A161680 -id:A161680 - cp's OEIS frontend

A081553 Sum of n-th row of A081551.

1, 21, 303, 4006, 50010, 600015, 7000021, 80000028, 900000036, 10000000045, 110000000055, 1200000000066, 13000000000078, 140000000000091, 1500000000000105, 16000000000000120, 170000000000000136, 1800000000000000153, 19000000000000000171, 200000000000000000190, 2100000000000000000210, 22000000000000000000231

Offset: 1

Amarnath Murthy, Apr 01 2003

G. C. Greubel, Table of n, a(n) for n = 1..500
Index entries for linear recurrences with constant coefficients, signature (23,-163,361,-320,100).

Cf. A081551, A081552.

Cf. A126431, A161680.

Table[n*10^(n-1) +n(n-1)/2, {n,40}] (* G. C. Greubel, May 27 2021 *)

[n*10^(n-1) +binomial(n,2) for n in (1..40)] # G. C. Greubel, May 27 2021

a(n) = n*10^(n-1) + n*(n-1)/2.
From G. C. Greubel, May 27 2021: (Start)
G.f.: x*(1 -2*x -17*x^2 +99*x^3)/((1-x)^3 * (1-10*x)^2).
E.g.f.: (1/2)*x*( x*exp(x) + 2*exp(10*x) ). (End)

Terms a(13) onward added by G. C. Greubel, May 27 2021

A139365 Array of digit sums of factorial representation of numbers 0,1,...,n!-1 for n >= 1.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 1, 2, 2, 3, 0, 1, 1, 2, 2, 3, 1, 2, 2, 3, 3, 4, 2, 3, 3, 4, 4, 5, 3, 4, 4, 5, 5, 6, 0, 1, 1, 2, 2, 3, 1, 2, 2, 3, 3, 4, 2, 3, 3, 4, 4, 5, 3, 4, 4, 5, 5, 6, 1, 2, 2, 3, 3, 4, 2, 3, 3, 4, 4, 5, 3, 4, 4, 5, 5, 6, 4, 5, 5, 6, 6, 7, 2, 3, 3, 4, 4, 5, 3, 4, 4, 5, 5, 6, 4, 5, 5, 6, 6, 7, 5, 6, 6, 7, 7

Offset: 0

Wolfdieter Lang, May 21 2008

The row lengths sequence is A000142 (factorials).
When the factorial representation is read as (D. N.) Lehmer code for permutations of n objects then the digit sums in row n count the inversions of the permutations arranged in lexicographic order.
Row n is the first n! terms of A034968. - Franklin T. Adams-Watters, May 13 2009

			n=3: The Lehmer codes for the permutations of {1,2,3} are [0,0,0], [0,1,0], [1,0,0], [1,1,0], [2,0,0] and [2,1,0]. These are the factorial representations for 0,1,...,5=3!-1. Therefore row n=3 has the digit sums 0,1,1,2,2,3, the number of inversions of the permutations [1,2,3], [1,3,2], [2,1,3], [2,3,1], [3,1,2] and [3,2,1] (lexicographic order).
Triangle begins:
  0;
  0;
  0, 1;
  0, 1, 1, 2, 2, 3;
  0, 1, 1, 2, 2, 3, 1, 2, 2, 3, 3, 4, 2, 3, 3, 4, 4, 5, 3, 4, 4, 5, 5, 6;
  ...

