A006003 -id:A006003 - cp's OEIS frontend

A061317 Split positive integers into extending even groups and sum: 1+2, 3+4+5+6, 7+8+9+10+11+12, 13+14+15+16+17+18+19+20, ...

0, 3, 18, 57, 132, 255, 438, 693, 1032, 1467, 2010, 2673, 3468, 4407, 5502, 6765, 8208, 9843, 11682, 13737, 16020, 18543, 21318, 24357, 27672, 31275, 35178, 39393, 43932, 48807, 54030, 59613, 65568, 71907, 78642, 85785, 93348, 101343, 109782

Offset: 0

Henry Bottomley, Feb 13 2002

5*a(n+1) is the sum of the products of the 10 distinct combinations of three consecutive numbers starting with n (using 1,2,3 the 10 combinations are 111 112 113 122 123 133 222 223 233 333; 1*1*1 + 1*1*2 + 1*1*3 + 1*2*2 + 1*2*3 + 1*3*3 + 2*2*2 + 2*2*3 + 2*3*3 + 3*3*3 = 90 = 5*a(2)). - J. M. Bergot, Mar 28 2014 [expanded by Jon E. Schoenfield, Feb 22 2015]

			1+2 = 3; 3+4+5+6 = 18; 7+8+9+10+11+12 = 57; 13+14+15+16+17+18+19+20 = 132.

Harry J. Smith, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A005898, A006003.

Cf. A000217, A002378, A110450.

A061317:=n->2*n^3+n; seq(A061317(n), n=0..100); # Wesley Ivan Hurt, Mar 20 2014

Table[2n^3+n,{n,0,5!}] (* Vladimir Joseph Stephan Orlovsky, Dec 04 2010 *)
LinearRecurrence[{4,-6,4,-1},{0,3,18,57},40] (* Harvey P. Dale, Aug 23 2015 *)
With[{nn=40},Total/@TakeList[Range[nn+nn^2],2Range[0,nn]]] (* Requires Mathematica version 11 or later *) (* Harvey P. Dale, Mar 10 2018 *)

a(n) = { 2*n^3 + n } \\ Harry J. Smith, Jul 21 2009

a(n) = 2*n^3 + n.
a(n) = A000217(A002378(n)) - A000217(A002378(n-1)).
a(n) = 3 * A005900(n).
a(n) = A001477(n) * A058331(n).
a(n) = A000578(n) + A034262(n).
G.f.: 3*x*(1+x)^2/(x-1)^4.
a(n) = A110450(n) - A110450(n-1). - J.S. Seneschal, Jul 01 2025

A078475 Determinant of rank n matrix of 1..n^2 filled successively back and forth along antidiagonals.

Original entry on oeis.org

1, -2, 15, -594, -5187, 23244, 122475, -279292, -1157143, 1850930, 6642839, -8529278, -27810555, 30741424, 93575187, -92784984, -268191855, 244875462, 679807583, -581798410, -1563707379, 1270245108, 3324627195, -2587197204, -6623079687, 4972012474, 12491212135

Offset: 1

Kit Vongmahadlek (kit119(AT)yahoo.com), Jan 03 2003

The matrix is formed by writing numbers 1 .. n^2 in zig-zag pattern as shown in examples below. Every other antidiagonal reads backwards from A069480.
Whereas each antidiagonal of A069480 begins with one more than a triangular number and ends with the next triangular number, here every other antidiagonal begins with one more than a triangular number and the next antidiagonal begins with a triangular number.
The trace of the matrix is the sequence A006003 (proved). - Stefano Spezia, Aug 07 2018
The matrix is defined by A[i,j] = (2 - i - j)*((i + j - 1) mod 2)+(j^2 + (2*i - 1)*j + i^2 - i)/2 + (j - 1)*(1 - 2*((i + j) mod 2)) if i + j <= n + 1 and A[i,j] = n^2 - ((4*n^2 + (- 4*j - 4*i + 6)*n + j^2 + (2*i - 3)*j + i^2 - 3*i + 2)/2 + (i + j - 2*n)*((2*n - i - j + 1) mod 2)) + 1 - (n - j)*(1 - 2*((i + j) mod 2)) if i + j > n + 1 (proved). - Stefano Spezia, Aug 11 2018

