecurrence[{5, -10, 10, -5, 1}, {1, 5, 15, 35, 70}, m = 10];
row1 = Join[{0}, row[1] // Most]; row[n_] := row[n] = row[n-1] + row1;
Table[row[n-k+1][[k]], {n, 1, m}, {k, 1, n}] // Flatten (* Jean-François Alcover, May 26 2016 *)

A(n, k) = binomial(k+3, 3) + n*binomial(k+3, 4)
table(n, k) = for(x=1, n, for(y=0, k-1, print1(A(x, y), ", ")); print(""))
/* Print initial 6 rows and 8 columns as follows: */
table(6, 8) \\ Felix Fröhlich, Jul 28 2016

G.f. for row n: (1 + (n-1)*x)/(1 - x)^5.
A(n,k) = C(k+3,3) + n * C(k+3,4) = A080852(n,k).
E.g.f. as array: exp(y)*(exp(x)*(24 + 24*(3 + x)*y + 36*(1 + x)*y^2 + 4*(1 + 3*x)*y^3 + x*y^4) - 4*(6 + 18*y + 9*y^2 + y^3))/24. - Stefano Spezia, Aug 15 2024

A333916 a(n) is the least integer that is pyramidal in exactly n ways.

Original entry on oeis.org

4, 10, 30, 550, 1540, 48070, 223300, 2634940, 402610950, 1570545340, 13960282700, 5677137442900, 248297918605660

Offset: 1

Ilya Gutkovskiy, Apr 09 2020

a(n) has exactly n representations as an m-gonal pyramidal number P(m, k) = k*(k + 1)*(k*(m - 2) - m + 5) / 6, with m > 2, k > 1.

a(12) > 5*10^11. - Giovanni Resta, Apr 11 2020

			a(3) = 30 because 30 is the least integer which is pyramidal in 3 ways (30 is the fourth square pyramidal number, the third octagonal pyramidal number and also the second 31-gonal pyramidal number).

Eric Weisstein's World of Mathematics, Pyramidal Number
Index to sequences related to pyramidal numbers

Cf. A063778, A080851, A261720, A333915.

a(9) from Jinyuan Wang, Apr 10 2020
a(10)-a(11) from Giovanni Resta, Apr 10 2020
a(12)-a(13) from Bert Dobbelaere, Apr 12 2020

A333914 a(n) is the least integer m >= 3 such that n is m-gonal pyramidal number.

Original entry on oeis.org

3, 4, 5, 6, 7, 8, 3, 10, 11, 12, 4, 14, 15, 16, 5, 18, 3, 20, 6, 22, 23, 24, 7, 26, 27, 28, 4, 30, 31, 32, 9, 3, 35, 36, 10, 38, 5, 40, 11, 42, 43, 44, 12, 46, 47, 48, 6, 50, 51, 52, 14, 4, 3, 56, 15, 58, 7, 60, 16, 62, 63, 64, 17, 66, 67, 68, 8, 70, 71, 72, 19, 5, 75, 76, 20, 78, 9

Offset: 4

Ilya Gutkovskiy, Apr 09 2020

nonn

			a(18) = 5 since 18 is a pentagonal pyramidal number, but not a square pyramidal or tetrahedral number.

Eric Weisstein's World of Mathematics, Pyramidal Number
Index to sequences related to pyramidal numbers

Cf. A080851, A176774, A261720.

A333915 Number of ways to represent n as a pyramidal number.

Original entry on oeis.org

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

Offset: 4

Ilya Gutkovskiy, Apr 09 2020

nonn

Frequency of n in the array A261720 of pyramidal numbers.

			a(10) = 2 because 10 is the third tetrahedral (or triangular pyramidal) number and also the second 9-gonal pyramidal number.
a(30) = 3 because 30 is the fourth square pyramidal number, the third octagonal pyramidal number and also the second 29-gonal pyramidal number.

Eric Weisstein's World of Mathematics, Pyramidal Number
Index to sequences related to pyramidal numbers

Cf. A080851, A129654, A177025, A261720, A333916.

cp's OEIS Frontend

A376353 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-pyramidal number A261720.

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A257199 a(n) = n*(n+1)*(n+2)*(n^2+2*n+17)/120.

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Magma

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A261721 Fourth-dimensional figurate numbers.

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A333916 a(n) is the least integer that is pyramidal in exactly n ways.

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A333914 a(n) is the least integer m >= 3 such that n is m-gonal pyramidal number.

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A333915 Number of ways to represent n as a pyramidal number.

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A257199 a(n) = n(n+1)(n+2)(n^2+2n+17)/120.