A027441 -id:A027441 - cp's OEIS frontend

A006003 a(n) = n*(n^2 + 1)/2.

0, 1, 5, 15, 34, 65, 111, 175, 260, 369, 505, 671, 870, 1105, 1379, 1695, 2056, 2465, 2925, 3439, 4010, 4641, 5335, 6095, 6924, 7825, 8801, 9855, 10990, 12209, 13515, 14911, 16400, 17985, 19669, 21455, 23346, 25345, 27455, 29679, 32020, 34481, 37065, 39775

Offset: 0

N. J. A. Sloane, Simon Plouffe

Write the natural numbers in groups: 1; 2,3; 4,5,6; 7,8,9,10; ... and add the groups. In other words, "sum of the next n natural numbers". - Felice Russo
Number of rhombi in an n X n rhombus, if 'crossformed' rhombi are allowed. - Matti De Craene (Matti.DeCraene(AT)rug.ac.be), May 14 2000
Also the sum of the integers between T(n-1)+1 and T(n), the n-th triangular number (A000217). Sum of n-th row of A000027 regarded as a triangular array.
Unlike the cubes which have a similar definition, it is possible for 2 terms of this sequence to sum to a third. E.g., a(36) + a(37) = 23346 + 25345 = 48691 = a(46). Might be called 2nd-order triangular numbers, thus defining 3rd-order triangular numbers (A027441) as n(n^3+1)/2, etc. - Jon Perry, Jan 14 2004
Also as a(n)=(1/6)*(3*n^3+3*n), n > 0: structured trigonal diamond numbers (vertex structure 4) (cf. A000330 = alternate vertex; A000447 = structured diamonds; A100145 for more on structured numbers). - James A. Record (james.record(AT)gmail.com), Nov 07 2004
The sequence M(n) of magic constants for n X n magic squares (numbered 1 through n^2) from n=3 begins M(n) = 15, 34, 65, 111, 175, 260, ... - Lekraj Beedassy, Apr 16 2005 [comment corrected by Colin Hall, Sep 11 2009]
The sequence Q(n) of magic constants for the n-queens problem in chess begins 0, 0, 0, 0, 34, 65, 111, 175, 260, ... - Paul Muljadi, Aug 23 2005
Alternate terms of A057587. - Jeremy Gardiner, Apr 10 2005
Also partial differences of A063488(n) = (2*n-1)*(n^2-n+2)/2. a(n) = A063488(n) - A063488(n-1) for n>1. - Alexander Adamchuk, Jun 03 2006
In an n X n grid of numbers from 1 to n^2, select -- in any manner -- one number from each row and column. Sum the selected numbers. The sum is independent of the choices and is equal to the n-th term of this sequence. - F.-J. Papp (fjpapp(AT)umich.edu), Jun 06 2006
Nonnegative X values of solutions to the equation (X-Y)^3 - (X+Y) = 0. To find Y values: b(n) = (n^3-n)/2. - Mohamed Bouhamida, May 16 2006
For the equation: m*(X-Y)^k - (X+Y) = 0 with X >= Y, k >= 2 and m is an odd number the X values are given by the sequence defined by a(n) = (m*n^k+n)/2. The Y values are given by the sequence defined by b(n) = (m*n^k-n)/2. - Mohamed Bouhamida, May 16 2006
If X is an n-set and Y a fixed 3-subset of X then a(n-3) is equal to the number of 4-subsets of X intersecting Y. - Milan Janjic, Jul 30 2007
(m*(2n)^k+n, m*(2n)^k-n) solves the Diophantine equation: 2m*(X-Y)^k - (X+Y) = 0 with X >= Y, k >= 2 where m is a positive integer. - Mohamed Bouhamida, Oct 02 2007
Also c^(1/2) in a^(1/2) + b^(1/2) = c^(1/2) such that a^2 + b = c. - Cino Hilliard, Feb 09 2008
a(n) = n*A000217(n) - Sum_{i=0..n-1} A001477(i). - Bruno Berselli, Apr 25 2010
a(n) is the number of triples (w,x,y) having all terms in {0,...,n} such that at least one of these inequalities fails: x+y < w, y+w < x, w+x < y. - Clark Kimberling, Jun 14 2012
Sum of n-th row of the triangle in A209297. - Reinhard Zumkeller, Jan 19 2013
The sequence starting with "1" is the third partial sum of (1, 2, 3, 3, 3, ...). - Gary W. Adamson, Sep 11 2015
a(n) is the largest eigenvalue of the matrix returned by the MATLAB command magic(n) for n > 0. - Altug Alkan, Nov 10 2015
a(n) is the number of triples (x,y,z) having all terms in {1,...,n} such that all these triangle inequalities are satisfied: x+y > z, y+z > x, z+x > y. - Heinz Dabrock, Jun 03 2016
Shares its digital root with the stella octangula numbers (A007588). See A267017. - Peter M. Chema, Aug 28 2016
Can be proved to be the number of nonnegative solutions of a system of three linear Diophantine equations for n >= 0 even: 2*a_{11} + a_{12} + a_{13} = n, 2*a_{22} + a_{12} + a_{23} = n and 2*a_{33} + a_{13} + a_{23} = n. The number of solutions is f(n) = (1/16)*(n+2)*(n^2 + 4n + 8) and a(n) = n*(n^2 + 1)/2 is obtained by remapping n -> 2*n-2. - Kamil Bradler, Oct 11 2016
For n > 0, a(n) coincides with the trace of the matrix formed by writing the numbers 1...n^2 back and forth along the antidiagonals (proved, see A078475 for the examples of matrix). - Stefano Spezia, Aug 07 2018
The trace of an n X n square matrix where the elements are entered on the ascending antidiagonals. The determinant is A069480. - Robert G. Wilson v, Aug 07 2018
Bisections are A317297 and A005917. - Omar E. Pol, Sep 01 2018
Number of achiral colorings of the vertices (or faces) of a regular tetrahedron with n available colors. An achiral coloring is identical to its reflection. - Robert A. Russell, Jan 22 2020
a(n) is the n-th centered triangular pyramidal number. - Lechoslaw Ratajczak, Nov 02 2021
a(n) is the number of words of length n defined on 4 letters {b,c,d,e} that contain one or no b's, one c or two d's, and any number of e's. For example, a(3) = 15 since the words are (number of permutations in parentheses): bce (6), bdd (3), cee (3), and dde (3). - Enrique Navarrete, Jun 21 2025

