A016061 -id:A016061 - cp's OEIS frontend

A014105 Second hexagonal numbers: a(n) = n(2n + 1).

0, 3, 10, 21, 36, 55, 78, 105, 136, 171, 210, 253, 300, 351, 406, 465, 528, 595, 666, 741, 820, 903, 990, 1081, 1176, 1275, 1378, 1485, 1596, 1711, 1830, 1953, 2080, 2211, 2346, 2485, 2628, 2775, 2926, 3081, 3240, 3403, 3570, 3741, 3916, 4095, 4278

Offset: 0

N. J. A. Sloane, Jun 14 1998

Note that when starting from a(n)^2, equality holds between series of first n+1 and next n consecutive squares: a(n)^2 + (a(n) + 1)^2 + ... + (a(n) + n)^2 = (a(n) + n + 1)^2 + (a(n) + n + 2)^2 + ... + (a(n) + 2*n)^2; e.g., 10^2 + 11^2 + 12^2 = 13^2 + 14^2. - Henry Bottomley, Jan 22 2001; with typos fixed by Zak Seidov, Sep 10 2015
a(n) = sum of second set of n consecutive even numbers - sum of the first set of n consecutive odd numbers: a(1) = 4-1, a(3) = (8+10+12) - (1+3+5) = 21. - Amarnath Murthy, Nov 07 2002
Partial sums of odd numbers 3 mod 4, that is, 3, 3+7, 3+7+11, ... See A001107. - Jon Perry, Dec 18 2004
If Y is a fixed 3-subset of a (2n+1)-set X then a(n) is the number of (2n-1)-subsets of X intersecting Y. - Milan Janjic, Oct 28 2007
More generally (see the first comment), for n > 0, let b(n,k) = a(n) + k*(4*n + 1). Then b(n,k)^2 + (b(n,k) + 1)^2 + ... + (b(n,k) + n)^2 = (b(n,k) + n + 1 + 2*k)^2 + ... + (b(n,k) + 2*n + 2*k)^2 + k^2; e.g., if n = 3 and k = 2, then b(n,k) = 47 and 47^2 + ... + 50^2 = 55^2 + ... + 57^2 + 2^2. - Charlie Marion, Jan 01 2011
Sequence found by reading the line from 0, in the direction 0, 10, ..., and the line from 3, in the direction 3, 21, ..., in the square spiral whose vertices are the triangular numbers A000217. - Omar E. Pol, Nov 09 2011
a(n) is the number of positions of a domino in a pyramidal board with base 2n+1. - César Eliud Lozada, Sep 26 2012
Differences of row sums of two consecutive rows of triangle A120070, i.e., first differences of A016061. - J. M. Bergot, Jun 14 2013 [In other words, the partial sums of this sequence give A016061. - Leo Tavares, Nov 23 2021]
a(n)*Pi is the total length of half circle spiral after n rotations. See illustration in links. - Kival Ngaokrajang, Nov 05 2013
For corresponding sums in first comment by Henry Bottomley, see A059255. - Zak Seidov, Sep 10 2015
a(n) also gives the dimension of the simple Lie algebras B_n (n >= 2) and C_n (n >= 3). - Wolfdieter Lang, Oct 21 2015
With T_(i+1,i)=a(i+1) and all other elements of the lower triangular matrix T zero, T is the infinitesimal generator for unsigned A130757, analogous to A132440 for the Pascal matrix. - Tom Copeland, Dec 13 2015
Partial sums of squares with alternating signs, ending in an even term: a(n) = 0^2 - 1^2 +- ... + (2*n)^2, cf. Example & Formula from Berselli, 2013. - M. F. Hasler, Jul 03 2018
Also numbers k with the property that in the symmetric representation of sigma(k) the smallest Dyck path has a central peak and the largest Dyck path has a central valley, n > 0. (Cf. A237593.) - Omar E. Pol, Aug 28 2018
a(n) is the area of a triangle with vertices at (0,0), (2*n+1, 2*n), and ((2*n+1)^2, 4*n^2). - Art Baker, Dec 12 2018
This sequence is the largest subsequence of A000217 such that gcd(a(n), 2*n) = a(n) mod (2*n) = n, n > 0 up to a given value of n. It is the interleave of A033585 (a(n) is even) and A033567 (a(n) is odd). - Torlach Rush, Sep 09 2019
A generalization of Hasler's Comment (Jul 03 2018) follows. Let P(k,n) be the n-th k-gonal number. Then for k > 1, partial sums of {P(k,n)} with alternating signs, ending in an even term, = n*((k-2)*n + 1). - Charlie Marion, Mar 02 2021
Let U_n(H) = {A in M_n(H): A*A^H = I_n} be the group of n X n unitary matrices over the quaternions (A^H is the conjugate transpose of A. Note that over the quaternions we still have A*A^H = I_n <=> A^H*A = I_n by mapping A and A^H to (2n) X (2n) complex matrices), then a(n) is the dimension of its Lie algebra u_n(H) = {A in M_n(H): A + A^H = 0} as a real vector space. A basis is given by {(E_{st}-E_{ts}), i*(E_{st}+E_{ts}), j*(E_{st}+E_{ts}), k*(E_{st}+E_{ts}): 1 <= s < t <= n} U {i*E_{tt}, j*E_{tt}, k*E_{tt}: t = 1..n}, where E_{st} is the matrix with all entries zero except that its (st)-entry is 1. - Jianing Song, Apr 05 2021

			For n=6, a(6) = 0^2 - 1^2 + 2^2 - 3^2 + 4^2 - 5^2 + 6^2 - 7^2 + 8^2 - 9^2 + 10^2 - 11^2 + 12^2 = 78. - _Bruno Berselli_, Aug 29 2013

Louis Comtet, Advanced Combinatorics, Reidel, 1974, pp. 77-78. (In the integral formula on p. 77 a left bracket is missing for the cosine argument.)