			n=2, det=-2: {1 2 / 3 4 }
n=3, det=15: {1 2 6 / 3 5 7 / 4 8 9 }
n=4, det=-594: { 1 2 6 7 / 3 5 8 13 / 4 9 12 14 / 10 11 15 16 }
n=5, det=-5187: { 1 2 6 7 15 / 3 5 8 14 16 / 4 9 13 17 22 / 10 12 18 21 23 / 11 19 20 24 25 }

Vaclav Kotesovec, Table of n, a(n) for n = 1..1450
Index entries for linear recurrences with constant coefficients, signature (0,-9,0,-36,0,-84,0,-126,0,-126,0,-84,0,-36,0,-9,0,-1).

Cf. A023999, A069480, A079340.

A078475 := function(k)
local i, j, n;
for n in [1 .. k] do
   A:=NullMat(n,n);
   for i in [1 .. n] do
      for j in [1 .. n] do
         if i+j<=n+1 then
            A[i][j] := (2-i-j)*RemInt(i+j-1,2)+(j^2+(2*i-1)*j+i^2-i)/2+(j-1)*(1-2*RemInt(i+j,2));;
         else
            A[i][j] := n^2-((4*n^2+(-4*j-4*i+6)*n+j^2+(2*i-3)*j+i^2-3*i+2)/2+(i+j-2*n)*RemInt(2*n-i-j+1,2))+1-(n-j)*(1-2*RemInt(i+j,2));
         fi;
      od;
   od;
   Print(n," ",Determinant(A),"\n");
od;
end;
A078475(27); # Stefano Spezia, Aug 12 2018

a[i_, j_, n_] := If[i+j<=n+1, (2-i-j)*Mod[i+j-1,2]+(j^2+(2*i-1)*j+i^2-i)/2+(j-1)*(1-2*Mod[i+j,2]),n^2-((4*n^2+(-4*j-4*i+6)*n+j^2+(2*i-3)*j+i^2-3*i+2)/2+(i+j-2*n)*Mod[2*n-i-j+1,2])+1-(n-j)*(1-2*Mod[i+j,2])]; f[n_] := Det[ Table[a[i, j, n], {i, n}, {j, n}]]; Array[f, 27] (* Stefano Spezia, Aug 11 2018 *)

A(i,j,n) = if (i + j <= n + 1, (2 - i - j)*((i + j - 1) % 2)+(j^2 + (2*i - 1)*j + i^2 - i)/2 + (j - 1)*(1 - 2*((i + j) % 2)), n^2 - ((4*n^2 + (- 4*j - 4*i + 6)*n + j^2 + (2*i - 3)*j + i^2 - 3*i + 2)/2 + (i + j - 2*n)*((2*n - i - j + 1) % 2)) + 1 - (n - j)*(1 - 2*((i + j) % 2)));
a(n) = matdet(matrix(n, n, i, j, A(i, j, n))); \\ Michel Marcus, Aug 11 2018
(MATLAB, FreeMat and Octave)
for(n=1:27)
   A=zeros(n,n);
   for(i=1:n)
      for(j=1:n)
         if(i+j<=n+1)
            A(i,j)=(2-i-j)*mod(i+j-1,2)+(j^2+(2*i-1)*j+i^2-i)/2+(j-1)*(1-2*mod(i+j,2));
         else
            A(i,j)=n^2-((4*n^2+(-4*j-4*i+6)*n+j^2+(2*i-3)*j+i^2-3*i+2)/2+(i+j-2*n)*mod(2*n-i-j+1,2))+1-(n-j)*(1-2*mod(i+j,2));
         end
      end
   end
   fprintf('%d %0.f\n',n,det(A));
end # Stefano Spezia, Aug 12 2018