			G.f. = x + 5*x^2 + 15*x^3 + 34*x^4 + 65*x^5 + 111*x^6 + 175*x^7 + 260*x^8 + ...
For a(2)=5, the five tetrahedra have faces AAAA, AAAB, AABB, ABBB, and BBBB with colors A and B. - _Robert A. Russell_, Jan 31 2020

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 15, p. 5, Ellipses, Paris 2008.
F.-J. Papp, Colloquium Talk, Department of Mathematics, University of Michigan-Dearborn, March 6, 2005.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
J. D. Bell, A translation of Leonhard Euler's "De Quadratis Magicis", E795, arXiv:math/0408230 [math.CO], 2004-2005.
James Grime and Brady Haran, Magic Hexagon, Numberphile video (2014).
Milan Janjic, Two Enumerative Functions.
Milan Janjic and Boris Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - _N. J. A. Sloane_, Feb 13 2013
Milan Janjic and Boris Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, Journal of Integer Sequences, 17 (2014), Article 14.3.5. - _Felix Fröhlich_, Oct 11 2016
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (11).
Ashish Kumar Pandey and Brajesh Kumar Sharma, A Note On Magic Squares And Magic Constants, Appl. Math. E-Notes (2023) Vol. 23, Art. No. 53, 577-582. See p. 577.
A. J. Turner and J. F. Miller, Recurrent Cartesian Genetic Programming Applied to Famous Mathematical Sequences, preprint, Proceedings of the Companion Publication of the 2015 Annual Conference on Genetic and Evolutionary Computation.
Eric Weisstein's World of Mathematics, Magic Constant.
Wikipedia, Floyd's triangle. - _Paul Muljadi_, Jan 25 2010
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
Index entries for sequences related to magic squares.
Index to sequences related to polygonal numbers.