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Matthew Cho, Anton Dochtermann, Ryota Inagaki, Suho Oh, Dylan Snustad, and Bailee Zacovic, Chip-firing and critical groups of signed graphs, arXiv:2306.09315 [math.CO], 2023. See p. 22.
Robert FERREOL, Illustration: triangular numbers of even order
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
Milan Janjic, Two Enumerative Functions, University of Banja Luka (Bosnia and Herzegovina, 2017).
Ângela Mestre and José Agapito, Square Matrices Generated by Sequences of Riordan Arrays, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
Kival Ngaokrajang, Illustration of half circle spiral.
Markus Scheuer, show that; strange sum yields triangular numbers, Mathematics StackExchange.
Amelia Carolina Sparavigna, The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences, Politecnico di Torino, Italy (2019).
Amelia Carolina Sparavigna, The groupoid of the Triangular Numbers and the generation of related integer sequences, Politecnico di Torino, Italy (2019).
Leo Tavares, Illustration: Squared Hexagons.
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A000290, A000384, A002378, A002943, A100040, A100041, A081266, A144312.
Cf. A033567, A033585, A059255, A130757, A132440, A027907.
Second column of array A094416.
Equals A033586(n) divided by 4.
See Comments of A132124.
Second n-gonal numbers: A005449, A147875, A045944, A179986, A033954, A062728, A135705.
Row sums in triangle A253580.
Cf. A016061, A003215, A000290, A188859.

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

a014105 n = n * (2 * n + 1)
a014105_list = scanl (+) 0 a004767_list  -- Reinhard Zumkeller, Oct 03 2012

[ n*(2*n+1) : n in [0..50] ]; // Wesley Ivan Hurt, Jun 14 2014

seq(binomial(2*n+1,2), n=0..46); # Zerinvary Lajos, Jan 21 2007

Table[n*(2*n+1), {n,0,100}] (* Vladimir Joseph Stephan Orlovsky, Nov 16 2008 *)
LinearRecurrence[{3,-3,1},{0,3,10},50] (* Harvey P. Dale, Feb 10 2015 *)
CoefficientList[Series[x*(3 + x)/(1 - x)^3,{x, 0, 50}], x] (* Stefano Spezia, Sep 02 2018 *)

PARI
```
a(n)=n*(2*n+1)
```

[n*(2*n+1) for n in range(50)] # G. C. Greubel, Dec 16 2018

a(n) = 3*Sum_{k=1..n} tan^2(k*Pi/(2*(n + 1))). - Ignacio Larrosa Cañestro, Apr 17 2001
a(n)^2 = n*(a(n) + 1 + a(n) + 2 + ... + a(n) + 2*n); e.g., 10^2 = 2*(11 + 12 + 13 + 14). - Charlie Marion, Jun 15 2003
From N. J. A. Sloane, Sep 13 2003: (Start)
G.f.: x*(3 + x)/(1 - x)^3.
E.g.f.: exp(x)*(3*x + 2*x^2).
a(n) = A000217(2*n) = A000384(-n). (End)
a(n) = A084849(n) - 1; A100035(a(n) + 1) = 1. - Reinhard Zumkeller, Oct 31 2004
a(n) = A126890(n, k) + A126890(n, n-k), 0 <= k <= n. - Reinhard Zumkeller, Dec 30 2006
a(2*n) = A033585(n); a(3*n) = A144314(n). - Reinhard Zumkeller, Sep 17 2008
a(n) = a(n-1) + 4*n - 1 (with a(0) = 0). - Vincenzo Librandi, Dec 24 2010
a(n) = Sum_{k=0.2*n} (-1)^k*k^2. - Bruno Berselli, Aug 29 2013
a(n) = A242342(2*n + 1). - Reinhard Zumkeller, May 11 2014
a(n) = Sum_{k=0..2} C(n-2+k, n-2) * C(n+2-k, n), for n > 1. - J. M. Bergot, Jun 14 2014
a(n) = floor(Sum_{j=(n^2 + 1)..((n+1)^2 - 1)} sqrt(j)). Fractional portion of each sum converges to 1/6 as n -> infinity. See A247112 for a similar summation sequence on j^(3/2) and references to other such sequences. - Richard R. Forberg, Dec 02 2014
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3, with a(0) = 0, a(1) = 3, and a(2) = 10. - Harvey P. Dale, Feb 10 2015
Sum_{n >= 1} 1/a(n) = 2*(1 - log(2)) = 0.61370563888010938... (A188859). - Vaclav Kotesovec, Apr 27 2016
From Wolfdieter Lang, Apr 27 2018: (Start)
a(n) = trinomial(2*n, 2) = trinomial(2*n, 2*(2*n-1)), for n >= 1, with the trinomial irregular triangle A027907; i.e., trinomial(n,k) = A027907(n,k).
a(n) = (1/Pi) * Integral_{x=0..2} (1/sqrt(4 - x^2)) * (x^2 - 1)^(2*n) * R(4*(n-1), x), for n >= 0, with the R polynomial coefficients given in A127672, and R(-m, x) = R(m, x). [See Comtet, p. 77, the integral formula for q = 3, n -> 2*n, k = 2, rewritten with x = 2*cos(phi).] (End)
a(n) = A002943(n)/2. - Ralf Steiner, Jul 23 2019
a(n) = A000290(n) + A002378(n). - Torlach Rush, Nov 02 2020
a(n) = A003215(n) - A000290(n+1). See Squared Hexagons illustration. Leo Tavares, Nov 23 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/2 + log(2) - 2. - Amiram Eldar, Nov 28 2021

Link added and minor errors corrected by Johannes W. Meijer, Feb 04 2010

A001845 Centered octahedral numbers (crystal ball sequence for cubic lattice).