Alois P. Heinz, Rows n = 0..8, flattened
FindStat - Combinatorial Statistic Finder, The number of inversions of a permutation
A. Kohnert, Kombinatorische Algorithmen in C, Skript, Uni Bayreuth, 1997, pp. 5-7 [Broken link]
Wolfdieter Lang, First 6 rows. Factorial representations or Lehmer code for permutations.
D. N. Lehmer, On the orderly listing of substitutions, Bull. AMS 12 (1906), 81-84.
Index entries for sequences related to factorial base representation

Cf. A034968. - Franklin T. Adams-Watters, May 13 2009
Cf. A008302.
Cf. A000142, A161680.
Row sums give A001809.

nn = 5; m = 1; While[Factorial@ m < nn! - 1, m++]; m; Table[Total@ IntegerDigits[k, MixedRadix[Reverse@ Range[2, m]]], {n, 0, 5}, {k, 0, n! - 1}] // Flatten (* Version 10.2, or *)
f[n_] := Block[{a = {{0, n}}}, Do[AppendTo[a, {First@ #, Last@ #} &@ QuotientRemainder[a[[-1, -1]], Times @@ Range[# - i]]], {i, 0, #}] &@ NestWhile[# + 1 &, 0, Times @@ Range[# + 1] <= n &]; Most@ Rest[a][[All, 1]]]; Table[Total@ f@ k, {n, 0, 5}, {k, 0, n! - 1}] // Flatten (* Michael De Vlieger, Aug 29 2016 *)

Row n >= 1: sum(facrep(n,m)[n-j],j=1..n), m=0,1,...,n!-1, with the factorial representation facrep(n,m) of m for given n.

T(n,n!-1) = A161680(n). - Alois P. Heinz, Jan 20 2025

Zeroth row added by Franklin T. Adams-Watters, May 13 2009

A253145 Triangular numbers (A000217) omitting the term 1.

Original entry on oeis.org

0, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225, 1275

Offset: 0

Paul Curtz, Mar 23 2015

The full triangle of the inverse Akiyama-Tanigawa transform applied to (-1)^n*A062510(n)=3*(-1)^n*A001045(n) yielding a(n) is
0,      3,    6,  10,   15,  21,  28, 36, ...
-3,    -6,  -12, -20,  -30, -42, -56, ...    essentially -A002378
3,     12,   24,  40,   60,  84, ...         essentially  A046092
-9,   -24,  -48, -80, -120, ...              essentially -A033996
15,    48,   96, 160, ...
-33,  -96, -192, ...
63,   192, ...
-129, ...
etc.
First column: (-1)^n*A062510(n).
The following columns are multiples of A122803(n)=(-2)^n. See A007283(n), A091629(n), A020714(n+1), A110286, A175805(n), 4*A005010(n).
An autosequence of the first kind is a sequence whose main diagonal is A000004 = 0's.
b(n) = 0, 0 followed by a(n) is an autosequence of the first kind.
The successive differences of b(n) are
0,   0,  0, 3, 6, 10, 15, 21, ...
0,   0,  3, 3, 4,  5,  6,  7, ...  see A194880(n)
0,   3,  0, 1, 1,  1,  1,  1, ...
3,  -3,  1, 0, 0,  0,  0,  0, ...
-6,  4, -1, 0, 0,  0,  0,  0, ...
10, -5,  1, 0, 0,  0,  0,  0, ...
-15, 6, -1, 0, 0,  0,  0,  0, ...
21, -7,  1, 0, 0,  0,  0,  0, ...
The inverse binomial transform (first column) is the signed sequence. This is general.
Also generalized hexagonal numbers without 1. - Omar E. Pol, Mar 23 2015

Muniru A Asiru, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A179865, A001045, A062510, A002378, A046092, A033996, A122803, A007283, A091629, A020714, A110286, A161680, A175805, A005010, A194880, A001840, A022003, A255935.

Concatenation([0],List([1..50],n->(n+1)*(n+2)/2)); # Muniru A Asiru, Oct 31 2018

Prepend[Table[(n + 1) (n + 2)/2, {n, 49}], 0] (* Michael De Vlieger, Mar 23 2015 *)

a(n)=if(n,(n+1)*(n+2)/2,0) \\ Charles R Greathouse IV, Mar 23 2015

Inverse Akiyama-Tanigawa transform of (-1)^n*A062510(n).
a(n) = (n+1)*(n+2)/2 for n > 0. - Charles R Greathouse IV, Mar 23 2015
a(n+1) = 3*A001840(n+1) + A022003(n).
a(n) = A161680(n+2) for n >= 1. - Georg Fischer, Oct 30 2018
From Stefano Spezia, May 28 2025: (Start)
G.f.: x*(3 - 3*x + x^2)/(1 - x)^3.
E.g.f.: exp(x)*(2 + 4*x + x^2)/2 - 1. (End)

A255935 Triangle read by rows: a(n) = Pascal's triangle A007318(n) + A197870(n+1).

