A107985 -id:A107985 - cp's OEIS frontend

A002024 k appears k times; a(n) = floor(sqrt(2n) + 1/2).

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

Offset: 1

N. J. A. Sloane

Integer inverse function of the triangular numbers A000217. The function trinv(n) = floor((1+sqrt(1+8n))/2), n >= 0, gives the values 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, ..., that is, the same sequence with offset 0. - N. J. A. Sloane, Jun 21 2009
Array T(k,n) = n+k-1 read by antidiagonals.
Eigensequence of the triangle = A001563. - Gary W. Adamson, Dec 29 2008
Can apparently also be defined via a(n+1)=b(n) for n >= 2 where b(0)=b(1)=1 and b(n) = b(n-b(n-2))+1. Tested to be correct for all n <= 150000. - José María Grau Ribas, Jun 10 2011
For any n >= 0, a(n+1) is the least integer m such that A000217(m)=m(m+1)/2 is larger than n. This is useful when enumerating representations of n as difference of triangular numbers; see also A234813. - M. F. Hasler, Apr 19 2014
Number of binary digits of A023758, i.e., a(n) = ceiling(log_2(A023758(n+2))). - Andres Cicuttin, Apr 29 2016
a(n) and A002260(n) give respectively the x(n) and y(n) coordinates of the sorted sequence of points in the integer lattice such that x(n) > 0, 0 < y(n) <= x(n), and min(x(n), y(n)) < max(x(n+1), y(n+1)) for n > 0. - Andres Cicuttin, Dec 25 2016
Partial sums (A060432) are given by S(n) = (-a(n)^3 + a(n)*(1+6n))/6. - Daniel Cieslinski, Oct 23 2017
As an array, T(k,n) is the number of digits columns used in carryless multiplication between a k-digit number and an n-digit number. - Stefano Spezia, Sep 24 2022
a(n) is the maximum number of possible solutions to an n-statement Knights and Knaves Puzzle, where each statement is of the form "x of us are knights" for some 1 <= x <= n, knights can only tell the truth and knaves can only lie. - Taisha Charles and Brittany Ohlinger, Jul 29 2023

			From _Clark Kimberling_, Sep 16 2008: (Start)
As a rectangular array, a northwest corner:
  1 2 3 4 5 6
  2 3 4 5 6 7
  3 4 5 6 7 8
  4 5 6 7 8 9
This is the weight array (cf. A144112) of A107985 (formatted as a rectangular array). (End)
G.f. = x + 2*x^2 + 2*x^3 + 3*x^4 + 3*x^5 + 3*x^6 + 4*x^7 + 4*x^9 + 4*x^9 + 4*x^10 + ...

Edward S. Barbeau, Murray S. Klamkin, and William O. J. Moser, Five Hundred Mathematical Challenges, Prob. 441, pp. 41, 194. MAA 1995.
R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 97.
K. Hardy and K. S. Williams, The Green Book of Mathematical Problems, p. 59, Solution to Prob. 14, Dover NY, 1985
R. Honsberger, Mathematical Morsels, pp. 133-134, MAA 1978.
J. F. Hurley, Litton's Problematical Recreations, pp. 152; 313-4 Prob. 22, VNR Co., NY, 1971.
D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 1, p. 43.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 129.