From Vaclav Kotesovec, Jan 08 2019: (Start)
Recurrence: (5*n^16 - 176*n^15 + 2888*n^14 - 29332*n^13 + 206454*n^12 - 1068276*n^11 + 4205934*n^10 - 12861022*n^9 + 30891328*n^8 - 58524140*n^7 + 87229074*n^6 - 101275380*n^5 + 89823673*n^4 - 58824210*n^3 + 26795412*n^2 - 7559784*n + 985608)*a(n) = 8*(n^14 - 20*n^13 + 169*n^12 - 754*n^11 + 1630*n^10 + 564*n^9 - 15184*n^8 + 52244*n^7 - 109015*n^6 + 167071*n^5 - 202816*n^4 + 191592*n^3 - 125145*n^2 + 45333*n - 5832)*a(n-1) - (5*n^16 - 96*n^15 + 848*n^14 - 4580*n^13 + 16966*n^12 - 45892*n^11 + 94310*n^10 - 151266*n^9 + 192520*n^8 - 195196*n^7 + 155666*n^6 - 94052*n^5 + 39329*n^4 - 6798*n^3 - 4572*n^2 + 5400*n - 1944)*a(n-2).
a(n) ~ ((-1)^n - 3) * (cos(Pi*n/2) + sin(Pi*n/2)) * n^8 / 72. (End)
a(n) = -9*a(n-2) - 36*a(n-4) - 84*a(n-6) - 126*a(n-8) - 126*a(n-10) - 84*a(n-12) - 36*a(n-14) - 9*a(n-16) - a(n-18) for n > 18. - Stefano Spezia, Apr 25 2021, simplified by Boštjan Gec, Sep 21 2023

Edited and extended by Robert G. Wilson v, May 08 2003

A081422 Triangle read by rows in which row n consists of the first n+1 n-gonal numbers.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 6, 10, 1, 4, 9, 16, 25, 1, 5, 12, 22, 35, 51, 1, 6, 15, 28, 45, 66, 91, 1, 7, 18, 34, 55, 81, 112, 148, 1, 8, 21, 40, 65, 96, 133, 176, 225, 1, 9, 24, 46, 75, 111, 154, 204, 261, 325, 1, 10, 27, 52, 85, 126, 175, 232, 297, 370, 451

Offset: 0

Paul Barry, Mar 21 2003

			The array starts
  1  1  3 10 ...
  1  2  6 16 ...
  1  3  9 22 ...
  1  4 12 28 ...
The triangle starts
  1;
  1,  1;
  1,  2,  3;
  1,  3,  6, 10;
  1,  4,  9, 16, 25;
  ...

T. D. Noe, Rows n = 0..100 of triangle, flattened
Eric Weisstein's World of Mathematics, Polygonal Number

Rows include A060354, A064808, A006000, A006003, A002411.
Diagonals include A001093, A053698, A069778, A000578, A002414, A081423, A081435, A081436, A081437, A081438, A081441.
Antidiagonals are composed of n-gonal numbers.

Flat(List([0..10], n-> List([1..n+1], k-> k*((n-2)*k-(n-4))/2 ))); # G. C. Greubel, Aug 14 2019

[[k*((n-2)*k-(n-4))/2: k in [1..n+1]]: n in [0..10]]; // G. C. Greubel, Oct 13 2018

Table[PolygonalNumber[n,i],{n,0,10},{i,n+1}]//Flatten (* Requires Mathematica version 10.4 or later *) (* Harvey P. Dale, Aug 27 2016 *)

tabl(nn) = {for (n=0, nn, for (k=1, n+1, print1(k*((n-2)*k-(n-4))/2, ", ");); print(););} \\ Michel Marcus, Jun 22 2015

[[k*((n-2)*k -(n-4))/2 for k in (1..n+1)] for n in (0..10)] # G. C. Greubel, Aug 14 2019

Array of coefficients of x in the expansions of T(k, x) = (1 + k*x -(k-2)*x^2)/(1-x)^4, k > -4.