Original entry on oeis.org

0, 1, 2, 1, 2, 0, 1, 3, 3, 2, 1, 4, 6, 4, 0, 1, 5, 10, 10, 5, 2, 1, 6, 15, 20, 15, 6, 0, 1, 7, 21, 35, 35, 21, 7, 2, 1, 8, 28, 56, 70, 56, 28, 8, 0, 1, 9, 36, 84, 126, 126, 84, 36, 9, 2, 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 0

Offset: 0

Paul Curtz, Mar 11 2015

Consider the difference table of a sequence with A000004(n)=0's as main diagonal. (Example: A000045(n).) We call this sequence an autosequence of the first kind.
Based on Pascal's triangle, a(n) =
0,                   T1
1, 2,
1, 2, 0,
1, 3, 3, 2,
etc.
transforms every sequence s(n) in an autosequence of the first kind via the multiplication by the triangle
s0,                  T2
s0, s1,
s0, s1, s2,
s0, s1, s2, s3,
etc.
Examples.
1) s(n) = A198631(n)/A006519(n+1), the second fractional Euler numbers (see A209308). This yields 0*1, 1*1+2*1/2=2, 1*1+2*1/2+0*0=2, 1*1+3*1/2++3*0+2*(-1/4)=2, ... .
The autosequence is 0 followed by 2's or 2*(0,1,1,1,1,1,1,1,... = b(n)).
b(n), the basic autosequence of the first kind, is not in the OEIS (see A140575 and A054977).
2) s(n) = A164555(n)/A027642(n), the second Bernoulli numbers, yields 0,2,2,3,4,5,6,7,... = A254667(n).
Row sums of T1: A062510(n) = 3*A001045(n).
Antidiagonal sums of T1: A111573(n).
With 0's instead of the spaces, every column, i.e.,
0, 1, 1, 1, 1,  1,  1,  1,  1,  1,   1, ...
0, 2, 2, 3, 4,  5,  6,  7,  8,  9,  10, ... = A001477(n) with 0 instead of 1 = A254667(n)
0, 0, 0, 3, 6, 10, 15, 21, 28, 36,  45, ... = A161680(n) with 0 instead of 1
0, 0, 0, 2, 4, 10, 20, 35, 56, 84, 120, ...
etc., is an autosequence of the first kind.
With T(0,0) = 1, it is (1, 0, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, May 24 2015

			Triangle starts:
0;
1, 2;
1, 2, 0;
1, 3, 3, 2;
1, 4, 6, 4, 0;
1, 5, 10, 10, 5, 2;
1, 6, 15, 20, 15, 6, 0;
...

Cf. A001045, A001477, A010673, A011973, A054977, A062510, A007318, A197870, A006519, A198631, A140575, A209308, A164555, A027642, A111573, A161680, A254667.

a[n_, k_] := If[k == n, 2*Mod[n, 2], Binomial[n, k]]; Table[a[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 23 2015 *)

a(n) = Pascal's triangle A007318(n) with main diagonal A010673(n) (= period 2: repeat 0, 2) instead of 1's=A000012(n).
a(n) = reversal abs(A140575(n)).
a(n) = A007318(n) + A197870(n+1).
T(n,k) = T(n-1,k) + T(n-2,k-1) + T(n-2,k-2), T(0,0) = 0, T(1,0) = 1, T(1,1) = 2, T(n,k) = 0 if k>n or if k<0 . - Philippe Deléham, May 24 2015
G.f.: (-1-2*x*y+x^2*y+x^2*y^2)/((x*y+1)*(x*y+x-1)) - 1. - R. J. Mathar, Aug 12 2015