Franklin T. Adams-Watters, Table of n, a(n) for n = 1..5050
Jaegug Bae and Sungjin Choi, A generalization of a subset-sum-distinct sequence, J. Korean Math. Soc. 40 (2003), no. 5, 757--768. MR1996839 (2004d:05198). See b(n).
Jonathan H. B. Deane and Guido Gentile, A diluted version of the problem of the existence of the Hofstadter sequence, arXiv:2311.13854 [math.NT], 2023. See p. 10.
Nathan Fox, Connecting Slow Solutions to Nested Recurrences with Linear Recurrent Sequences, arXiv:2203.09340 [math.CO], 2022.
H. T. Freitag and H. W. Gould, Solution to Problem 571, Math. Mag., 38 (1965), 185-187.
H. T. Freitag (Proposer) and H. W. Gould (Solver), Problem 571: An Ordering of the Rationals, Math. Mag., 38 (1965), 185-187 [Annotated scanned copy]
Mikel Garcia-de-Andoin, Álvaro Saiz, Pedro Pérez-Fernández, Lucas Lamata, Izaskun Oregi, and Mikel Sanz, Digital-Analog Quantum Computation with Arbitrary Two-Body Hamiltonians, arXiv:2307.00966 [quant-ph], 2023.
S. W. Golomb, Discrete chaos: sequences satisfying "strange" recursions, unpublished manuscript, circa 1990 [cached copy, with permission (annotated)]
Henry W. Gould, Letters to N. J. A. Sloane, Oct 1973 and Jan 1974.
Abraham Isgur, Vitaly Kuznetsov, and Stephen Tanny, A combinatorial approach for solving certain nested recursions with non-slow solutions, arXiv:1202.0276 [math.CO], 2012.
Stanley Wu-Wei Liu, Closed form expressions for A002024(n).
M. A. Nyblom, Some curious sequences involving floor and ceiling functions, Am. Math. Monthly 109 (#6, 200), 559-564.
Boris Putievskiy, Transformations [Of] Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.
Raphael Schumacher, Extension of Summation Formulas involving Stirling series, arXiv:1605.09204 [math.NT], 2016.
Raphael Schumacher, The self-counting identity, Fib. Quart., 55 (No. 2 2017), 157-167.
Raphael Schumacher, The Self-Counting Flow, Fibonacci Quart. 60 (2022), no. 5, 324-343.
N. J. A. Sloane, Handwritten notes on Self-Generating Sequences, 1970. (note that A1148 has now become A005282)
Michael Somos, Sequences used for indexing triangular or square arrays.
L. J. Upton, Letter to N. J. A. Sloane, May 22 1991.
Eric Weisstein's World of Mathematics, Self-Counting Sequence.
Index entries for Hofstadter-type sequences

a(n+1) = 1+A003056(n), A022846(n)=a(n^2), a(n+1)=A002260(n)+A025581(n).
Cf. A001462, A002262, A003881, A004736, A127899, A107985, A001563, A014132, A000194, A005145, A131507, A093995, A060432 (partial sums).
A123578 is an essentially identical sequence.

a002024 n k = a002024_tabl !! (n-1) !! (k-1)
a002024_row n = a002024_tabl !! (n-1)
a002024_tabl = iterate (\xs@(x:_) -> map (+ 1) (x : xs)) [1]
a002024_list = concat a002024_tabl
a002024' = round . sqrt . (* 2) . fromIntegral
-- Reinhard Zumkeller, Jul 05 2015, Feb 12 2012, Mar 18 2011

a002024_list = [1..] >>= \n -> replicate n n

a002024 = (!!) $ [1..] >>= \n -> replicate n n
-- Sascha Mücke, May 10 2016

[Floor(Sqrt(2*n) + 1/2): n in [1..80]]; // Vincenzo Librandi, Nov 19 2014

A002024 := n-> ceil((sqrt(1+8*n)-1)/2); seq(A002024(n), n=1..100);

a[1] = 1; a[n_] := a[n] = a[n - a[n - 1]] + 1 (* Branko Curgus, May 12 2009 *)
Table[n, {n, 13}, {n}] // Flatten (* Robert G. Wilson v, May 11 2010 *)
Table[PadRight[{},n,n],{n,15}]//Flatten (* Harvey P. Dale, Jan 13 2019 *)

t1(n)=floor(1/2+sqrt(2*n)) /* A002024 = this sequence */

t2(n)=n-binomial(floor(1/2+sqrt(2*n)),2) /* A002260(n-1) */

t3(n)=binomial(floor(3/2+sqrt(2*n)),2)-n+1 /* A004736 */

t4(n)=n-1-binomial(floor(1/2+sqrt(2*n)),2) /* A002260(n-1)-1 */

A002024(n)=(sqrtint(n*8)+1)\2 \\ M. F. Hasler, Apr 19 2014

PARI
```
a(n)=(sqrtint(8*n-7)+1)\2
```

a(n)=my(k=1);while(binomial(k+1,2)+1<=n,k++);k \\ R. J. Cano, Mar 17 2014

from math import isqrt
def A002024(n): return (isqrt(8*n)+1)//2 # Chai Wah Wu, Feb 02 2022