Original entry on oeis.org

1, 7, 25, 63, 129, 231, 377, 575, 833, 1159, 1561, 2047, 2625, 3303, 4089, 4991, 6017, 7175, 8473, 9919, 11521, 13287, 15225, 17343, 19649, 22151, 24857, 27775, 30913, 34279, 37881, 41727, 45825, 50183, 54809, 59711, 64897, 70375, 76153, 82239

Offset: 0

N. J. A. Sloane

Number of points in simple cubic lattice at most n steps from origin.
If X is an n-set and Y_i (i=1,2,3) mutually disjoint 2-subsets of X then a(n-6) is equal to the number of 6-subsets of X intersecting each Y_i (i=1,2,3). - Milan Janjic, Aug 26 2007
Equals binomial transform of [1, 6, 12, 8, 0, 0, 0, ...] where (1, 6, 12, 8) = row 3 of the Chebyshev triangle A013609. - Gary W. Adamson, Jul 19 2008
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=2,(i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n >= 4, a(n-2) = -coeff(charpoly(A,x),x^(n-3)). - Milan Janjic, Jan 26 2010
a(n) = A005408(n) * A097080(n-1) / 3. - Reinhard Zumkeller, Dec 15 2013
a(n) = D(3,n) where D are the Delannoy numbers (A008288). As such, a(n) gives the number of grid paths from (0,0) to (3,n) using steps that move one unit north, east, or northeast. - David Eppstein, Sep 07 2014
The first comment above can be re-expressed and generalized as follows: a(n) is the number of points in Z^3 that are L1 (Manhattan) distance <= n from any given point. Equivalently, due to a symmetry that is easier to see in the Delannoy numbers array (A008288), as a special case of Dmitry Zaitsev's Dec 10 2015 comment on A008288, a(n) is the number of points in Z^n that are L1 (Manhattan) distance <= 3 from any given point. - Shel Kaphan, Jan 02 2023

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 81.
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).

T. D. Noe, Table of n, a(n) for n = 0..1000
Luciano Ancora, The Square Pyramidal Number and other figurate numbers, ch. 4.
Bela Bajnok, Additive Combinatorics: A Menu of Research Problems, arXiv:1705.07444 [math.NT], May 2017. See Section 2.3.
D. Bump, K. Choi, P. Kurlberg, and J. Vaaler, A local Riemann hypothesis, I pages 16 and 17
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
Milan Janjić, Two Enumerative Functions
Milan Janjić, Hessenberg Matrices and Integer Sequences, J. Int. Seq. 13 (2010) # 10.7.8.
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
G. Kreweras, Sur les hiérarchies de segments, Cahiers Bureau Universitaire Recherche Opérationnelle, Cahier 20, Inst. Statistiques, Univ. Paris, 1973.
G. Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973). (Annotated scanned copy)
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (10).
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
R. G. Stanton and D. D. Cowan, Note on a "square" functional equation, SIAM Rev., 12 (1970), 277-279.
Eric Weisstein's World of Mathematics, Haüy Construction
Eric Weisstein's World of Mathematics, Octahedral Number
Index entries for crystal ball sequences
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Sums of 2 consecutive terms give A008412.
(1/12)*t*(2*n^3 - 3*n^2 + n) + 2*n - 1 for t = 2, 4, 6, ... gives A049480, A005894, A063488, A001845, A063489, A005898, A063490, A057813, A063491, A005902, A063492, A005917, A063493, A063494, A063495, A063496.
Partial sums of A005899.
Cf. A001846, A001847, A001848, etc., A014820, A013609.
Cf. also A240876, A002412, A016061, A005899.
The 28 uniform 3D tilings: cab: A299266, A299267; crs: A299268, A299269; fcu: A005901, A005902; fee: A299259, A299265; flu-e: A299272, A299273; fst: A299258, A299264; hal: A299274, A299275; hcp: A007899, A007202; hex: A005897, A005898; kag: A299256, A299262; lta: A008137, A299276; pcu: A005899, A001845; pcu-i: A299277, A299278; reo: A299279, A299280; reo-e:  A299281, A299282; rho: A008137, A299276; sod: A005893, A005894; sve: A299255, A299261; svh: A299283, A299284; svj: A299254, A299260; svk: A010001, A063489; tca: A299285, A299286; tcd: A299287, A299288; tfs: A005899, A001845; tsi: A299289, A299290; ttw: A299257, A299263; ubt: A299291, A299292; bnn: A007899, A007202. See the Proserpio link in A299266 for overview.
Row/column 3 of A008288.

a001845 n = (2 * n + 1) * (2 * n ^ 2 + 2 * n + 3) `div` 3
-- Reinhard Zumkeller, Dec 15 2013