A271023 Number T(n,k) of set partitions of [n] having exactly k pairs (i,j) with i=0, 0<=k<=n*(n-1)/2 read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 0, 1, 1, 6, 3, 4, 0, 0, 1, 1, 10, 15, 10, 10, 0, 5, 0, 0, 0, 1, 1, 15, 45, 35, 60, 0, 25, 15, 0, 0, 6, 0, 0, 0, 0, 1, 1, 21, 105, 140, 210, 105, 105, 105, 0, 35, 21, 21, 0, 0, 0, 7, 0, 0, 0, 0, 0, 1, 1, 28, 210, 476, 665, 840, 350, 700, 210

Offset: 0

Alois P. Heinz, Mar 28 2016

			T(3,0) = 1: 1|2|3.
T(3,1) = 3: 12|3, 13|2, 1|23.
T(3,3) = 1: 123.
Triangle T(n,k) begins:
  1;
  1;
  1,  1;
  1,  3,  0,  1;
  1,  6,  3,  4,  0, 0,  1;
  1, 10, 15, 10, 10, 0,  5,  0, 0, 0, 1;
  1, 15, 45, 35, 60, 0, 25, 15, 0, 0, 6, 0, 0, 0, 0, 1;

Alois P. Heinz, Rows n = 0..40, flattened

Columns k=0-2 give: A000012, A161680, A050534(n-1) for n>0.
Row sums give A000110.
Cf. A105488, A161680, A271024.

b:= proc(n, l) option remember; `if`(n=0, x^
      add(j*(j-1)/2, j=l), b(n-1, [l[], 1])+
      add(b(n-1, subsop(j=l[j]+1, l)), j=1..nops(l)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [])):
seq(T(n), n=0..10);

b[n_, l_] := b[n, l] = If[n == 0, x^Sum[j*(j-1)/2, {j, l}], b[n-1, Append[l, 1]] + Sum[b[n-1, ReplacePart[l, j -> l[[j]]+1]], {j, 1, Length[l]}]];
T[n_] := CoefficientList[b[n, {}], x];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Apr 29 2022, after Alois P. Heinz *)

T(n,k) = A271024(n,n*(n-1)/2-k).

Sum_{k=0..n*(n-1)/2} k * T(n,k) = A105488(n+2) for n > 1.

A271371 Total number of inversions in all partitions of n into distinct parts.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 5, 6, 9, 13, 22, 26, 38, 48, 66, 89, 113, 142, 185, 230, 289, 368, 449, 554, 679, 831, 1003, 1224, 1474, 1767, 2117, 2528, 2996, 3568, 4206, 4967, 5855, 6862, 8027, 9391, 10943, 12724, 14785, 17124, 19807, 22898, 26376, 30345, 34893, 40013

Offset: 0

Alois P. Heinz, Apr 05 2016

nonn

			a(3) = 1: 21.
a(4) = 1: 31.
a(5) = 2: 32, 41.
a(6) = 5: 42, 51, 321 (three inversions).

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..8950 (terms 0..5000 from Alois P. Heinz)
Wikipedia, Inversion (discrete mathematics)

Cf. A000009, A008289, A161680, A271370, A271372.

b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
     `if`(n=0, [1, 0], b(n, i-1, t)+`if`(i>n, 0,
      (p-> p+[0, p[1]*t])(b(n-i, i-1, t+1)))))
    end:
a:= n-> b(n$2, 0)[2]:
seq(a(n), n=0..60);

b[n_, i_, t_] := b[n, i, t] = If[n > i*(i+1)/2, 0, If[n == 0, {1, 0}, b[n, i-1, t] + If[i>n, 0, Function[p, p+{0, p[[1]]*t}][b[n-i, i-1, t+1]]]]]; a[n_] := b[n, n, 0][[2]]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Feb 05 2017, translated from Maple *)

a(n) = Sum_{k>=1} A161680(k) * A008289(n,k).