[floor(sqrt(2*n) +1/2) for n in (1..80)] # G. C. Greubel, Dec 10 2018

a(n) = floor(1/2 + sqrt(2n)). Also a(n) = ceiling((sqrt(1+8n)-1)/2). [See the Liu link for a large collection of explicit formulas. - N. J. A. Sloane, Oct 30 2019]
a((k-1)*k/2 + i) = k for k > 0 and 0 < i <= k. - Reinhard Zumkeller, Aug 30 2001
a(n) = a(n - a(n-1)) + 1, with a(1)=1. - Ian M. Levitt (ilevitt(AT)duke.poly.edu), Aug 18 2002
a(n) = round(sqrt(2n)). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Nov 01 2002
T(n,k) = A003602(A118413(n,k)); = T(n,k) = A001511(A118416(n,k)). - Reinhard Zumkeller, Apr 27 2006
G.f.: (x/(1-x))*Product_{k>0} (1-x^(2*k))/(1-x^(2*k-1)). - Vladeta Jovovic, Oct 06 2003
Equals A127899 * A004736. - Gary W. Adamson, Feb 09 2007
Sum_{i=1..n} Sum_{j=i..n+i-1} T(j,i) = A000578(n); Sum_{i=1..n} T(n,i) = A000290(n). - Reinhard Zumkeller, Jun 24 2007
a(n) + n = A014132(n). - Vincenzo Librandi, Jul 08 2010
a(n) = ceiling(-1/2 + sqrt(2n)). - Branko Curgus, May 12 2009
a(A169581(n)) = A038567(n). - Reinhard Zumkeller, Dec 02 2009
a(n) = round(sqrt(2*n)) = round(sqrt(2*n-1)); there exist a and b greater than zero such that 2*n = 2+(a+b)^2 -(a+3*b) and a(n)=(a+b-1). - Fabio Civolani (civox(AT)tiscali.it), Feb 23 2010
A005318(n+1) = 2*A005318(n) - A205744(n), A205744(n) = A005318(A083920(n)), A083920(n) = n - a(n). - N. J. A. Sloane, Feb 11 2012
Expansion of psi(x) * x / (1 - x) in powers of x where psi() is a Ramanujan theta function. - Michael Somos, Mar 19 2014
G.f.: (x/(1-x)) * Product_{n>=1} (1 + x^n) * (1 - x^(2*n)). - Paul D. Hanna, Feb 27 2016
a(n) = 1 + Sum_{i=1..n/2} ceiling(floor(2(n-1)/(i^2+i))/(2n)). - José de Jesús Camacho Medina, Jan 07 2017
a(n) = floor((sqrt(8*n-7)+1)/2). - Néstor Jofré, Apr 24 2017
a(n) = floor((A000196(8*n)+1)/2). - Pontus von Brömssen, Dec 10 2018
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/4 (A003881). - Amiram Eldar, Oct 01 2022
G.f. as array: (x^2*(1 - y)^2 + y^2 + x*y*(1 - 2*y))/((1 - x)^2*(1 - y)^2). - Stefano Spezia, Apr 22 2024

A098737 Triangle read by rows: number of triangles formed by lines from two vertices of a triangle to the opposite side that segment the opposite sides into m and n segments. Since f(m,n) = f(n,m), it suffices to give the results in a triangular table.

Original entry on oeis.org

1, 3, 8, 6, 15, 27, 10, 24, 42, 64, 15, 35, 60, 90, 125, 21, 48, 81, 120, 165, 216, 28, 63, 105, 154, 210, 273, 343, 36, 80, 132, 192, 260, 336, 420, 512, 45, 99, 162, 234, 315, 405, 504, 612, 729, 55, 120, 195, 280, 375, 480, 595, 720, 855, 1000, 66, 143, 231, 330