Cf. A000330, A000537, A066886, A057587, A027480, A002817 (partial sums).
Cf. A000578 (cubes).
Cf. A007742, A005449, A005448, A118465, A226449, A034262, A080992, A267017.
(1/12)*t*(n^3-n)+n for t = 2, 4, 6, ... gives A004006, A006527, this sequence, A005900, A004068, A000578, A004126, A000447, A004188, A004466, A004467, A007588, A062025, A063521, A063522, A063523.
Antidiagonal sums of array in A000027. Row sums of the triangular view of A000027.
Cf. A063488 (sum of two consecutive terms), A005917 (bisection), A317297 (bisection).
Cf. A105374 / 8.
Cf. A000292, A011863, A001620, A248177.
Tetrahedron colorings: A006008 (oriented), A000332(n+3) (unoriented), A000332 (chiral), A037270 (edges).
Other polyhedron colorings: A337898 (cube faces, octahedron vertices), A337897 (octahedron faces, cube vertices), A337962 (dodecahedron faces, icosahedron vertices), A337960 (icosahedron faces, dodecahedron vertices).
Row 3 of A325001 (simplex vertices and facets) and A337886 (simplex faces and peaks).

a_n:=List([0..nmax], n->n*(n^2 + 1)/2); # Stefano Spezia, Aug 12 2018

a006003 n = n * (n ^ 2 + 1) `div` 2
a006003_list = scanl (+) 0 a005448_list
-- Reinhard Zumkeller, Jun 20 2013

% Also works with FreeMat.
for(n=0:nmax); tm=n*(n^2 + 1)/2; fprintf('%d\t%0.f\n', n, tm); end
% Stefano Spezia, Aug 12 2018

[n*(n^2 + 1)/2 : n in [0..50]]; // Wesley Ivan Hurt, Sep 11 2015

[Binomial(n,3)+Binomial(n-1,3)+Binomial(n-2,3): n in [2..60]]; // Vincenzo Librandi, Sep 12 2015

Table[ n(n^2 + 1)/2, {n, 0, 45}]
LinearRecurrence[{4,-6,4,-1}, {0,1,5,15},50] (* Harvey P. Dale, May 16 2012 *)
CoefficientList[Series[x (1 + x + x^2)/(x - 1)^4, {x, 0, 45}], x] (* Vincenzo Librandi, Sep 12 2015 *)
With[{n=50},Total/@TakeList[Range[(n(n^2+1))/2],Range[0,n]]] (* Requires Mathematica version 11 or later *) (* Harvey P. Dale, Nov 28 2017 *)

a(n):=n*(n^2 + 1)/2$ makelist(a(n), n, 0, nmax); /* Stefano Spezia, Aug 12 2018 */

{a(n) = n * (n^2 + 1) / 2}; /* Michael Somos, Dec 24 2011 */

concat(0, Vec(x*(1+x+x^2)/(x-1)^4 + O(x^20))) \\ Felix Fröhlich, Oct 11 2016

def A006003(n): return n*(n**2+1)>>1 # Chai Wah Wu, Mar 25 2024

a(n) = binomial(n+2, 3) + binomial(n+1, 3) + binomial(n, 3). [corrected by Michel Marcus, Jan 22 2020]
G.f.: x*(1+x+x^2)/(x-1)^4. - Floor van Lamoen, Feb 11 2002
Partial sums of A005448. - Jonathan Vos Post, Mar 16 2006
Binomial transform of [1, 4, 6, 3, 0, 0, 0, ...] = (1, 5, 15, 34, 65, ...). - Gary W. Adamson, Aug 10 2007
a(n) = -a(-n) for all n in Z. - Michael Somos, Dec 24 2011
a(n) = Sum_{k = 1..n} A(k-1, k-1-n) where A(i, j) = i^2 + i*j + j^2 + i + j + 1. - Michael Somos, Jan 02 2012
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), with a(0)=0, a(1)=1, a(2)=5, a(3)=15. - Harvey P. Dale, May 16 2012
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 3. - Ant King, Jun 13 2012
a(n) = A000217(n) + n*A000217(n-1). - Bruno Berselli, Jun 07 2013
a(n) = A057145(n+3,n). - Luciano Ancora, Apr 10 2015
E.g.f.: (1/2)*(2*x + 3*x^2 + x^3)*exp(x). - G. C. Greubel, Dec 18 2015; corrected by Ilya Gutkovskiy, Oct 12 2016
a(n) = T(n) + T(n-1) + T(n-2), where T means the tetrahedral numbers, A000292. - Heinz Dabrock, Jun 03 2016
From Ilya Gutkovskiy, Oct 11 2016: (Start)
Convolution of A001477 and A008486.
Convolution of A000217 and A158799.
Sum_{n>=1} 1/a(n) = H(-i) + H(i) = 1.343731971048019675756781..., where H(k) is the harmonic number, i is the imaginary unit. (End)
a(n) = A000578(n) - A135503(n). - Miquel Cerda, Dec 25 2016
Euler transform of length 3 sequence [5, 0, -1]. - Michael Somos, Dec 25 2016
a(n) = A037270(n)/n for n > 0. - Kritsada Moomuang, Dec 15 2018
a(n) = 3*A000292(n-1) + n. - Bruce J. Nicholson, Nov 23 2019
a(n) = A011863(n) - A011863(n-2). - Bruce J. Nicholson, Dec 22 2019
From Robert A. Russell, Jan 22 2020: (Start)
a(n) = C(n,1) + 3*C(n,2) + 3*C(n,3), where the coefficient of C(n,k) is the number of tetrahedron colorings using exactly k colors.
a(n) = C(n+3,4) - C(n,4).
a(n) = 2*A000332(n+3) - A006008(n) = A006008(n) - 2*A000332(n) = A000332(n+3) - A000332(n).
a(n) = A325001(3,n). (End)
From Amiram Eldar, Aug 21 2023: (Start)
Sum_{n>=1} 1/a(n) = 2 * (A248177 + A001620).
Product_{n>=2} (1 - 1/a(n)) = cosh(sqrt(7)*Pi/2)*cosech(Pi)/4.
Product_{n>=1} (1 + 1/a(n)) = cosh(sqrt(7)*Pi/2)*cosech(Pi). (End)