A321124 a(n) = (4n^3 - 6n^2 + 14*n + 3)/3.

Original entry on oeis.org

1, 5, 13, 33, 73, 141, 245, 393, 593, 853, 1181, 1585, 2073, 2653, 3333, 4121, 5025, 6053, 7213, 8513, 9961, 11565, 13333, 15273, 17393, 19701, 22205, 24913, 27833, 30973, 34341, 37945, 41793, 45893, 50253, 54881, 59785, 64973, 70453, 76233, 82321, 88725

Offset: 0

Franck Maminirina Ramaharo, Dec 17 2018

For n >= 5, a(n) is the number of evaluation points on the n-dimensional cube in Phillips-Dobrodeev's degree 7 cubature rule.

Ronald Cools, Encyclopaedia of Cubature Formulas
Ronald Cools, Monomial cubature rules since "Stroud": a compilation - part 2, Journal of Computational and Applied Mathematics - Numerical evaluation of integrals Vol. 112 (1999), 21-27.
Ronald Cools and Philip Rabinowitz, Monomial cubature rules since "Stroud": a compilation, Journal of Computational and Applied Mathematics Vol. 48 (1993), 309-326.
L. N. Dobrodeev, Cubature formulas of the seventh order of accuracy for a hypersphere and a hypercube, USSR Computational Mathematics and Mathematical Physics Vol. 10 (1970), 252-253.
G. M. Phillips, Numerical integration over an N-dimensional rectangular region, The Computer Journal Vol. 10 (1967), 297-299.
Rudolf M. Schürer, High-Dimensional Numerical Integration on Parallel Computers, Phd Dissertation, 2001. p. 80.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000292, A128445, A161680, A174794, A322594, A322595.

Table[(4*n^3 - 6*n^2 + 14*n + 3)/3, {n, 0, 50}]

makelist((4*n^3 - 6*n^2 + 14*n + 3)/3, n, 0, 50);

a(n) = 8*binomial(n, 3) + 4*binomial(n, 2) + 4*binomial(n, 1) + 1.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), n >= 4.
a(n) = a(n-1) + A128445(n+1), n >= 1.
E.g.f.: (1/3)*(3 + 12*x + 6*x^2 + 4*x^3)*exp(x).
G.f.: (1 + x - x^2 + 7*x^3)/(1 - x)^4.

A341736 a(n) is the label of the square of the n-th element in the semigroup S = {(0,0), (i,j): i >= j >= 1}.

Original entry on oeis.org

0, 1, 7, 10, 37, 40, 45, 121, 124, 129, 136, 301, 304, 309, 316, 325, 631, 634, 639, 646, 655, 666, 1177, 1180, 1185, 1192, 1201, 1212, 1225, 2017, 2020, 2025, 2032, 2041, 2052, 2065, 2080, 3241, 3244, 3249, 3256, 3265, 3276, 3289, 3304, 3321, 4951, 4954, 4959

Offset: 0

Alois P. Heinz, Feb 17 2021

nonn

The product in S is computed componentwise.

For the labeling of the elements in S and further information see A341317.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Main diagonal of A341317 and of A341318.

Cf. A000217, A000290, A037270, A123578, A161680.

t:= n-> n*(n-1)/2:
f:= n-> ceil((sqrt(1+8*n)-1)/2):
g:= n-> (x-> [x, n-t(x)])(f(n)):
a:= n-> (h-> t(h[1]^2)+h[2]^2)(g(n)):
seq(a(n), n=0..60);

t[n_] := n*(n - 1)/2;
f[n_] := Ceiling[(Sqrt[1 + 8*n] - 1)/2];
g[n_] := Function[x, {x, n - t[x]}][f[n]];
a[n_] := Function[h, t[h[[1]]^2] + h[[2]]^2][g[n]];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Feb 26 2022, after Alois P. Heinz *)

a(n) = A341317(n,n) = A341318(n,n).

a(A000217(n)) = A037270(n) = A000217(A000290(n)).

