A056106 -id:A056106 - cp's OEIS frontend

A188947 a(n) = n^3 - 2n^2 + 2n + 1.

2, 5, 16, 41, 86, 157, 260, 401, 586, 821, 1112, 1465, 1886, 2381, 2956, 3617, 4370, 5221, 6176, 7241, 8422, 9725, 11156, 12721, 14426, 16277, 18280, 20441, 22766, 25261, 27932, 30785, 33826, 37061, 40496, 44137, 47990, 52061, 56356, 60881, 65642, 70645

Offset: 1

Adeniji, Adenike, Apr 14 2011

The original definition was "Identity difference partial one - one transformation semigroup is a semigroup having the property that the difference between max im(alpha) and min im(alpha) is not greater than 1. This is denoted by S = IDI_n for each n." [Needs editing.]

For all n >= 3, a(n) expressed in base n has the three digits n-2, 2, and 1; for example, a(16) in hexadecimal is "E21". For all n >= 3, a(n+1) expressed in base n is "1112". For all n >= 7, a(n+2) expressed in base n is "1465". - Mathew Englander, Jan 07 2021

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A027444, A053698, A056106 (first differences), A060354, A162607, A188377, A188716.

[n^3 - 2*n^2 + 2*n + 1: n in [1..30]]; // Vincenzo Librandi, May 01 2011

A188947[n_] := n^3 - 2*n^2 + 2*n + 1; Table[A188947[n], {n, 1, 42}] (* Robert P. P. McKone, Jan 31 2021 *)

a(n)=n^3-2*n^2+2*n+1 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = (n+1) + n*(n-1)^2 = n^3 - 2*n^2 + 2*n + 1 = 1 + A053698(n-1).
G.f.: ( -x*(-2 + 3*x - 8*x^2 + x^3) ) / ( (x-1)^4 ). - R. J. Mathar, Apr 14 2011
a(n) = A060354(n) + A162607(n+1). - Lechoslaw Ratajczak, Sep 24 2020
E.g.f.: exp(x)*(1 + x)*(1 + x^2) - 1. - Stefano Spezia, Apr 10 2022

Edited by N. J. A. Sloane, Apr 23 2011

A251599 Centers of rows of the triangular array formed by the natural numbers.

Original entry on oeis.org

1, 2, 3, 5, 8, 9, 13, 18, 19, 25, 32, 33, 41, 50, 51, 61, 72, 73, 85, 98, 99, 113, 128, 129, 145, 162, 163, 181, 200, 201, 221, 242, 243, 265, 288, 289, 313, 338, 339, 365, 392, 393, 421, 450, 451, 481, 512, 513, 545, 578, 579, 613, 648, 649, 685, 722, 723

Offset: 1

Dave Durgin, Dec 05 2014

nonn

Forms a cascade of 3-number triangles down the center of the triangle array. Related to A000124 (left/west bank of same triangular array), A000217 (right/east bank) and A001844 (center column).
Sums of the mentioned cascading triangles: a(3*n-2) + a(3*n-1) + a(3*n) = A058331(n) + A001105(n) + A001844(n-1) = 2*A056106(n) = 2*(3*n^2-n+1). - Reinhard Zumkeller, Dec 13 2014
Union of A080827 and A000982. - David James Sycamore, Aug 09 2018

			First ten terms (1,2,3,5,8,9,13,18,19,25) may be read down the center of the triangular formation:
               1
             2   3
           4   5   6
         7   8   9  10
      11  12  13  14  15
    16  17  18  19  20  21
  22  23  24  25  26  27  28

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
David James Sycamore, A080827 & A000982
Index entries for linear recurrences with constant coefficients, signature (1,0,2,-2,0,-1,1).