T(n, k) = k*((n-2)*k -(n-4))/2 (see MathWorld link). - Michel Marcus, Jun 22 2015

A337897 Number of achiral colorings of the 8 triangular faces of a regular octahedron or the 8 vertices of a cube using n or fewer colors.

Original entry on oeis.org

1, 21, 201, 1076, 4025, 11901, 29841, 66256, 134001, 251725, 445401, 750036, 1211561, 1888901, 2856225, 4205376, 6048481, 8520741, 11783401, 16026900, 21474201, 28384301, 37055921, 47831376, 61100625, 77305501, 96944121

Offset: 1

Robert A. Russell, Sep 28 2020

nonn

An achiral coloring is identical to its reflection. The Schläfli symbols for the cube and regular octahedron are {4,3} and {3,4} respectively. They are mutually dual.
There are 24 elements in the automorphism group of the regular octahedron/cube that are not in the rotation group. They divide into five conjugacy classes. The first formula is obtained by averaging the cube vertex (octahedron face) cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
  Conjugacy Class     Count    Odd Cycle Indices
  Inversion              1     x_2^4
  Vertex rotation*       8     x_2^1x_6^1      Asterisk indicates that the
  Edge rotation*         6     x_1^4x_2^2      operation is followed by an
  Small face rotation*   3     x_4^2           inversion.
  Large face rotation*   6     x_2^4

Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).

Cf. A000543 (oriented), A128766 (unoriented), A337896 (chiral).
Other elements: A331351 (edges), A337898 (cube faces, octahedron vertices).
Other polyhedra: A006003 (tetrahedron), A337962  (dodecahedron faces, icosahedron vertices), A337960 (icosahedron faces, dodecahedron vertices).
Row 3 of A337894 (orthoplex faces, orthotope peaks) and A325015 (orthotope vertices, orthoplex facets).

Mathematica
```
Table[n^2(7+2n^2+3n^4)/12, {n,30}]
```

a(n) = n^2 * (7 + 2*n^2 + 3*n^4) / 12.
a(n) = 1*C(n,1) + 19*C(n,2) + 141*C(n,3) + 394*C(n,4) + 450*C(n,5) + 180*C(n,6), where the coefficient of C(n,k) is the number of achiral colorings using exactly k colors.
a(n) = 2*A128766(n) - A000543(n) = A000543(n) - 2*A337896(n) = A128766(n) - A337896(n).
G.f.: x * (1+x) * (1 + 13*x + 62*x^2 + 13*x^3 + x^4) / (1-x)^7.

A034958 Divide primes into groups with prime(n) elements and add together.

Original entry on oeis.org

5, 23, 101, 311, 931, 1895, 3875, 6349, 10643, 18335, 25873, 39593, 55607, 71301, 94559, 127315, 167495, 204063, 258283, 315087, 369749, 451635, 533015, 640097, 779283, 902789, 1013795, 1159073, 1295871, 1457935, 1786691, 2002645, 2272221

Offset: 1

Patrick De Geest, Oct 15 1998

nonn

			a(1) = 5 because the first 2 primes are 2 and 3 and 2 + 3 = 5.
a(2) = 23 because the next 3 primes are 5, 7, 11, and they add up to 23.
a(3) = 101 because the next 5 primes are 13, 17, 19, 23, 29 which add up to 101.
a(4) = 311 because the next 7 primes are 31, 37, 41, 43, 47, 53, 59 and they add up to 311.

Hieronymus Fischer, Table of n, a(n) for n = 1..1000

Cf. A006003, A027441, A034956.

Cf. A007504, A046992, A034959, A034960, A180302.