Better description from Albert Rich (Albert_Rich(AT)msn.com), Mar 1997

A068239 1/2 the number of colorings of a 3 X 3 square array with n colors.

Original entry on oeis.org

1, 123, 4806, 71410, 583455, 3232341, 13675228, 47502036, 141991245, 377162335, 910842306, 2033854758, 4253012491, 8411348505, 15856955640, 28673921896, 49991146713, 84387303171, 138412872190, 221253017370, 345558093111, 528471784093, 792890261076

Offset: 2

R. H. Hardin, Feb 24 2002

nonn

Alois P. Heinz, Table of n, a(n) for n = 2..1000

Cf. A068239-A068305, A000332, A002417, A027441, A182368, A182406.

a:= n-> (79+(-323+(594+(-648+(459+(-216+(66+(-12+n)*n)*n) *n)*n)*n)*n)*n) *n/2:
seq(a(n), n=2..30); # Alois P. Heinz, Apr 27 2012

From Alois P. Heinz, Apr 27 2012: (Start)
G.f.: x^2*(1199*x^7 +16567*x^6 +60099*x^5 +71075*x^4 +28765*x^3 +3621*x^2 +113*x+1) / (x-1)^10.
a(n) = (79*n -323*n^2 +594*n^3 -648*n^4 +459*n^5 -216*n^6 +66*n^7 -12*n^8 +n^9) / 2.
(End)

A068305 1/16 the number of colorings of an n X n octagonal array with 16 colors.

Original entry on oeis.org

1, 2730, 1097599230, 65013773510046270, 567344965666217922692546310, 729405912578031916581095228654013174510, 138156586653036397048665899068285784224754055444205830, 3855264256604335001270711977271948484283006596889356139648293174199350

Offset: 1

R. H. Hardin, Feb 24 2002

nonn

Cf. A068239-A068305, A000332, A002417, A027441, A212208, A212209.

a(5)-a(8) from Alois P. Heinz, May 04 2012

A091940 Given n colors, sequence gives number of ways to color the vertices of a square such that no edge has the same color on both of its vertices.

Original entry on oeis.org

0, 2, 18, 84, 260, 630, 1302, 2408, 4104, 6570, 10010, 14652, 20748, 28574, 38430, 50640, 65552, 83538, 104994, 130340, 160020, 194502, 234278, 279864, 331800, 390650, 457002, 531468, 614684, 707310, 810030, 923552, 1048608, 1185954, 1336370, 1500660

Offset: 1

Ryan Witko (witko(AT)nyu.edu), Mar 11 2004

Also equals the number of pairs of pairs ((a_1,a_2),(b_1,b_2)) that are disjoint (a_i != b_j) where all elements belong to {1,...,n}. See A212085. - Lewis Baxter, Mar 06 2023

			a(4) = 84 since there are 84 different ways to color the vertices of a square with 4 colors such that no two vertices that share an edge are the same color.
There are 4 possible colors for the first vertex and 3 for the second vertex. For the third vertex, divide into two cases: the third vertex can be the same color as the first vertex, and then the fourth vertex has 3 possible colors (4 * 3 * 1 * 3 = 36 colorings). Or the third vertex can be a different color from the first vertex, and then the fourth vertex has 2 possible colors (4 * 3 * 2 * 2 = 48 colorings). So there are a total of 36 + 48 = 84. - _Michael B. Porter_, Jul 24 2016