Cf. A000124, A000217, A001844.
Cf. A092942 (first differences).
Cf. A001105, A056106, A058331.
Cf. A080827, A000982.

a251599 n = a251599_list !! (n-1)
a251599_list = f 0 $ g 1 [1..] where
   f i (us:vs:wss) = [head $ drop i us] ++ (take 2 $ drop i vs) ++
                     f (i + 1) wss
   g k zs = ys : g (k + 1) xs where (ys,xs) = splitAt k zs
-- Reinhard Zumkeller, Dec 12 2014

a:= n-> (m-> 2*(m+1)^2-[2*m+1, 0, -1][1+r])(iquo(n-1, 3, 'r')):
seq(a(n), n=1..100);  # Alois P. Heinz, Dec 10 2014

LinearRecurrence[{1, 0, 2, -2, 0, -1, 1}, {1, 2, 3, 5, 8, 9, 13}, 60] (* Jean-François Alcover, Jan 09 2016 *)

Vec(-x*(x^2+1)*(x^4-x^3+x+1)/((x^2+x+1)^2*(x-1)^3) + O(x^80)) \\ Michel Marcus, Jan 09 2016

Terms for n=1 (mod 3): 2m^2+2m+1, for n=2 (mod 3): 2m^2+4m+2, for n=0 (mod 3): 2m^2+4m+3, where m = floor((n-1)/3).

G.f.: -x*(x^2+1)*(x^4-x^3+x+1)/((x^2+x+1)^2*(x-1)^3). - Alois P. Heinz, Dec 10 2014

A185877 Array T given by T(n,k) = k^2 +(2n-3)k -2*n +3, by antidiagonals.

Original entry on oeis.org

1, 3, 1, 7, 5, 1, 13, 11, 7, 1, 21, 19, 15, 9, 1, 31, 29, 25, 19, 11, 1, 43, 41, 37, 31, 23, 13, 1, 57, 55, 51, 45, 37, 27, 15, 1, 73, 71, 67, 61, 53, 43, 31, 17, 1, 91, 89, 85, 79, 71, 61, 49, 35, 19, 1, 111, 109, 105, 99, 91, 81, 69, 55, 39, 21, 1, 133, 131, 127, 121, 113, 103, 91, 77, 61, 43, 23, 1, 157, 155, 151, 145, 137, 127, 115, 101, 85, 67, 47, 25, 1, 183, 181, 177, 171, 163, 153, 141, 127, 111, 93, 73, 51, 27, 1

Offset: 1

Clark Kimberling, Feb 05 2011

A member of the accumulation chain ... < A185879 < A185877 < A185878 < A185880 < ... (See A144112 for the definition of accumulation array).

			Northwest corner:
  1, 3,  7, 13, 21
  1, 5, 11, 19, 29
  1, 7, 15, 25, 45
  1, 9, 19, 31, 45

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A144112, A185878, A185879, A185880.
Row 1 to 3: A002061, A028387, A082111.
diag (1,5,...): A056108;
diag (3,11,...): A056106;
diag (7,19,...): A003215;
diag (13,29,...): A144391;
diag (1,7,...): A003215;
diag (1,9,...): A144390.

(* This program generates A185877, its accumulation array A185878, and its weight array A185879. *)
f[n_,0]:=0;f[0,k_]:=0;
f[n_,k_]:=k^2+(2n-3)k-2n+3;
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]] (* A185877 *)
Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}]; (* accumulation array of {f(n,k)} *)
FullSimplify[s[n,k]]  (* formula for A185878 *)
TableForm[Table[s[n,k],{n,1,10},{k,1,15}]]
Table[s[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
w[m_,n_]:=f[m,n]+f[m-1,n-1]-f[m,n-1]-f[m-1,n]/;Or[m>0,n>0];
TableForm[Table[w[n,k],{n,1,10},{k,1,15}]] (* A185879 *)
Table[w[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten

T(n,k) = k^2 + (2*n-3)*k - 2*n + 3, k>=1, n>=1.

A185880 Second accumulation array of A185877, by antidiagonals.

Original entry on oeis.org

1, 5, 3, 16, 17, 6, 40, 56, 38, 10, 85, 140, 128, 70, 15, 161, 295, 320, 240, 115, 21, 280, 553, 670, 600, 400, 175, 28, 456, 952, 1246, 1250, 1000, 616, 252, 36, 705, 1536, 2128, 2310, 2075, 1540, 896, 348, 45, 1045, 2355, 3408, 3920, 3815, 3185, 2240, 1248, 465, 55, 1496, 3465, 5190, 6240, 6440, 5831, 4620, 3120, 1680, 605, 66, 2080, 4928, 7590, 9450, 10200, 9800, 8428, 6420, 4200, 2200

Offset: 1

Clark Kimberling, Feb 05 2011

A member of the accumulation chain ... < A185879 < A185877 < A185878 < A185880 < ... See A144112 for the definition of accumulation array.