A370945 Number T(n,k) of partitions of [n] whose singletons sum to k; triangle T(n,k), n>=0, 0<=k<=A000217(n), read by rows.

Original entry on oeis.org

1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 4, 1, 1, 2, 2, 2, 1, 1, 0, 0, 1, 11, 4, 4, 5, 5, 6, 3, 3, 3, 3, 2, 1, 1, 0, 0, 1, 41, 11, 11, 15, 15, 19, 20, 13, 10, 11, 8, 8, 5, 4, 4, 3, 2, 1, 1, 0, 0, 1, 162, 41, 41, 52, 52, 63, 67, 78, 41, 45, 39, 39, 33, 30, 20, 17, 14, 10, 10, 6, 5, 4, 3, 2, 1, 1, 0, 0, 1

Offset: 0

Alois P. Heinz, Mar 06 2024

			T(4,0) = 4: 1234, 12|34, 13|24, 14|23.
T(4,1) = 1: 1|234.
T(4,2) = 1: 134|2.
T(4,3) = 2: 124|3, 1|2|34.
T(4,4) = 2: 123|4, 1|24|3.
T(4,5) = 2: 1|23|4, 14|2|3.
T(4,6) = 1: 13|2|4.
T(4,7) = 1: 12|3|4.
T(4,10) = 1: 1|2|3|4.
Triangle T(n,k) begins:
   1;
   0, 1;
   1, 0, 0, 1;
   1, 1, 1, 1, 0, 0, 1;
   4, 1, 1, 2, 2, 2, 1, 1, 0, 0, 1;
  11, 4, 4, 5, 5, 6, 3, 3, 3, 3, 2, 1, 1, 0, 0, 1;
  ...

Alois P. Heinz, Rows n = 0..50, flattened
Wikipedia, Partition of a set

Column k=0 gives A000296.
Row sums give A000110.
Row lengths give A000124.
Reversed rows converge to A370946.
T(n,n) gives A370947.
Cf. A000217, A053632, A105479, A369596.

h:= proc(n) option remember; `if`(n=0, 1,
      add(h(n-j)*binomial(n-1, j-1), j=2..n))
    end:
b:= proc(n, i, m) option remember; `if`(n>i*(i+1)/2, 0,
     `if`(n=0, h(m), b(n, i-1, m)+b(n-i, min(n-i, i-1), m-1)))
    end:
T:= (n, k)-> b(k, min(n, k), n):
seq(seq(T(n, k), k=0..n*(n+1)/2), n=0..7);

h[n_] := h[n] = If[n == 0, 1,
    Sum[h[n-j]*Binomial[n-1, j-1], {j, 2, n}]];
b[n_, i_, m_] := b[n, i, m] = If[n > i*(i + 1)/2, 0,
    If[n == 0, h[m], b[n, i - 1, m] + b[n - i, Min[n - i, i - 1], m - 1]]];
T[n_, k_] := b[k, Min[n, k], n];
Table[Table[T[n, k], { k, 0, n*(n + 1)/2}], {n, 0, 7}] // Flatten (* Jean-François Alcover, Mar 12 2024, after Alois P. Heinz *)

Sum_{k=0..A000217(n)} k * T(n,k) = A105479(n+1).
T(n,A161680(n)) = A370946(n).
T(n,A000217(n)) = 1.

A087125 Indices k of hex numbers H(k) that are also triangular.