Alois P. Heinz, Table of n, a(n) for n = 1..1000
OEIS Wiki, Colorings of square grid graphs
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Column k=2 of A212085.
Cf. A027441, A182368, A182406.
Cf. A000217, A002061.

a := n -> (n-1)+(n-1)^4; for n to 35 do a(n) end do; # Rainer Rosenthal, Dec 03 2006

Table[2Binomial[n, 2] + 12Binomial[n, 3] + 24Binomial[n, 4], {n, 35}] (* Robert G. Wilson v, Mar 16 2004 *)
Table[(n-1)^4+(n-1),{n,1,60}] (* Vladimir Joseph Stephan Orlovsky, May 12 2011 *)

a(n) = 2*C(n,2) + 12*C(n,3) + 24*C(n,4) = n*(n-1)*(n^2-3*n+3).
a(n) = (n-1) + (n-1)^4. - Rainer Rosenthal, Dec 03 2006
G.f.: 2*x^2*(1+4*x+7*x^2)/(1-x)^5. a(n) = 2*A027441(n-1). - R. J. Mathar, Sep 09 2008
For n > 1, a(n) = floor(n^7/(n^3-1)). - Gary Detlefs, Feb 10 2010
a(n) = 2 * A000217(n-1) * A002061(n-1), n >= 1. - Daniel Forgues, Jul 14 2016
E.g.f.: exp(x)*x^2*(1 + x)^2. - Stefano Spezia, Oct 08 2022

More terms from Robert G. Wilson v, Mar 16 2004

A068248 1/6 the number of colorings of a 5 X 5 staggered hexagonal array with n colors.

Original entry on oeis.org

1, 673072, 24674450670, 47695073906240, 16222886703881375, 1842996310592836896, 98798502888215704812, 3068393794369671728640, 62960689505171989129005, 933100312771109288146000, 10639781342848431789710266, 97779035987698387480546752, 750090455960001686602653035

Offset: 3

R. H. Hardin, Feb 24 2002

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..1000
Alois P. Heinz, Formula for A068248

Cf. A068239-A068305, A000332, A002417, A027441, A212194, A212195.

a:= n-> (3006792824+ (-26845691044+ (115537440058+ (-319333174471+ (636781496832+ (-975359012827+ (1192518013138+ (-1193724499144+ (995462037353+ (-699932345254+ (418375639535+ (-213720396671+ (93568819963+ (-35133625647+ (11298632584+
(-3101089710+ (722137763+ (-141421592+ (23000726+ (-3051871+ (321994+ (-25992+ (1508+(-56+n)*n) *n)*n) *n)*n) *n)*n) *n)*n) *n)*n) *n)*n) *n) *n) *n) *n) *n) *n) *n) *n) *n) *n) *n/6:
seq (a(n), n=3..40);  # Alois P. Heinz, May 03 2012

Extended beyond a(10) by Alois P. Heinz, May 03 2012

A068293 a(1) = 1; thereafter a(n) = 6*(2^(n-1) - 1).

Original entry on oeis.org

1, 6, 18, 42, 90, 186, 378, 762, 1530, 3066, 6138, 12282, 24570, 49146, 98298, 196602, 393210, 786426, 1572858, 3145722, 6291450, 12582906, 25165818, 50331642, 100663290, 201326586, 402653178, 805306362, 1610612730, 3221225466, 6442450938, 12884901882

Offset: 1

R. H. Hardin, Feb 24 2002

1/4 the number of colorings of an n X n octagonal array with 4 colors.
Consider the planar net 3^6 (as in the top left figure in the uniform planar nets link). Then a(n) is the total number of ways that a spider starting at a point P can reach any point n steps away by using a path of length n. - N. J. A. Sloane, Feb 20 2016
From Gary W. Adamson, Jan 13 2009: (Start)
Equals inverse binomial transform of A091344: (1, 7, 31, 115, 391, ...).
Equals binomial transform of (1, 5, 7, 5, 7, 5, ...). (End)
For n > 1, number of ternary strings of length n with exactly 2 different digits. - Enrique Navarrete, Nov 20 2020

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Ana Rechtman, Février 2016, 3e défi, Images des Mathématiques, CNRS, 2016.
N. J. A. Sloane, The uniform planar nets and their A-numbers [Annotated scanned figure from Gruenbaum and Shephard (1977)]
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A068239-A068305, A000332, A002417, A027441.