Join[{5},Total[Prime[Range[#[[1]]+1,#[[2]]]]]&/@Partition[ Accumulate[ Prime[ Range[40]]],2,1]] (* Harvey P. Dale, Oct 03 2013 *)
Module[{nn=33},Total/@TakeList[Prime[Range[Total[Prime[Range[nn]]]]], Prime[ Range[ nn]]]] (* Requires Mathematica version 11 or later *) (* Harvey P. Dale, Mar 16 2018 *)
s = 0; Total[Table[s = s + 1; Prime[s], {j, 33}, {n, Prime[j]}], {2}] (* Horst H. Manninger, Jan 17 2019 *)

s(n) = sum(k=1, n, prime(k)); \\ A007504
a(n) = s(s(n)) - s(s(n-1)); \\ Michel Marcus, Oct 12 2018

From Hieronymus Fischer, Sep 26 2012: (Start)
a(n) = Sum_{k=A007504(n-1)+1..A007504(n)} A000040(k), n > 1.
a(n) = A007504(A007504(n)) - A007504(A007504(n-1)), n > 1.
If we define A007504(0) := 0, then the formulas are also true for n = 1.
(End)

A074149 Sum of terms in each group in A074147.

Original entry on oeis.org

1, 6, 15, 36, 65, 114, 175, 264, 369, 510, 671, 876, 1105, 1386, 1695, 2064, 2465, 2934, 3439, 4020, 4641, 5346, 6095, 6936, 7825, 8814, 9855, 11004, 12209, 13530, 14911, 16416, 17985, 19686, 21455, 23364, 25345, 27474, 29679, 32040, 34481, 37086

Offset: 1

Amarnath Murthy, Aug 28 2002

The odd-indexed entries are the sums pertaining to the corresponding magic squares.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A000035, A074147, A074148, A061925, A006003, A061804.

LinearRecurrence[{2,1,-4,1,2,-1},{1,6,15,36,65,114},50] (* Harvey P. Dale, Jun 22 2016 *)

a(n)=n^3/2 + n*(3+(-1)^n)/4 \\ Charles R Greathouse IV, Jun 11 2015

def A074149(n): return (n*(n**2-(n&1))>>1)+n # Chai Wah Wu, Aug 30 2022

a(2n-1) = 4n^3 - 6n^2 + 4n - 1, a(2n) = 4n^3 + 2n. a(n) = (n^3 + n)/2 if n odd, n^3/2 + n if n even. a(n) = n^3/2 + n(3 + (-1)^n)/4. - Franklin T. Adams-Watters, Jul 17 2006
G.f.: x*(x^2+1)*(x^2+4*x+1) / ( (1+x)^2*(x-1)^4 ). - R. J. Mathar, Mar 07 2011
E.g.f.: x*((2 + 3*x + x^2)*cosh(x) + (3 + 3*x + x^2)*sinh(x))/2. - Stefano Spezia, May 07 2021
a(n) = n*(n^2-A000035(n))/2 + n. - Chai Wah Wu, Aug 30 2022

More terms from Franklin T. Adams-Watters, Jul 17 2006

A125860 Rectangular table where column k equals row sums of matrix power A097712^k, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 17, 12, 4, 1, 1, 86, 69, 22, 5, 1, 1, 698, 612, 178, 35, 6, 1, 1, 9551, 8853, 2251, 365, 51, 7, 1, 1, 226592, 217041, 46663, 5990, 651, 70, 8, 1, 1, 9471845, 9245253, 1640572, 161525, 13131, 1057, 92, 9, 1, 1, 705154187

Offset: 0

Paul D. Hanna, Dec 13 2006

Triangle A097712 satisfies: A097712(n,k) = A097712(n-1,k) + [A097712^2](n-1,k-1) for n > 0, k > 0, with A097712(n,0)=A097712(n,n)=1 for n >= 0. Column 1 equals A016121, which counts the sequences (a_1, a_2, ..., a_n) of length n with a_1 = 1 satisfying a_i <= a_{i+1} <= 2*a_i.