			Northwest corner:
   1,    5,   16,   40,   85
   3,   17,   56,  140,  295
   6,   38,  128,  320,  670
  10,   70,  240,  600, 1250

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A144112, A185877, A185878.
Antidiagonal sums: A037235.
diag (1,5,...): A056108 (4th spoke on hexagonal wheel);
diag (3,11,...): A056106 (2nd spoke on hexagonal wheel);
diag (7,19,...): A003215 (hex numbers);
diag (13,29,...): A144391.

(* This program generates A185878 first and then generates A185880 as the accumulation array of A185878. *)
f[n_,k_]:=(k*n/6)(7-3k+2k^2-3n+3kn);
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]] (* A185878 *)
Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}];
FullSimplify[s[n,k]]
TableForm[Table[s[n,k],{n,1,10},{k,1,15}]] (* A185880 *)
f[n_, k_] := (1/72)*k*(1 + k)*n*(1 + n)*(16 - k + 3 *k^2 + 4 *(-1 + k) *n); Table[f[n - k + 1, k], {n, 10}, {k, n, 1, -1}] // Flatten (* G. C. Greubel, Jul 21 2017 *)

T(n,k) = C(k,2)*C(n,2)*(3*k^2+4*k*n-k-4*n+16)/18, k>=1, n>=1.

A113653 Isolated semiprimes in the hexagonal spiral.

Original entry on oeis.org

6, 51, 69, 82, 91, 183, 194, 221, 249, 265, 287, 289, 309, 314, 319, 323, 355, 371, 403, 417, 437, 469, 478, 511, 517, 519, 533, 579, 589, 649, 681, 689, 731, 749, 758, 807, 838, 849, 926, 943, 951, 961, 965, 979, 1011, 1018, 1037, 1055, 1057, 1067, 1077, 1099, 1126, 1145, 1149, 1154, 1159

Offset: 1

Jonathan Vos Post, Jan 16 2006

nonn

Isolated semiprimes in the hexagonal spiral of A003215 and A001399, which is centered on 0. Of course such a spiral can be constructed beginning with any integer. Centering on 0 gives the interesting partition and multigraph equalities of A001399.
Integers in A001358 which are not adjacent in any of six directions to any other integer in A001358 when arranged in the hexagonal spiral.
An analog of  A113688 "Isolated semiprimes in the [square] spiral," and of the hexagonal prime spiral of [Abbott 2005; Weisstein, "Prime Spiral", MathWorld].
Unfortunately the original submission (which has been preserved as the "dead" sequence A335704) omitted the number 44 from the spiral, which has caused an enormous amount of trouble. - N. J. A. Sloane, Jun 27 2020

			The spiral begins:
                120-119-118-117-116-115-114
                 /                         \
              121  85--84--83-*82*-81--80 113
               /   /                     \   \
            122  86  56--55--54--53--52  79 112
             /   /   /                 \   \   \
          123  87  57  33--32--31--30 *51* 78 111
           /   /   /   /             \   \   \   \
        124  88  58  34  16--15--14  29  50  77 110
         /   /   /   /   /         \   \   \   \   \
      125  89  59  35  17   5---4  13  28  49  76 109
       /   /   /   /   /   /     \   \   \   \   \   \
    126  90  60  36  18  *6*  0   3  12  27  48  75 108
     /   /   /   /   /   /   /   /   /   /   /   /   /
  127 *91* 61  37  19   7   1---2  11  26  47  74 107 146
     \   \   \   \   \   \         /   /   /   /   /   /
    128  92  62  38  20   8---9--10  25  46  73 106 145
       \   \   \   \   \             /   /   /   /   /
      129  93  63  39  21--22--23--24  45  72 105 144
         \   \   \   \                 /   /   /   /
        130  94  64  40--41--42--43--44  71 104 143
           \   \   \                     /   /   /
          131  95  65--66--67--68-*69*-70 103 142
             \   \                         /   /
            132  96--97--98--99-100-101-102 141
               \                             /
              133-134-135-136-137-138-139-140

Abbott, P. (Ed.). "Mathematica One-Liners: Spiral on an Integer Lattice." Mathematica J. 1, 39, 1990.