Offset: 1

Eugene McDonnell (eemcd(AT)mac.com), Oct 29 2004

Frank Buss gave this as a puzzle; K. L. Metlov solved it, submitting his result in the J language created by Kenneth Iverson. The program given below is only five tokens long. J defines a series of three functions to be a "fork" defined by x (f g h ) y = (x f y) g (f h y) - a generalization of the usual mathematical practice of writing (f + g) y to mean (f y) + (g y). J also has a primitive "half" and has a dummy function "cap" whose purpose is to permit more forks to be written. 3 (* * +) 5 is thus (3 * 5) * (3 + 5) or 120. cap half 3 (* * +) 5 is thus 60.
This sequence is the dimensions of the various irreducible representations of SU(3). In the language of physics, the integers m and n are one more than the numbers of quarks or antiquarks, respectively, that label the representation. - Alex Meiburg, Dec 13 2020 =
Comment on the previous one: D(n, m) = f(m+1, n+1) = (n+1)*(m+1)*(n+m+2), for 0 <= m <= n, (given as array D(n,m) as example in A212331) is the dimension of the irreducible SU(3) multiplet (n, m), denoted also by D(n, m). The multiplet (m, n) is denoted also by a bar over D(n, m). The irreducuble tensor t(n, m) is symmetric in n upper indices from {1,2,3}, symmetric in m lower indices, and traceless in every pair of an upper and a lower index. See the Coleman reference for a derivation. - Wolfdieter Lang, Dec 18 2020

			f(3, 5) is 60, from 1/2 * (3 * 5) * (3 + 5) or 1/2 * 15 * 8.
The triangle f(m, n) starts:
m\n     1   2   3   4   5   6   7   8   9   10   11 ...
1:      1
2:      3   8
3:      6  15  27
4:     10  24  42  64
5:     15  35  60  90 125
6:     21  48  81 120 165 216
7:     28  63 105 154 210 273 343
8:     36  80 132 192 260 336 420 512
9:     45  99 162 234 315 405 504 612 729
10:    55 120 195 280 375 480 595 720 855 1000
11:    66 143 231 330 440 561 693 836 990 1155 1331
... reformatted and extended by _Wolfdieter Lang_, Dec 18 2020

Sidney Coleman, Quantum Field Theory, Eds. Bryan Gin-ge Chen et al., World Scientific, 2019, eq. (37.8), p. 799.

Wikipedia, Clebsch-Gordan coefficients for SU(3)

Cf. A000217, A005563, A140091, A067728, A212331, A140681 (columns), A000578, A059270, A331433 (diagonals).
(diagonal).
See also A107985, A212331 (array as example).

J
```
cap half * * +
```

t[m_, n_] := (m*n)(m + n)/2; Flatten[ Table[ t[m, n], {m, 10}, {n, m}]] (* Robert G. Wilson v, Nov 04 2004 *)

f(m, n) = 1/2 * (m * n) * (m + n).

G.f.: x*y*(1 + 4*x*y + x^2*(y - 9)*y - 3*x^3*(y - 1)*y + 3*x^4*y^2)/((1 - x)^3*(1 - x*y)^4). - Stefano Spezia, Oct 01 2023

More terms from Robert G. Wilson v, Nov 04 2004

A362679 a(n) is the permanent of the n X n symmetric matrix M(n) defined by M[i, j, n] = min(i, j)(n + 1) - ij.

Original entry on oeis.org

1, 1, 5, 72, 2309, 140400, 14495641, 2347782144, 562385930985, 190398813728000, 87889475202276461, 53726132414026874880, 42454821207656237294381, 42495322215073539046387712, 52954624815227996007075890625, 80932107560443542398970529579008, 149736953621087625813286348913927569

Offset: 0

Stefano Spezia, Apr 29 2023

nonn

M(n-1)/n is the inverse of the Cartan matrix for SU(n): the special unitary group of degree n.
The elements sum of the matrix M(n) is A002415(n+1).
The antidiagonal sum of the matrix M(n) is A005993(n-1).
The n-th row of A107985 gives the row or column sums of the matrix M(n+1).

			a(3) = 72:
           [3, 2, 1]
    M(3) = [2, 4, 2]
           [1, 2, 3]
a(5) = 140400:
           [5, 4, 3, 2, 1]
           [4, 8, 6, 4, 2]
    M(5) = [3, 6, 9, 6, 3]
           [2, 4, 6, 8, 4]
           [1, 2, 3, 4, 5]

E. B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Am. Math. Soc. Translations, Series 2, Vol. 6, 1957.

Chao Ju, Chern-Simons Theory, Ehrhart Polynomials, and Representation Theory, arXiv:2304.11830 [math-ph], 2023. See p. 14.
Stefano Spezia, A determinantal formula for the number of trees on n labeled nodes
Wikipedia, Special unitary group

Cf. A000272, A000292 (trace), A002415, A003983, A003991, A005993, A106314 (antidiagonals), A107985.