Cf. A091344. - Gary W. Adamson, Jan 13 2009

[1] cat [6*(2^(n-1)-1): n in [2..40]]; // Vincenzo Librandi, Feb 20 2016

a=0; lst={1}; k=6; Do[a+=k; AppendTo[lst, a]; k+=k, {n, 0, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Dec 16 2008 *)
Transpose[NestList[{First[#]+1,6(2^First[#]-1)}&,{1,1},30]][[2]] (* or *) Join[{1},LinearRecurrence[{3,-2},{6,18},30]] (* Harvey P. Dale, Nov 27 2011 *)

a(n)=polcoeff(prod(i=1,2,(1+i*x))/(prod(i=1,2,(1-i*x))+x*O(x^n)),n)
for(n=0,50,print1(a(n),","))

G.f.: (1+x)*(1+2*x)/((1-x)*(1-2*x)). - Benoit Cloitre, Apr 13 2002
a(n) = 3*a(n-1) - 2*a(n-2); a(1)=1, a(2)=6, a(3)=18. - Harvey P. Dale, Nov 27 2011
E.g.f.: 1 - 6*exp(x)*(exp(x) - 1). - Stefano Spezia, May 18 2024

More terms from Benoit Cloitre, Apr 13 2002

Old definition (which is now a comment) replaced with explicit formula by N. J. A. Sloane, May 12 2010

A034956 Divide natural numbers in groups with prime(n) elements and add together.

Original entry on oeis.org

3, 12, 40, 98, 253, 455, 850, 1292, 2047, 3335, 4495, 6623, 8938, 11180, 14335, 18815, 24249, 28731, 35845, 42884, 49348, 59408, 69139, 81791, 98164, 112211, 124939, 141026, 155434, 173681, 210439, 233966, 263040, 286062, 328098, 355152, 393442, 434558, 472777

Offset: 1

Patrick De Geest, Oct 15 1998

nonn

Natural numbers starting from 1,2,3,4,...

			{1,2} #2 S=3;
{3,4,5} #3 S=12;
{6,7,8,9,10} #5 S=40;
{11,12,13,14,15,16,17} #7 S=98.

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

Cf. A006003, A027441, A034957.
Cf. A046992, A034958, A034959, A034960.
Cf. A000040, A000217, A007504.

s:= proc(n) s(n):= `if`(n<1, 0, s(n-1)+ithprime(n)) end:
a:= n-> (t-> t(s(n))-t(s(n-1)))(i-> i*(i+1)/2):
seq(a(n), n=1..40);  # Alois P. Heinz, Mar 22 2023

Module[{nn=50,pr},pr=Prime[Range[nn]];Total/@TakeList[Range[ Total[ pr]], pr]](* Requires Mathematica version 11 or later *) (* Harvey P. Dale, Oct 01 2017 *)

from itertools import islice
from sympy import nextprime
def A034956_gen(): # generator of terms
    a, p = 0, 2
    while True:
        yield p*((a<<1)+p+1)>>1
        a, p = a+p, nextprime(p)
A034956_list = list(islice(A034956_gen(),20)) # Chai Wah Wu, Mar 22 2023

From Hieronymus Fischer, Sep 27 2012: (Start)
a(n) = Sum_{k=A007504(n-1)+1..A007504(n)} k, n > 1.
a(n) = (A007504(n) - A007504(n-1))*(A007504(n) + A007504(n-1) + 1)/2, n > 1.
a(n) = (A000217(A007504(n)) - A000217(A007504(n-1))), n > 0.
If we define A007504(0) := 0, then the formulas above are also true for n=1.
a(n) = (A034960(n) + A000040(n))/2.
a(n) = A034957(n) + A000040(n). (End)

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)

A034960 Divide odd numbers into groups with prime(n) elements and add together.