P. Abbott, Re: Hexagonal Spiral, (alt link), May 11, 2005
H. Bottomley, Spokes of a Hexagonal Spiral.
R. J. Mathar, Maple program for A113653
Eric Weisstein's World of Mathematics, Prime Spiral.

Cf. A001358, A001399, A113688.
For the sequence of isolated primes see A335916.
Related sequences:
A113519 Semiprimes in 1st spoke of a hexagonal spiral starting at 1 (A056105).
A113524 Semiprimes in 2nd spoke of a hexagonal spiral (A056106).
A113525 Semiprimes in 3rd spoke of a hexagonal spiral (A056107).
A113527 Semiprimes in 4th spoke of a hexagonal spiral (A056108).
A113528 Semiprimes in 5th spoke of a hexagonal spiral (A056109).
A113530 Semiprimes in 6th spoke of a hexagonal spiral (A003215).

Corrected and edited by N. J. A. Sloane, Jun 27 2020. Thanks to Jeffrey K. Aronson for pointing out the error in the original submission.

Terms a(4) onwards corrected by R. J. Mathar, Jun 29 2020

A362007 Fourth Lie-Betti number of a path graph on n vertices.

Original entry on oeis.org

0, 0, 3, 16, 48, 107, 203, 347, 551, 828, 1192, 1658, 2242, 2961, 3833, 4877, 6113, 7562, 9246, 11188, 13412, 15943, 18807, 22031, 25643, 29672, 34148, 39102, 44566, 50573, 57157, 64353, 72197, 80726, 89978, 99992, 110808

Offset: 1

Samuel J. Bevins, Apr 05 2023

Sequence T(n,4) in A360571.

Marco Aldi and Samuel Bevins, L_oo-algebras and hypergraphs, arXiv:2212.13608 [math.CO], 2022. See page 9.
Meera Mainkar, Graphs and two step nilpotent Lie algebras, arXiv:1310.3414 [math.DG], 2013. See page 1.
Eric Weisstein's World of Mathematics, Path Graph.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A360571 (path graph triangle), A088459 (second Lie-Betti number of path graphs), A361230.

Cf. A011379, A006002, A001105, A056106, A000027, A000332, A000217.

def A362007(n):
    values = [0,0,3]
    for i in range(4, n+1):
        result = (i**4 + 18*i**3 - 97*i**2 + 174*i - 168)/24
        values.append(int(result))
    return values

a(1) = a(2) = 0, a(3) = 3, a(n) = (n^4 + 18*n^3 - 97*n^2 + 174*n - 168)/24 for n >= 4.
a(n) = A011379(n-3) + A006002(n-4) + A001105(n-3) + A056106(n-2) + A000027(n-3) + A000332(n-3) + A000217(n-5) + A000027(n-4) for n >= 5.
From Stefano Spezia, Mar 02 2025: (Start)
G.f.: x^2*(3 + x - 2*x^2 - 3*x^3 + 3*x^4 - x^5)/(1 - x)^5.
E.g.f.: (12*(6 + 4*x + x^2) - exp(x)*(72 - 24*x - 36*x^2 - 28*x^3 - x^4))/24. (End)

a(34) and Python program corrected by Robert C. Lyons, Apr 17 2023

A275673 List of numbers that are in a spoke of a hexagonal spiral.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 11, 13, 15, 17, 19, 22, 25, 28, 31, 34, 37, 41, 45, 49, 53, 57, 61, 66, 71, 76, 81, 86, 91, 97, 103, 109, 115, 121, 127, 134, 141, 148, 155, 162, 169, 177, 185, 193, 201, 209, 217, 226, 235, 244, 253, 262, 271, 281, 291, 301, 311, 321

Offset: 1

Peter Kagey, Aug 04 2016

nonn

This sequence contains k if and only if k is in one of the following sequences: A056105, A056106, A056107, A056108, A056109, A003215.