Table[(4 n^3 - 6 n^2 + 8 n - 3)/3, {n, 100}] (* Vladimir Joseph Stephan Orlovsky, Jan 15 2011 *)
LinearRecurrence[{4, -6, 4, -1}, {1, 7, 25, 63}, 40] (* Harvey P. Dale, Jun 05 2013 *)
CoefficientList[Series[(1 + x)^3/(-1 + x)^4, {x, 0, 20}], x] (* Eric W. Weisstein, Sep 27 2017 *)

a(n)=(2*n+1)*(2*n^2+2*n+3)/3 \\ Charles R Greathouse IV, Dec 06 2011

G.f.: (1+x)^3 /(1-x)^4. [conjectured (correctly) by Simon Plouffe in his 1992 dissertation]
a(n) = (2*n+1)*(2*n^2 + 2*n + 3)/3.
First differences of A014820(n). - Alexander Adamchuk, May 23 2006
a(n) = a(n-1) + 4*n^2 + 2, a(0)=1. - Vincenzo Librandi, Mar 27 2011
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), with a(0)=1, a(1)=7, a(2)=25, a(3)=63. - Harvey P. Dale, Jun 05 2013
a(n) = Sum_{k=0..min(3,n)} 2^k * binomial(3,k) * binomial(n,k). See Bump et al. - Tom Copeland, Sep 05 2014
From Luciano Ancora, Jan 08 2015: (Start)
a(n) = 2 * A000330(n) + A000330(n+1) + A000330(n-1).
a(n) = A005900(n) + A005900(n+1).
a(n) = A005900(n) + A000330(n) + A000330(n+1).
a(n) = A000330(n-1) + A000330(n) + A005900(n+1). (End)
a(n) = A002412(n+1) + A016061(n-1) for n > 0. - Bruce J. Nicholson, Nov 12 2017
E.g.f.: exp(x)*(3 + 18*x + 18*x^2 + 4*x^3)/3. - Stefano Spezia, Mar 14 2024
Sum_{n >= 1} (-1)^(n+1)/(n*a(n-1)*a(n)) = 5/6 - log(2) = (1 - 1/2 + 1/3) - log(2). - Peter Bala, Mar 21 2024

A002412 Hexagonal pyramidal numbers, or greengrocer's numbers.

Original entry on oeis.org

0, 1, 7, 22, 50, 95, 161, 252, 372, 525, 715, 946, 1222, 1547, 1925, 2360, 2856, 3417, 4047, 4750, 5530, 6391, 7337, 8372, 9500, 10725, 12051, 13482, 15022, 16675, 18445, 20336, 22352, 24497, 26775, 29190, 31746, 34447, 37297, 40300

Offset: 0

N. J. A. Sloane

Binomial transform of (1, 6, 9, 4, 0, 0, 0, ...). - Gary W. Adamson, Oct 16 2007
a(n) is the sum of the maximum(m,n) over {(m,n): m,n in positive integers, m<=n}. - Geoffrey Critzer, Oct 11 2009
We obtain these numbers for d=2 in the identity n*(n*(d*n-d+2)/2)-sum(k*(d*k-d+2)/2, k=0..n-1) = n*(n+1)*(2*d*n-2*d+3)/6 (see Klaus Strassburger in Formula lines). - Bruno Berselli, Apr 21 2010, Nov 16 2010
q^a(n) is the Hankel transform of the q-Catalan numbers. - Paul Barry, Dec 15 2010
Row 1 of the convolution array A213835. - Clark Kimberling, Jul 04 2012
From Ant King, Oct 24 2012: (Start)
For n>0, the digital roots of this sequence A010888(A002412(n)) form the purely periodic 27-cycle {1,7,4,5,5,8,9,3,3,4,1,7,8,8,2,3,6,6,7,4,1,2,2,5,6,9,9}.
For n>0, the units' digits of this sequence A010879(A002412(n)) form the purely periodic 20-cycle {1,7,2,0,5,1,2,2,5,5,6,2,7,5,0,6,7,7,0,0}.
(End)
Partial sums of A000384. - Omar E. Pol, Jan 12 2013
Row sums of A094728. - J. M. Bergot, Jun 14 2013
Number of orbits of Aut(Z^7) as function of the infinity norm (n+1) of the representative integer lattice point of the orbit, when the cardinality of the orbit is  equal to 40320. - Philippe A.J.G. Chevalier, Dec 28 2015
Coefficients in the hypergeometric series identity 1 - 7*(x - 1)/(3*x + 1) + 22*(x - 1)*(x - 2)/((3*x + 1)*(3*x + 2)) - 50*(x - 1)*(x - 2)*(x - 3)/((3*x + 1)*(3*x + 2)*(3*x + 3)) + ... = 0, valid for Re(x) > 1. Cf. A000326 and A002418. Column 3 of A103450. - Peter Bala, Mar 14 2019

			Let n=5, 2*n=10. Since 10 = 1 + 9 = 2 + 8 = 3 + 7 = 4 + 6 = 5 + 5, a(5) = 1*9 + 2*8 + 3*7 + 4*6 + 5*5 = 95. - _Vladimir Shevelev_, May 11 2012

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 194.
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 2.
T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.
I. Siap, Linear codes over F_2 + u*F_2 and their complete weight enumerators, in Codes and Designs (Ohio State, May 18, 2000), pp. 259-271. De Gruyter, 2002.
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).

T. D. Noe and William A. Tedeschi, Table of n, a(n) for n = 0..10000 (first 1000 terms computed by T. D. Noe)
Abdullah Atmaca and A. Yavuz Oruç, On the size of two families of unlabeled bipartite graphs, AKCE International Journal of Graphs and Combinatorics, Vo. 16, No. 2 (2019), pp. 222-229.
Bruno Berselli, A description of the transform in Comments lines: website Matem@ticamente (in Italian).
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Luis Verde-Star, A Matrix Approach to Generalized Delannoy and Schröder Arrays, J. Int. Seq., Vol. 24 (2021), Article 21.4.1.
Eric Weisstein's World of Mathematics, Hexagonal Pyramidal Number.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Bisection of A002623. Equals A000578(n) - A000330(n-1).
Cf. A000292, A016061.
a(n) = A093561(n+2, 3), (4, 1)-Pascal column.
Cf. A220084 for a list of numbers of the form n*P(k,n)-(n-1)*P(k,n-1), where P(k,n) is the n-th k-gonal pyramidal number (see Adamson's formula).
Cf. similar sequences listed in A237616.
Orbits of Aut(Z^7) as function of the infinity norm A000579, A154286, A102860, A002412, A045943, A115067, A008585, A005843, A001477, A000217.