T(2, n) = (n+1)*A005408(n) - Sum_{i=0..n} A001477(i) = (n+1)*(2*n+1) - A000217(n) = (n+1)*(3*n+2)/2; T(3, n) = (n+1)*A001106(n+1) - Sum_{i=0..n} A001477(i) = (n+1)*((n+1)*(7*n+2)/2) - A000217(n) = (n+1)*(7*n^2 + 8*n + 2)/2. - Bruno Berselli, Apr 25 2010

			Recurrence is illustrated by:
  T(4,1) = T(3,1) + T(3,2) = 17 + 69 = 86;
  T(4,2) = T(3,2) + T(3,3) + T(3,4) = 69 + 178 + 365 = 612;
  T(4,3) = T(3,3) + T(3,4) + T(3,5) + T(3,6) = 178 + 365 + 651 + 1057 = 2251.
Rows of this table begin:
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...;
  1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,...;
  1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, 330, ...;
  1, 17, 69, 178, 365, 651, 1057, 1604, 2313, 3205, 4301, 5622, 7189,..;
  1, 86, 612, 2251, 5990, 13131, 25291, 44402, 72711, 112780, 167486,..;
  1, 698, 8853, 46663, 161525, 435801, 996583, 2025458, 3768273, ...;
  1, 9551, 217041, 1640572, 7387640, 24530016, 66593821, 156664796, ...;
  1, 226592, 9245253, 100152049, 586285040, 2394413286, 7713533212, ...;
  1, 9471845, 695682342, 10794383587, 82090572095, 412135908606, ...;
  1, 705154187, 93580638024, 2079805452133, 20540291522675, ...;
  1, 94285792211, 22713677612832, 723492192295786, 9278896006526795,...;
  1, 22807963405043, 10025101876435413, 458149292979837523, ...;
  ...
where column k equals the row sums of matrix power A097712^k for k >= 0.
Triangle A097712 begins:
  1;
  1,      1;
  1,      3,       1;
  1,      8,       7,       1;
  1,     25,      44,      15,       1;
  1,    111,     346,     208,      31,      1;
  1,    809,    4045,    3720,     912,     63,     1;
  1,  10360,   77351,   99776,   35136,   3840,   127,   1;
  1, 236952, 2535715, 4341249, 2032888, 308976, 15808, 255; ...
where A097712(n,k) = A097712(n-1,k) + [A097712^2](n-1,k-1);
e.g., A097712(5,2) = A097712(4,2) + [A097712^2](4,1) = 44 + 302 = 346.
Matrix square A097712^2 begins:
     1;
     2,     1;
     5,     6,     1;
    17,    37,    14,     1;
    86,   302,   193,    30,    1;
   698,  3699,  3512,   881,   62,   1;
  9551, 73306, 96056, 34224, 3777, 126, 1; ...
Matrix cube A097712^3 begins:
       1;
       3,      1;
      12,      9,      1;
      69,     87,     21,      1;
     612,   1146,    447,     45,    1;
    8853,  22944,  12753,   2019,   93,   1;
  217041, 744486, 549453, 120807, 8595, 189, 1; ...

Olivier Danvy, Summa Summarum: Moessner's Theorem without Dynamic Programming, arXiv:2412.03127 [cs.DM], 2024. See p. 16.

Cf. A097712; columns: A016121, A125862, A125863, A125864, A125865; A125861 (diagonal), A125859 (antidiagonal sums). Variants: A125790, A125800.

Cf. for recursive method [Ar(m) is the m-th term of a sequence in the OEIS] a(n) = n*Ar(n) - A000217(n-1) or a(n) = (n+1)*Ar(n+1) - A000217(n) and similar: A081436, A005920, A005945, A006003. - Bruno Berselli, Apr 25 2010

T[n_, k_] := T[n, k] = If[Or[n == 0, k == 0], 1, Sum[T[n - 1, j + k], {j, 0, k}]];
Table[T[#, k] &[n - k + 1], {n, 0, 9}, {k, 0, n + 1}] (* Michael De Vlieger, Dec 10 2024, after PARI *)

T(n,k)=if(n==0 || k==0,1,sum(j=0,k,T(n-1,j+k)))

T(n,k) = Sum_{j=0..k} T(n-1, j+k) for n > 0, with T(0,n)=T(n,0)=1 for n >= 0.