Alternatively, this sequence consists of the numbers of the form 3k^2 + bk + 1 for nonnegative k and -2 <= b <= 3.

Peter Kagey, Table of n, a(n) for n = 1..10003

Cf. A056105, A056106, A056107, A056108, A056109, A003215.

a275673 n = a275673_list !! (n - 1)
a275673_list = scanl (+) 1 $ concatMap (replicate 6) [1..]

Conjectures from Colin Barker, Aug 05 2016: (Start)
a(n) = 2*a(n-1)-a(n-2)+a(n-6)-2*a(n-7)+a(n-8) for n>8.
G.f.: x*(1-x^6+x^7) / ((1-x)^3*(1+x)*(1-x+x^2)*(1+x+x^2)).
(End)
Conjecture: a(n) = (n+4-3*floor((n+4)/6)-2)*floor((n+4)/6)+1. - Luce ETIENNE, May 25 2017

A329482 Interleave 1 - n + 3n^2, 1 + 3n*(1+n) for n >= 0.

Original entry on oeis.org

1, 1, 3, 7, 11, 19, 25, 37, 45, 61, 71, 91, 103, 127, 141, 169, 185, 217, 235, 271, 291, 331, 353, 397, 421, 469, 495, 547, 575, 631, 661, 721, 753, 817, 851, 919, 955, 1027, 1065, 1141, 1181, 1261

Offset: 0

Paul Curtz, Nov 14 2019

a(n+1) - 2*a(n) = -1, 1, 1, -3, -3, -13, -13, -29, -29, ...
Hexagonal spiral for A000265:
.
                   17--35---9--37
                   /
                 33  17---9--19---5
                 /   /             \
                1   1   3---7---1  21
               /   /   /         \   \
             31  15   5   1---1   9  11
               \   \   \     /   /   /
               15   7   1---3   5  23
                 \   \         /   /
                 29  13---3--11   3
                   \             /
                    7--27--13--25
.
The two sequences are perpendicular.
a(n+1) - a(n) = 0, 2, 4, 4, 8, 6, 12, ... = 2*A029578(n+2).
A003215 is a bisection of 1, 1, 13, 7, 49, 19, 109, 37, ... .

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A003215, A029578, A056106, A056108.

LinearRecurrence[{1, 2, -2, -1, 1}, {1, 1, 3, 7, 11}, 42] (* Amiram Eldar, Nov 23 2019 *)
Module[{nn=20,a,b},a=Table[1-n+3 n^2,{n,0,nn}];b=Table[1+3n(1+n),{n,0,nn}];Riffle[a,b]] (* Harvey P. Dale, Apr 30 2023 *)

Vec((1 + 4*x^3 + x^4) / ((1 - x)^3*(1 + x)^2) + O(x^40)) \\ Colin Barker, Nov 15 2019

From Colin Barker, Nov 14 2019: (Start)
G.f.: (1 + 4*x^3 + x^4) / ((1 - x)^3*(1 + x)^2).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 4.
a(n) = (5 + 3*(-1)^n - 2*(1 + (-1)^n)*n + 6*n^2) / 8.
(End)
E.g.f.: (1/8)*exp(-x)*(3 + 2*x + exp(2*x)*(5 + 4*x + 6*x^2)). - Stefano Spezia, Nov 14 2019 after Colin Barker
a(-n) = 1, 1, 5, 7, 15, 19, ... = interleave 1 + n + 3*n^2, 1 + 3*n*(1+n), both in the spiral.

A335704 Erroneous version of A113653.

Original entry on oeis.org

6, 51, 55, 69, 82, 183, 194, 249, 259, 287, 309, 314, 319

Offset: 1

dead

This is the erroneous version of A113653 that was submitted to the OEIS by Jonathan Vos Post on Jan 16 2006. Because 44 was omitted from the spiral, not only are the terms here incorrect, but a large number of other sequences will also need to be corrected. For this reason the whole of the original submission has been preserved here with a different A-number. - N. J. A. Sloane, Jun 27 2020