List([0..40],n->n*(n+1)*(4*n-1)/6); # Muniru A Asiru, Mar 18 2019

[n*(n+1)*(4*n-1)/6: n in [0..40]]; // Vincenzo Librandi, Nov 28 2015

seq(sum(i*(2*k-i), i=1..k), k=0..100); # Wesley Ivan Hurt, Sep 25 2013

Figurate[ ngon_, rank_, dim_] := Binomial[rank + dim - 2, dim - 1] ((rank - 1)*(ngon - 2) + dim)/dim; Table[ Figurate[6, r, 3], {r, 0, 40}] (* Robert G. Wilson v, Aug 22 2010 *)
Table[n(n+1)(4n-1)/6, {n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1}, {0,1,7,22}, 40] (* Harvey P. Dale, Jul 16 2011 *)

A002412(n):=n*(n+1)*(4*n-1)/6$ makelist(A002412(n),n,0,20); /* Martin Ettl, Dec 12 2012 */

v=vector(40,i,(i*(i+1))\2); s=0; print1(s","); forstep(i=1,40,2,s+=v[i]; print1(s","))

print([n*(n+1)*(4*n-1)//6 for n in range(40)]) # Michael S. Branicky, Mar 28 2022

a(n) = n(n + 1)(4n - 1)/6.
G.f.: x*(1+3*x)/(1-x)^4. - Simon Plouffe in his 1992 dissertation.
a(n) = n^3 - Sum_{i=1..n-1} i^2. - Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de)
Partial sums of n odd-indexed triangular numbers, e.g., a(3) = t(1)+t(3)+t(5) = 1+6+15 = 22. - Jon Perry, Jul 23 2003
a(n) = Sum_{i=0..n-1} (n - i)*(n + i). - Jon Perry, Sep 26 2004
a(n) = n*A000292(n) - (n-1)*A000292(n-1) = n*binomial((n+2),3) - (n-1)*binomial((n+1),3); e.g., a(5) = 95 = 5*35 - 4*20. - Gary W. Adamson, Dec 28 2007
a(n) = Sum_{i=0..n} (2i^2 + 3i + 1), for n >= 0 (Omits the leading 0). - William A. Tedeschi, Aug 25 2010
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)=7, a(3)=22. - Harvey P. Dale, Jul 16 2011
a(n) = sum a*b, where the summing is over all unordered partitions 2*n = a+b. - Vladimir Shevelev, May 11 2012
From Ant King, Oct 24 2012: (Start)
a(n) = a(n-1) + n*(2*n-1).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 4.
a(n) = (n+1)*(2*A000384(n) + n)/6 = (4*n-1)*A000217(n)/3.
a(n) = A000292(n) + 3*A000292(n-1) = A002411(n) + A000292(n-1).
a(n) = binomial(n+2,3) + 3*binomial(n+1,3) = (4*n-1)*binomial(n+1,2)/3.
Sum_{n>=1} 1/a(n) = 6*(12*log(2)-2*Pi-1)/5 = 1.2414...
(End)
a(n) = Sum_{i=1..n} Sum_{j=1..n} max(i,j) = Sum_{i=1..n} i*(2*n-i). - Enrique Pérez Herrero, Jan 15 2013
a(n) = A005900(n+1) - A000326(n+1) = Octahedral - Pentagonal Numbers. - Richard R. Forberg, Aug 07 2013
a(n) = n*A000217(n) + Sum_{i=0..n-1} A000217(i). - Bruno Berselli, Dec 18 2013
a(n) = 2n * A000217(n) - A000330(n). - J. M. Bergot, Apr 05 2014
a(n) = A080851(4,n-1). - R. J. Mathar, Jul 28 2016
E.g.f.: x*(6 + 15*x + 4*x^2)*exp(x)/6. - Ilya Gutkovskiy, May 12 2017
Sum_{n>=1} (-1)^(n+1)/a(n) = 6*(1 + 2*sqrt(2)*Pi - 2*(3+sqrt(2))*log(2) + 4*sqrt(2)*log(2-sqrt(2)))/5. - Amiram Eldar, Jan 04 2022

A002623 Expansion of 1/((1-x)^4*(1+x)).

Original entry on oeis.org

1, 3, 7, 13, 22, 34, 50, 70, 95, 125, 161, 203, 252, 308, 372, 444, 525, 615, 715, 825, 946, 1078, 1222, 1378, 1547, 1729, 1925, 2135, 2360, 2600, 2856, 3128, 3417, 3723, 4047, 4389, 4750, 5130, 5530, 5950, 6391, 6853, 7337, 7843, 8372, 8924, 9500