A226449 a(n) = n(5n^2-8*n+5)/2.

Original entry on oeis.org

0, 1, 9, 39, 106, 225, 411, 679, 1044, 1521, 2125, 2871, 3774, 4849, 6111, 7575, 9256, 11169, 13329, 15751, 18450, 21441, 24739, 28359, 32316, 36625, 41301, 46359, 51814, 57681, 63975, 70711, 77904, 85569, 93721, 102375, 111546, 121249, 131499, 142311, 153700

Offset: 0

Bruno Berselli, Jun 07 2013

Sequences of the type b(m)+m*b(m-1), where b is a polygonal number:
A006003(n) = A000217(n) + n*A000217(n-1)      (b = triangular numbers);
A069778(n) = A000290(n+1) + (n+1)*A000290(n)  (b = square numbers);
A143690(n) = A000326(n+1) + (n+1)*A000326(n)  (b = pentagonal numbers);
A212133(n) = A000384(n) + n*A000384(n-1)      (b = hexagonal numbers);
a(n)       = A000566(n) + n*A000566(n-1)      (b = heptagonal numbers);
A226450(n) = A000567(n) + n*A000567(n-1)      (b = octagonal numbers);
A226451(n) = A001106(n) + n*A001106(n-1)      (b = nonagonal numbers);
A204674(n) = A001107(n+1) + (n+1)*A001107(n)  (b = decagonal numbers).

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. (see the comment) A000566, A006003, A069778, A143690, A204674, A212133, A226450, A226451.

Magma
```
[n*(5*n^2-8*n+5)/2: n in [0..40]];
```

I:=[0,1,9,39]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..45]]; // Vincenzo Librandi, Aug 18 2013

Table[n (5 n^2 - 8 n + 5)/2, {n, 0, 40}]
CoefficientList[Series[x (1 + 5 x + 9 x^2)/(1 - x)^4, {x, 0, 45}], x] (* Vincenzo Librandi, Aug 18 2013 *)
LinearRecurrence[{4,-6,4,-1},{0,1,9,39},50] (* Harvey P. Dale, May 19 2017 *)

a(n)=n*(5*n^2-8*n+5)/2 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: x*(1+5*x+9*x^2)/(1-x)^4.

a(n) - a(-n) = A008531(n) for n>0.

A229183 a(n) = n*(n^2 + 3)/2.

Original entry on oeis.org

0, 2, 7, 18, 38, 70, 117, 182, 268, 378, 515, 682, 882, 1118, 1393, 1710, 2072, 2482, 2943, 3458, 4030, 4662, 5357, 6118, 6948, 7850, 8827, 9882, 11018, 12238, 13545, 14942, 16432, 18018, 19703, 21490, 23382, 25382, 27493, 29718, 32060, 34522, 37107, 39818

Offset: 0

Derek Orr, Sep 15 2013

Numbers a(n) such that (a(n) + B)^(1/3) + (a(n) - B)^(1/3) = n, where B = sqrt(a(n)^2 + 1).
4*a(n) is the sum of two cubes. In fact: 2*n*(n^2 + 3) = (n-1)^3 + (n+1)^3. - Bruno Berselli, Apr 11 2016
From Olivier Gérard, Aug 07 2016 (Start)
Row sums of n consecutive integers, starting at 2, seen as a triangle:
.
2   | 2
7   | 3  4
18  | 5  6  7
38  | 8  9  10 11
70  | 12 13 14 15 16
117 | 17 18 19 20 21 22
(End)
Take a long horizontal strip of paper 1 unit high and mark two points on the top edge, n/2 and n units from the top left corner. Then fold over the top left corner so that the fold line passes through the bottom left corner and the point n units along the top edge. Then draw a line from the bottom left corner of the strip through the new position of the n/2 point. The point at which that shallow diagonal line meets the top edge of the strip of paper will be a(n) from the top left corner. - Elliott Line, Jul 09 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A006003 (row sums of integers, starting with 1).