Isolated semiprimes in the hexagonal spiral, embedded in the triangular lattice, are the analogy to A113688 "Isolated semiprimes in the [square] spiral," as well as analogous in another way to the hexagonal prime spiral of [Abbott 2005; Weisstein, "Prime Spiral", MathWorld]. A113519 Semiprimes in first spoke of a hexagonal spiral (A056105). A113524 Semiprimes in second spoke of a hexagonal spiral (A056106). A113525 Semiprimes in third spoke of a hexagonal spiral (A056107). A113527 Semiprimes in fourth spoke of a hexagonal spiral (A056108). A113528 Semiprimes in fifth spoke of a hexagonal spiral (A056109). A113530 Semiprimes in sixth spoke of a hexagonal spiral (A003215). This is embedded in the hexagonal spiral of A003215 and A001399, which is centered on zero; of course such a spiral can be constructed beginning with any integer. Centering on zero gives the interesting partition and multigraph equalities of A001399.

			Copy this as proportionally spaced text, make semiprimes bold, draw boundaries around clumps of adjacent semiprimes. For example, there is a triangular clump of three semiprimes: {4, 14, 15}; a linear clump of three semiprimes {49, 77, 111}; a linear clump of two semiprimes {247, 305}; an irregular clump of seven {115, 155, 201, 202, 203, 253, 254}; a clump of eighteen whose least element is 33 and greatest is 206; and a long branching clump of sixteen whose least element is 9 and greatest is 129.
.................209.208.207.206.205.204.203.202.201
................210.162.161.160.159.158.157.156.155.200
..............211.163.121.120.119.118.117.116.115.154.199
............212.164.122.86..85..84..83..82..81.114.153.198
..........213.165.123.87..57..56..55..54..53..80.113.152.197
........214.166.124.88..58..33..32..31..30..52..79.112.151.196
......215.167.125.89..59..34..16..15..14..29..51..78.111.150.195
....216.168.126.90..60..35..17..5...4...13..28..50..77.110.149.194
..217.169.127.91..61..36..18..6...0...3...12..27..49..76.109.148.193
218.170.128.92..62..37..19..7...1...2...11..26..48..75.108.147.192.243
..219.171.129.93..63..38..20..8...9...10..25..47..74.107.146.191.242
....220.172.130.94..64..39..21..22..23..24..46..73.106.145.190.241
......221.173.131.95..65..40..41..42..43..45..72.105.144.189.240
........222.174.132.96..66..67..68..69..70..71.104.143.188.239
..........223.175.133.97..98..99.100.101.102.103.142.187.238
............224.176.134.135.136.137.138.139.140.141.186.237
..............225.177.178.179.180.181.182.183.184.185.236
................226.227.228.229.230.231.232.233.234.235

Abbott, P. (Ed.). "Mathematica One-Liners: Spiral on an Integer Lattice." Mathematica J. 1, 39, 1990.

P. Abbott, Re: Hexagonal Spiral, (alt link), May 11, 2005
H. Bottomley, Spokes of a Hexagonal Spiral.
Eric Weisstein's World of Mathematics, Prime Spiral.

Cf. A001358, A001399, A003215, A056105-A056109, A113688, A113519, A113524, A113525, A113528, A113527, A113530, A113688.

{a(n)} = {integers in A001358 which are not adjacent in any of six directions to any other integers in A001358 when arranged as the hexagonal spiral of A003215}.

cp's OEIS Frontend

A188947 a(n) = n^3 - 2*n^2 + 2*n + 1.

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A251599 Centers of rows of the triangular array formed by the natural numbers.

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Haskell

Maple

Mathematica

PARI

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A185877 Array T given by T(n,k) = k^2 +(2*n-3)*k -2*n +3, by antidiagonals.

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Mathematica

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A185880 Second accumulation array of A185877, by antidiagonals.

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Mathematica

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A113653 Isolated semiprimes in the hexagonal spiral.

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A362007 Fourth Lie-Betti number of a path graph on n vertices.

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Python

Formula

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A275673 List of numbers that are in a spoke of a hexagonal spiral.

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Haskell

A188947 a(n) = n^3 - 2n^2 + 2n + 1.

A185877 Array T given by T(n,k) = k^2 +(2n-3)k -2*n +3, by antidiagonals.

A329482 Interleave 1 - n + 3n^2, 1 + 3n*(1+n) for n >= 0.