Offset: 0

N. J. A. Sloane

Also a(n) is the number of nondegenerate triangles that can be made from rods of lengths 1 to n+1. - Alfred Bruckstein; corrected by Hans Rudolf Widmer, Nov 02 2023
Also number of circumscribable (or escrible) quadrilaterals that can be made from rods of length 1,2,3,4,...,n. - Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr)
Also number of 2 X n binary matrices up to row and column permutation (see the link: Binary matrices up to row and column permutations). - Vladeta Jovovic
Also partial sum of alternate triangular numbers (1, 3, 1+6, 3+10, 1+6+15, 3+10+21, etc.); and also number of triangles pointing in opposite direction to largest triangle in triangular matchstick arrangement of side n+2 (cf. A002717, also the Larsen article). - Henry Bottomley, Aug 08 2000
Ordered union of A002412(n+1) and A016061(n+1). - Lekraj Beedassy, Oct 13 2003
Also Molien series for certain 4-D representation of cyclic group of order 2. - N. J. A. Sloane, Jun 12 2004
From Radu Grigore (radugrigore(AT)gmail.com), Jun 19 2004: (Start)
a(n) = floor( (n+2)*(n+4)*(2n+3) / 24 ). E.g., a(2) = floor(4*6*7/24) = 7 because there are 7 upside down triangles (6 of size one and 1 of size two) in the matchstick figure:
     /\
    /\/\
   /\/\/\
  /\/\/\/\
(End)
Number of non-congruent non-parallelogram trapezoids with positive integer sides (trapezints) and perimeter 2n+5. Also with perimeter 2n+8. - Michael Somos, May 12 2005
a(n) = A108561(n+4,n) for n > 0. - Reinhard Zumkeller, Jun 10 2005
Also number of nonisomorphic planes with n points and 2 lines. E.g., a(0)=1 because with no points, we just have two empty lines. a(1)=3 because the one point may belong to 0, 1 or 2 lines. a(2)=7 because there are 7 ways to determine which of 2 points belong to which of 2 lines, up to isomorphism, i.e., up to a bijection f on the sets of points and a bijection g on the sets of lines, such that A belongs to a iff f(A) belongs to g(a). - Bjorn Kjos-Hanssen (bjorn(AT)math.uconn.edu), Nov 10 2005
a(n-2) is the number of ways to pick two non-overlapping subwords of equal nonzero length from a word of length n. E.g., a(5-2)=a(3)=13 since the word 12345 of length 5 has the following subword pairs: 1,2; 1,3; 1,4; 1,5; 2,3; 2,4; 2,5; 3,4; 3,5; 4,5; 12,34; 12,45; 23,45. - Michael Somos, Oct 22 2006
Partial sums of A002620. - G.H.J. van Rees (vanrees(AT)cs.umanitoba.ca), Feb 16 2007
From Philippe LALLOUET (philip.lallouet(AT)orange.fr), Oct 19 2007: (Start)
Also number of squares of any size in a staircase of n steps built with unit squares:
   
  ||__
  ||__|
  ||__||
For a staircase of 3 steps 6 squares of size 1 and 1 square of size 2, hence c(3)=7.
Columns sums of:
  1 3 6 10 15 21 28 ...
      1  3  6 10 15 ...
            1  3  6 ...
                  1 ...
  ---------------------
  1 3 7 13 22 34 50 ...
(End)
a(n) = sum of row n+1 of triangle A134446. Also, binomial transform of [1, 2, 2, 0, 1, -2, 4, -8, 16, -32, ...]. - Gary W. Adamson, Oct 25 2007
Let b(n) be the number of 4-tuples (w,x,y,z) having all terms in {1,...,n} and 2w=x+y+z+n; then b(n+3) = a(n) for n >= 0. - Clark Kimberling, May 08 2012
a(n) is the number of 3-tuples (w,x,y) having all terms in {0,...,n} and w >= x+y and x <= y. - Clark Kimberling, Jun 04 2012
Also, number of unlabeled bipartite graphs with two left vertices and n right vertices. - Yavuz Oruc, Jan 14 2018
Also number of triples (x,y,z) with 0 < x <= y <= z <= n + 1, x + y > z. - Ralf Steiner, Feb 06 2020
Bisections A002412 and A016061: a(2*k) = k*(k+1)*(4*k-1)/3! and a(2*k+1) = (k+1)*(k+2)*(4*k+9)/3!, for k >= 0. See the Woolhouse link, II. Solution by Stephen Watson, p. 65, with index shifts. - Mo Li, Apr 02 2020
Also, Wiener index of the square of the path graph P_(n+2). - Allan Bickle, Aug 01 2020
Maximum Wiener index of all maximal 2-degenerate graphs with n+2 vertices.  (A maximal 2-degenerate graph can be constructed from a 2-clique by iteratively adding a new 2-leaf (vertex of degree 2) adjacent to two existing vertices.)  The extremal graphs are squares of paths, so the bound also applies to 2-trees and maximal outerplanar graphs. - Allan Bickle, Sep 15 2022

			G.f. = 1 + 3*x + 7*x^2 + 13*x^3 + 22*x^4 + 34*x^5 + 50*x^6 + 70*x^7 + 95*x^8 + ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 74, Problem 7.
P. Diaconis, R. L. Graham and B. Sturmfels, Primitive partition identities, in Combinatorics: Paul Erdős is Eighty, Vol. 2, Bolyai Soc. Math. Stud., 2, 1996, pp. 173-192.
H. Gupta, Partitions of j-partite numbers into twelve or a smaller number of parts. Collection of articles dedicated to Professor P. L. Bhatnagar on his sixtieth birthday. Math. Student 40 (1972), 401-441 (1974).
I. Siap, Linear codes over F_2 + u*F_2 and their complete weight enumerators, in Codes and Designs (Ohio State, May 18, 2000), pp. 259-271. De Gruyter, 2002.
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).