Original entry on oeis.org

0, 5, 54, 539, 5340, 52865, 523314, 5180279, 51279480, 507614525, 5024865774, 49741043219, 492385566420, 4874114620985, 48248760643434, 477613491813359, 4727886157490160, 46801248083088245, 463284594673392294, 4586044698650834699, 45397162391834954700

Offset: 0

Eric W. Weisstein, Aug 14 2003

From the law of cosines, the non-Pythagorean triple {a(n), a(n)+1=A253475(n+1), A072256(n+1)} forms a near-isosceles triangle with the angle bounded by the consecutive sides equal to the regular tetrahedron's central angle (see A156546 and A247412). This implies also that a(n) are those numbers k such that (16/3)*A000217(k)+1 is a perfect square. - Federico Provvedi, Apr 04 2023

Colin Barker, Table of n, a(n) for n = 0..1000
Editors, L'Intermédiaire des Mathématiciens, Query 4500: The equation x(x+1)/2 = y*(y+1)/3, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (I).
Editors, L'Intermédiaire des Mathématiciens, Query 4500: The equation x(x+1)/2 = y*(y+1)/3, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (II).
Editors, L'Intermédiaire des Mathématiciens, Query 4500: The equation x(x+1)/2 = y*(y+1)/3, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (III).
Editors, L'Intermédiaire des Mathématiciens, Query 4500: The equation x(x+1)/2 = y*(y+1)/3, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (IV).
Eric Weisstein's World of Mathematics, Hex Number
Index entries for linear recurrences with constant coefficients, signature (11,-11,1).

Cf. A006244, A031138, A072256, A054318.

[Round((-4-(5-2*Sqrt(6))^n*(-2+Sqrt(6)) + (2+Sqrt(6))*(5 + 2*Sqrt(6))^n)/8): n in [0..25]]; // G. C. Greubel, Nov 04 2017

CoefficientList[Series[(-x^2+5*x)/((1-x)*(1-10*x+x^2)), {x, 0, 25}], x] (* G. C. Greubel, Nov 04 2017 *)
LinearRecurrence[{11,-11,1},{0,5,54},30] (* Harvey P. Dale, Jun 14 2022 *)
Table[(x Sqrt[z^(2 n + 1) + z^-(2 n + 1) - 2] - 4) / 8 //. {x -> Sqrt[2], y -> Sqrt[3], z -> (5 + 2 x y)}, {n, 0, 100}] // Round (* Federico Provvedi, Apr 16 2023 *)

concat(0, Vec(x*(x-5)/((x-1)*(x^2-10*x+1)) + O(x^50))) \\ Colin Barker, Jun 23 2015

G.f.: (-x^2+5*x)/((1-x)*(1-10*x+x^2)).
a(n) = 11*a(n-1) - 11*a(n-2) + a(n-3) for n > 2. - Colin Barker, Jun 23 2015
a(n) = (-4 - (5-2*sqrt(6))^n*(-2 + sqrt(6)) + (2+sqrt(6))*(5+2*sqrt(6))^n)/8. - Colin Barker, Mar 05 2016
a(n) = 10*a(n-1) - a(n-2) + 4 for n > 1. - Charlie Marion, Feb 14 2023
a(n) = ((x^(n+1)+1)*(x^n-1))/(2*x^n*(x-1)), with x=5+2*sqrt(6). - Federico Provvedi, Apr 04 2023
a(n) = sqrt(3*A161680(A054318(n+1)) + 1/4) - 1/2 = floor(sqrt(3*A000217(A054318(n+1)-1) + 1/4)). - Federico Provvedi, Apr 16 2023

cp's OEIS Frontend

A081553 Sum of n-th row of A081551.

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A139365 Array of digit sums of factorial representation of numbers 0,1,...,n!-1 for n >= 1.

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A253145 Triangular numbers (A000217) omitting the term 1.

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A255935 Triangle read by rows: a(n) = Pascal's triangle A007318(n) + A197870(n+1).

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A271023 Number T(n,k) of set partitions of [n] having exactly k pairs (i,j) with i=0, 0<=k<=n*(n-1)/2 read by rows.

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A271371 Total number of inversions in all partitions of n into distinct parts.

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A321124 a(n) = (4*n^3 - 6*n^2 + 14*n + 3)/3.

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A341736 a(n) is the label of the square of the n-th element in the semigroup S = {(0,0), (i,j): i >= j >= 1}.

A321124 a(n) = (4n^3 - 6n^2 + 14*n + 3)/3.