a:= n-> `if`(n=0, 1, LinearAlgebra[Permanent](
    Matrix(n, (i, j)-> min(i, j)*(n+1)-i*j))):
seq(a(n), n=0..16);  # Alois P. Heinz, Apr 30 2023

M[i_, j_, n_]:=Min[i, j](n+1)-i j; Join[{1}, Table[Permanent[Table[M[i, j, n], {i, n}, {j, n}]], {n, 17}]]

a(n) = matpermanent(matrix(n, n, i, j, min(i, j)*(n + 1) - i*j)); \\ Michel Marcus, Apr 30 2023

Conjecture: det(M(n)) = A000272(n+1).

The conjecture is true (see proof in Links). - Stefano Spezia, May 24 2023

A157636 Triangle read by rows: T(n, k) = 1 if k=0 or k=n, otherwise = nk(n-k)/2.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 6, 8, 6, 1, 1, 10, 15, 15, 10, 1, 1, 15, 24, 27, 24, 15, 1, 1, 21, 35, 42, 42, 35, 21, 1, 1, 28, 48, 60, 64, 60, 48, 28, 1, 1, 36, 63, 81, 90, 90, 81, 63, 36, 1, 1, 45, 80, 105, 120, 125, 120, 105, 80, 45, 1

Offset: 0

Roger L. Bagula, Mar 03 2009

			Triangle begins as:
  1;
  1,  1;
  1,  1,  1;
  1,  3,  3,   1;
  1,  6,  8,   6,   1;
  1, 10, 15,  15,  10,   1;
  1, 15, 24,  27,  24,  15,   1;
  1, 21, 35,  42,  42,  35,  21,   1;
  1, 28, 48,  60,  64,  60,  48,  28,  1;
  1, 36, 63,  81,  90,  90,  81,  63, 36,  1;
  1, 45, 80, 105, 120, 125, 120, 105, 80, 45, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000578, A002415, A107985, A132411.

A157636:= func< n,k | k eq 0 or k eq n select 1 else n*k*(n-k)/2 >;
[A157636(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Dec 13 2021

T[n_, k_] = If[n*k*(n-k)==0, 1, n*k*(n-k)/2];
Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten

def A157636(n,k): return 1 if (k==0 or k==n) else n*k*(n-k)/2
flatten([[A157636(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Dec 13 2021

T(n, k) = 1 if k=0 or k=n, otherwise = n*k*(n-k)/2.
Sum_{k=0..n} T(n, k) = 2 + n^2*(n^2 - 1)/12 = 2 + A002415(n) if n>0.
From G. C. Greubel, Dec 13 2021: (Start)
T(n, k) = T(n, n-k).
T(n, 1) = [n<2] + binomial(n, 2).
T(n, 2) = A132411(n-1), for n >= 2.
T(2*n, n) = [n=0] + A000578(n). (End)

A346159 Number of n-dimensional representations of the group SU(3).

Original entry on oeis.org

1, 1, 1, 3, 3, 3, 8, 8, 9, 17, 19, 21, 35, 39, 44, 68, 79, 87, 127, 145, 162, 228, 261, 291, 395, 451, 506, 665, 760, 850, 1096, 1254, 1400, 1765, 2016, 2249, 2800, 3188, 3556, 4356, 4953, 5522, 6688, 7581, 8447, 10123, 11464, 12747, 15141, 17094, 18997, 22395

Offset: 0

Michel Marcus, Jul 08 2021

nonn

Kathrin Bringmann and Johann Franke, An asymptotic formula for the number of n-dimensional representations of SU(3), arXiv:2107.03261 [math.RT], 2021.
Dan Romik, On the number of n-dimensional representations of SU(3), the Bernoulli numbers, and the Witten zeta function, arXiv:1503.03776 [math.NT], 2015-2016.

Cf. A107985 (as a rectangular array).

fij(lim) = my(imax = ceil((sqrt(8*lim+1)-1)/2), list=List()); for (i=1, imax, for (j=1, imax, if ((p=i*j*(i+j)/2) <= lim, listput(list, p)))); list;
lista(nn) = my(v=fij(nn), x='x+O('x^nn), w=Vec(prod(k=1, #v, 1/(1-x^v[k])))); vector(nn, k, w[k]);

G.f.: Product_{j,k>=1} 1/(1-x^(j*k*(j+k)/2)).