T. D. Noe, Table of n, a(n) for n = 0..1000
A. Atmaca and A. Yavuz Oruç, On the size of two families of unlabeled bipartite graphs, AKCE International Journal of Graphs and Combinatorics, 2017.
M. Benoumhani and M. Kolli, Finite topologies and partitions, JIS 13 (2010) # 10.3.5, Lemma 6 5th line.
Allan Bickle and Zhongyuan Che, Wiener indices of maximal k-degenerate graphs, arXiv:1908.09202 [math.CO], 2019.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Shalosh B. Ekhad and Doron Zeilberger, Computerizing the Andrews-Fraenkel-Sellers Proofs on the Number of m-ary partitions mod m (and doing MUCH more!), arXiv preprint arXiv:1511.06791 [math.CO], 2015.
E. Fix and J. L. Hodges, Significance probabilities of the Wilcoxon test, Annals Math. Stat., 26 (1955), 301-312. [Annotated scanned copy]
E. Fix and J. L. Hodges, Jr., Significance probabilities of the Wilcoxon test, Annals Math. Stat., 26 (1955), 301-312.
Enrique González-Jiménez and Xavier Xarles, On a conjecture of Rudin on squares in Arithmetic Progressions, arXiv preprint arXiv:1301.5122 [math.NT], 2013.
H. Gupta, Partitions of j-partite numbers into twelve or a smaller number of parts, Math. Student 40 (1972), 401-441 (1974). [Annotated scanned copy]
M. A. Harrison, On the number of classes of binary matrices, IEEE Trans. Computers, 22 (1973), 1048-1051. doi:10.1109/T-C.1973.223649.
M. A. Harrison, On the number of classes of binary matrices, IEEE Transactions on Computers, C-22.12 (1973), 1048-1052. (Annotated scanned copy)
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 203
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 413
Vladeta Jovovic, Binary matrices up to row and column permutations
A. Kerber, Experimentelle Mathematik, Séminaire Lotharingien de Combinatoire. Institut de Recherche Math. Avancée, Université Louis Pasteur, Strasbourg, Actes 19 (1988), 77-83. [Annotated scanned copy]. See page 79.
W. Lanssens, B. Demoen, and P.-L. Nguyen, The Diagonal Latin Tableau and the Redundancy of its Disequalities, Report CW 666, July 2014, Department of Computer Science, KU Leuven
M. E. Larsen, The eternal triangle - a history of a counting problem, College Math. J., 20 (1989), 370-392.
P. Lisonek, Quasi-polynomials: A case study in experimental combinatorics, RISC-Linz Report Series No. 93-18, 1983. (Annotated scanned copy)
Math StackExchange, cycle index for S_2 X S_4, Apr. 2021
B. Misek, On the number of classes of strongly equivalent incidence matrices, (Czech with English summary) Casopis Pest. Mat. 89 1964 211-218. See page 217.
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
Brian O'Sullivan and Thomas Busch, Spontaneous emission in ultra-cold spin-polarised anisotropic Fermi seas, arXiv 0810.0231v1 [quant-ph], 2008. [Eq 10a, lambda=2]
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Giovanni Resta, Illustration for a(8)=70.
Marko Riedel, The O.g.f. of inequivalent colorings of a 2xN board with C colors in N is (1/2) 1/(1-w)^(C^2) + (1/2) 1/(1+w)^(C(C-1)/2)*1/(1-w)^(C(C+1)/2) by PET
Sebastian Sassi, Aula Al-Adulrazzaq, Matti Heikinheimo, and Kimmo Tuominen, Fast numerical evaluation of dark matter direct detection event rates, arXiv:2504.19714 [hep-ph], 2025. See p. 7.
Atsuto Seko, Tutorial: Systematic development of polynomial machine learning potentials for elemental and alloy systems, J. Appl. Phys. (2023) Vol. 133, 011101.
J. Silverman, V. E. Vickers and J. M. Mooney, On the number of Costas arrays as a function of array size, Proc. IEEE, 76 (1988), 851-853.
Eric Weisstein's World of Mathematics, Triangle Counting.
W. S. B. Woolhouse, Problem 2420. On the probability of the number of triangles, Mathematical questions with their solutions, v. 9 (June 1868), pp. 63-65.
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Cf. A002620 (first differences), A000292, A001752 (partial sums), A062109 (binomial transf.).
Bisections A002412, A016061.
Cf. also A002717 (a companion sequence), A002727, A006148, A057524, A134446, A014125, A122046, A122047.
Cf. A000217, A006918, A058187, A108561.
The maximum Wiener index of all maximal k-degenerate graphs for k=1..6 are given in A000292, A002623 (this sequence), A014125, A122046, A122047, A175724, respectively.

A002623 := n->(1/16)*(1+(-1)^n)+(n+1)/8+binomial(n+2,2)/4+binomial(n+3,3)/2;
seq( ((2*n+3)*(n+2)*(n+1)/6-floor((n+2)/2))/4,n=1..47); # Lewis
a := n -> ((-1)^n*3 + 45 + 68*n + 30*n^2 + 4*n^3) / 48:
seq(a(n), n=0..46); # Peter Luschny, Jan 22 2018

CoefficientList[Series[1/((1-x)^3(1-x^2)),{x,0,50}],x] (* or *) LinearRecurrence[{3,-2,-2,3,-1},{1,3,7,13,22},50] (* Harvey P. Dale, Jul 19 2011 *)
Table[((2 n^3 + 15 n^2 + 34 n + 45 / 2 + (3/2) (-1)^n) / 24), {n, 0, 100}] (* Vincenzo Librandi, Jan 15 2018 *)
a[ n_] := Floor[(n + 2)*(n + 4)*(2*n + 3)/24]; (* Michael Somos, Feb 19 2024 *)