Original entry on oeis.org

4, 21, 75, 189, 495, 897, 1683, 2565, 4071, 6641, 8959, 13209, 17835, 22317, 28623, 37577, 48439, 57401, 71623, 85697, 98623, 118737, 138195, 163493, 196231, 224321, 249775, 281945, 310759, 347249, 420751, 467801, 525943, 571985, 656047

Offset: 1

Patrick De Geest, Oct 15 1998

nonn

			{1,3} #2 S=4;
{5,7,9} #3 S=21;
{11,13,15,17,19} #5 S=75;
{21,23,25,27,29,31,33} #7 S=189.

Hieronymus Fischer, Table of n, a(n) for n = 1..10000

Cf. A006003, A027441, A034959.
Cf. A007504.
Cf. A000040, A034956, A034957, A034958, A046992, A061802.

S:= n-> sum(ithprime(k), k=1..n): seq(S(n+1)^2-S(n)^2, n=0..40); # Gary Detlefs, Dec 20 2011

Accumulate[Join[{2}, ListConvolve[{1, 1}, #]]]*# & [Prime[Range[50]]] (* Paolo Xausa, Jun 23 2025 *)

a0(n) = vecsum(primes(n))^2 - vecsum(primes(n-1))^2; \\ Michel Marcus, Jun 16 2024

from itertools import islice
from sympy import nextprime
def A034960_gen(): # generator of terms
    a, p = 0, 2
    while True:
        yield p*((a<<1)+p)
        a, p = a+p, nextprime(p)
A034960_list = list(islice(A034960_gen(),20)) # Chai Wah Wu, Mar 22 2023

From Hieronymus Fischer, Sep 26 2012: (Start)
a(n) = Sum_{k=A007504(n-1)+1..A007504(n)} (2*k-1).
a(n) = A007504(n)^2 - A007504(n-1)^2.
a(n) = 2*A034957(n) + A000040(n).
a(n) = 2*A034956(n) - A000040(n).
a(n) = A034959(n) + A000040(n). (End)
a(n) = A061802(n)*A000040(n). - Marco Zárate, May 12 2023

A021003 a(n) = (n/2)*(n^4 + 1).

Original entry on oeis.org

0, 1, 17, 123, 514, 1565, 3891, 8407, 16388, 29529, 50005, 80531, 124422, 185653, 268919, 379695, 524296, 709937, 944793, 1238059, 1600010, 2042061, 2576827, 3218183, 3981324, 4882825, 5940701, 7174467, 8605198, 10255589, 12150015, 14314591, 16777232, 19567713

Offset: 0

Arlin Anderson (starship1(AT)gmail.com)

Magic constant for n X n X n X n hypercube (magic hypercube may not exist).

Kelvin Voskuijl, Table of n, a(n) for n = 0..10000 (terms 0..595 from Vincenzo Librandi)
Eric Weisstein's World of Mathematics, Magic Constant.
Eric Weisstein's World of Mathematics, Magic Tesseract.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A000217, A006003, A027441.

[(n/2)*(n^4+1): n in [0..50]]; // Vincenzo Librandi, Apr 29 2011

Table[(n(n^4+1))/2,{n,0,40}] (* or *) LinearRecurrence[ {6,-15,20,-15,6,-1},{0,1,17,123,514,1565},40] (* Harvey P. Dale, Dec 18 2011 *)

{a(n) = (n^5 + n) / 2}; /* Michael Somos, Jul 11 2017 */

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), a(0)=0, a(1)=1, a(2)=17, a(3)=123, a(4)=514, a(5)=1565. - Harvey P. Dale, Dec 18 2011
a(n) = -a(-n) for all n in Z. - Michael Somos, Jul 11 2017
From Elmo R. Oliveira, Aug 31 2025: (Start)
G.f.: x*(1 + 11*x + 36*x^2 + 11*x^3 + x^4)/(x-1)^6.
E.g.f.: x*(2 + 15*x + 25*x^2 + 10*x^3 + x^4)*exp(x)/2. (End)

cp's OEIS Frontend

A006003 a(n) = n*(n^2 + 1)/2.

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A068239 1/2 the number of colorings of a 3 X 3 square array with n colors.

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A068305 1/16 the number of colorings of an n X n octagonal array with 16 colors.

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A091940 Given n colors, sequence gives number of ways to color the vertices of a square such that no edge has the same color on both of its vertices.

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A068248 1/6 the number of colorings of a 5 X 5 staggered hexagonal array with n colors.

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A068293 a(1) = 1; thereafter a(n) = 6*(2^(n-1) - 1).

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A034956 Divide natural numbers in groups with prime(n) elements and add together.

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