A362711 a(n) is the hafnian of the 2n X 2n symmetric matrix defined by M[i, j] = min(i, j)(2n + 1) - i*j.

Original entry on oeis.org

1, 1, 17, 1177, 210249, 76961257, 50203153993, 53127675356625, 85252003916011889, 197131843368693693937, 631233222450168374457057

Offset: 0

Stefano Spezia, Apr 30 2023

M(n-1)/n is the inverse of the Cartan matrix for SU(n): the special unitary group of degree n.
The elements sum of the matrix M(n) is A002415(n+1).
The antidiagonal sum of the matrix M(n) is A005993(n-1).
The n-th row of A107985 gives the row or column sums of the matrix M(n+1).

			a(2) = 17:
    [4, 3, 2, 1]
    [3, 6, 4, 2]
    [2, 4, 6, 3]
    [1, 2, 3, 4]

E. B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Am. Math. Soc. Translations, Series 2, Vol. 6, 1957.

Chao Ju, Chern-Simons Theory, Ehrhart Polynomials, and Representation Theory, arXiv:2304.11830 [math-ph], 2023. See p. 14.
Stefano Spezia, A determinantal formula for the number of trees on n labeled nodes
Wikipedia, Hafnian
Wikipedia, Special unitary group
Wikipedia, Symmetric matrix

Cf. A000272, A000292 (trace), A002415, A003983, A003991, A005993, A106314 (antidiagonals), A107985, A362679 (permanent).

M[i_, j_, n_]:=Part[Part[Table[Min[r,c](n+1)-r c, {r, n}, {c, n}], i], j]; a[n_]:=Sum[Product[M[Part[PermutationList[s, 2n], 2i-1], Part[PermutationList[s, 2n], 2i], 2n], {i, n}], {s, SymmetricGroup[2n]//GroupElements}]/(n!*2^n); Array[a, 6, 0]

tm(n) = matrix(n, n, i, j, min(i, j)*(n + 1) - i*j);
a(n) = my(m = tm(2*n), s=0); forperm([1..2*n], p, s += prod(j=1, n, m[p[2*j-1], p[2*j]]); ); s/(n!*2^n); \\ Michel Marcus, May 02 2023

Conjecture: det(M(n)) = A000272(n+1).

The conjecture is true (see proof in Links). - Stefano Spezia, May 24 2023

a(6) from Michel Marcus, May 02 2023

a(7)-a(10) from Pontus von Brömssen, Oct 15 2023

Let W be the array given by w(1,1)=1, w(2,2)=-1, and w(n,k)=0 for all other (n,k).

Write "A < B" to indicate that an array B is the accumulation array of A (defined at A144112). Then W < A103451 < A002024 < A107985 < A185784 < A185785 < A185786.

See A185784. The pattern established by the formulas for A185785, A185786, A185787, suggests that the H-th accumulation array of A107985 may be given by

T(n,k)=(n+k+H)C(n+H,H+1)C(k+H,H+1)/(H+2).

cp's OEIS Frontend

A185784 Accumulation array of A107985, by antidiagonals.

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A185785 Second accumulation array of A107985, by antidiagonals.

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A185786 Third accumulation array of A107985, by antidiagonals.

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A002024 k appears k times; a(n) = floor(sqrt(2n) + 1/2).

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Haskell

Haskell

Haskell

Magma

Maple

Mathematica

PARI

PARI

PARI

PARI

PARI

PARI

PARI

Python

Sage

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A098737 Triangle read by rows: number of triangles formed by lines from two vertices of a triangle to the opposite side that segment the opposite sides into m and n segments. Since f(m,n) = f(n,m), it suffices to give the results in a triangular table.

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J

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Extensions

A362679 a(n) is the permanent of the n X n symmetric matrix M(n) defined by M[i, j, n] = min(i, j)*(n + 1) - i*j.

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A362679 a(n) is the permanent of the n X n symmetric matrix M(n) defined by M[i, j, n] = min(i, j)(n + 1) - ij.

A157636 Triangle read by rows: T(n, k) = 1 if k=0 or k=n, otherwise = nk(n-k)/2.

A362711 a(n) is the hafnian of the 2n X 2n symmetric matrix defined by M[i, j] = min(i, j)(2n + 1) - i*j.