Cf. A027480 (row sums of integers, starting with 0).

[n*(n^2 + 3) div 2: n in [0..50]]; // Vincenzo Librandi, Sep 23 2013

A229183 := proc(n)
    n*(n^2+3) /2;
end proc:
[seq(A229183(n),n=0..30)] ; # R. J. Mathar, Aug 16 2019

Table[(n^3 + 3n)/2, {n, 0, 100}] (* T. D. Noe, Sep 16 2013 *)
CoefficientList[Series[x (2 - x + 2 x^2)/(x - 1)^4, {x, 0, 50}], x] (* Vincenzo Librandi, Sep 23 2013 *)

vector(100,n,((n-1)^3+3*n-3)/2) \\ Derek Orr, Mar 12 2015

{print((n**3+3*n)/2,end=', ') for n in range(0,100)} # Simplified by Derek Orr, Mar 12 2015

G.f.: x*(2 - x + 2*x^2) / (x-1)^4. - R. J. Mathar, Sep 22 2013
a(n)^2 + 1 = (n^2 + 1)^2 * ((n/2)^2 + 1). - Joerg Arndt, Jan 22 2015
E.g.f.: exp(x)*x*(4 + 3*x + x^2)/2. - Stefano Spezia, Jul 04 2021

A337898 Number of achiral colorings of the 6 square faces of a cube or the 6 vertices of a regular octahedron using n or fewer colors.

Original entry on oeis.org

1, 10, 55, 200, 560, 1316, 2730, 5160, 9075, 15070, 23881, 36400, 53690, 77000, 107780, 147696, 198645, 262770, 342475, 440440, 559636, 703340, 875150, 1079000, 1319175, 1600326, 1927485, 2306080, 2741950, 3241360

Offset: 1

Robert A. Russell, Sep 28 2020

An achiral coloring is identical to its reflection. The Schläfli symbols for the cube and regular octahedron are {4,3} and {3,4} respectively. They are mutually dual.
There are 24 elements in the automorphism group of the regular octahedron/cube that are not in the rotation group. They divide into five conjugacy classes. The first formula is obtained by averaging the cube face (octahedron vertex) cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
  Conjugacy Class     Count    Odd Cycle Indices
  Inversion              1     x_2^3
  Vertex rotation*       8     x_6^1           Asterisk indicates that the
  Edge rotation*         6     x_1^2x_2^2      operation is followed by an
  Small face rotation*   6     x_2^1x_4^1      inversion.
  Large face rotation*   3     x_1^4x_2^1

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A047780 (oriented), A198833 (unoriented), A093566(n+1) (chiral).
Other elements: A331351 (edges), A337897 (cube vertices/octahedron faces).
Other polyhedra: A006003 (simplex), A337962 (dodecahedron faces, icosahedron vertices), A337960 (icosahedron faces, dodecahedron vertices).
Row 3 of A325007 (orthotope facets, orthoplex vertices) and A337890 (orthotope faces, orthoplex peaks).

Table[n(1+n)(2+n)(4-3n+3n^2)/24, {n, 35}]
LinearRecurrence[{6,-15,20,-15,6,-1},{1,10,55,200,560,1316},40] (* Harvey P. Dale, Feb 15 2022 *)

a(n)=n*(n+1)*(n+2)*(3*n^2-3*n+4)/24 \\ Charles R Greathouse IV, Oct 21 2022

a(n) = n * (n+1) * (n+2) * (3*n^2 - 3*n + 4) / 24.
a(n) = 1*C(n,1) + 8*C(n,2) + 28*C(n,3) + 36*C(n,4) + 15*C(n,5), where the coefficient of C(n,k) is the number of achiral colorings using exactly k colors.
a(n) = 2*A198833(n) - A047780(n) = A047780(n) - 2*A093566(n+1) = A198833(n) - A093566(n+1).
G.f.: x * (x + 4*x^2 + 10*x^3) / (1-x)^6.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6). - Wesley Ivan Hurt, Sep 30 2020

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