{a(n) = (8 + 34/3*n + 5*n^2 + 2/3*n^3) \ 8}; /* Michael Somos, Sep 04 1999 */

x='x+O('x^50); Vec(1/((1 - x)^3 * (1 - x^2))) \\ Indranil Ghosh, Apr 04 2017

def A002623(n): return ((n+2)*(n+4)*((n<<1)+3)>>3)//3 # Chai Wah Wu, Mar 25 2024

a(n+1) = a(n) + {(k-1)*k if n=2*k} or {k*k if n=2*k+1}.
a(n)+a(n+1) = A000292(n+1).
a(n) = a(n-2) + A000217(n+1) = A002717(n+2) - A000292(n+1).
Also: a(n) = C(n+3, 3) - a(n-1) with a(0)=1. - Labos Elemer, Apr 26 2003
From Paul Barry, Jul 01 2003: (Start)
a(n) = Sum_{k=0..n} (-1)^(n-k)*C(k+3,3).
The signed version 1, -3, 7, ... has the formula:
a(n) = (4*n^3 + 30*n^2 + 68*n + 45)*(-1)^n/48 + 1/16.
This is the partial sums of the signed version of A000292. (End)
From Paul Barry, Jul 21 2003: (Start)
a(n) = Sum_{k=0..n} floor((k+2)^2/4).
a(n) = Sum_{k=0..n} Sum_{j=0..k} Sum_{i=0..j} (1+(-1)^i)/2. (End)
a(n) = a(n - 2) + (n*(n - 1))/2, with n>2, a(1)=0, a(2)=1; a(n) = (4*n^3+6*n^2-4*n+3*(-1)^n-3)/48, with offset 2. - Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 14 2004 (formula simplified by Bruno Berselli, Aug 29 2013)
a(n) = ((2*n+3)*(n+2)*(n+1)/6-floor((n+2)/2))/4, with offset 1. - Jerry W. Lewis (JLewis(AT)wyeth.com), Mar 23 2005
a(n) = 2*a(n-1) - a(n-2) + 1 + floor(n/2). - Bjorn Kjos-Hanssen (bjorn(AT)math.uconn.edu), Nov 10 2005
A002620(n+3) = a(n+1) - a(n). - Michael Somos, Sep 04 1999
Euler transform of length 2 sequence [ 3, 1]. - Michael Somos, Sep 04 2006
a(n) = -a(-5-n) for all n in Z. - Michael Somos, Sep 04 2006
Let P(i,k) be the number of integer partitions of n into k parts, then with k=2 we have a(n) = sum_{m=1}^{n} sum_{i=k}^{m} P(i,k). For k=1 we get A000217 = triangular numbers. - Thomas Wieder, Feb 18 2007
a(n) = (n+(3+(-1)^n)/2)*(n+(7+(-1)^n)/2)*(2*n+5-2*(-1)^n)/24. - Philippe LALLOUET (philip.lallouet(AT)orange.fr), Oct 19 2007 (corrected by Bruno Berselli, Aug 30 2013)
From Johannes W. Meijer, May 20 2011: (Start)
a(n) = A006918(n+1) + A006918(n).
a(n) = A058187(n-2) + 2*A058187(n-1) + A058187(n). (End)
a(0)=1, a(1)=3, a(2)=7, a(3)=13, a(4)=22; for n > 4, a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5). - Harvey P. Dale, Jul 19 2011
a(n) = Sum_{i=0..n+2} floor(i/2)*ceiling(i/2). - Bruno Berselli, Aug 30 2013
a(n) = 15/16 + (1/16)*(-1)^n + (17/12)*n + (5/8)*n^2 + (1/12)*n^3. - Robert Israel, Jul 07 2014
a(n) = Sum_{i=0..n+2} (n+1-i)*floor(i/2+1). - Bruno Berselli, Apr 04 2017
a(n) = 1 + floor((2*n^3 + 15*n^2 + 34*n) / 24). - Allan Bickle, Aug 01 2020
E.g.f.: ((24 + 51*x + 21*x^2 + 2*x^3)*cosh(x) + (21 + 51*x + 21*x^2 + 2*x^3)*sinh(x))/24. - Stefano Spezia, Jun 02 2021

A000447 a(n) = 1^2 + 3^2 + 5^2 + 7^2 + ... + (2n-1)^2 = n(4*n^2 - 1)/3.

Original entry on oeis.org

0, 1, 10, 35, 84, 165, 286, 455, 680, 969, 1330, 1771, 2300, 2925, 3654, 4495, 5456, 6545, 7770, 9139, 10660, 12341, 14190, 16215, 18424, 20825, 23426, 26235, 29260, 32509, 35990, 39711, 43680, 47905, 52394, 57155, 62196, 67525, 73150, 79079, 85320, 91881, 98770, 105995, 113564, 121485

Offset: 0

N. J. A. Sloane

4 times the variance of the area under an n-step random walk: e.g., with three steps, the area can be 9/2, 7/2, 3/2, 1/2, -1/2, -3/2, -7/2, or -9/2 each with probability 1/8, giving a variance of 35/4 or a(3)/4. - Henry Bottomley, Jul 14 2003
Number of standard tableaux of shape (2n-1,1,1,1) (n>=1). - Emeric Deutsch, May 30 2004
Also a(n) = (1/6)*(8*n^3-2*n), n>0: structured octagonal diamond numbers (vertex structure 9). Cf. A059722 = alternate vertex; A000447 = structured diamonds; and structured tetragonal anti-diamond numbers (vertex structure 9). Cf. A096000 = alternate vertex; A100188 = structured anti-diamonds. Cf. A100145 for more on structured numbers. - James A. Record (james.record(AT)gmail.com), Nov 07 2004
The n-th tetrahedral (or pyramidal) number is n(n+1)(n+2)/6. This sequence contains the tetrahedral numbers of A000292 obtained for n= 1,3,5,7,... (see A015219). - Valentin Bakoev, Mar 03 2009
Using three consecutive numbers u, v, w, (u+v+w)^3-(u^3+v^3+w^3) equals 18 times the numbers in this sequence. - J. M. Bergot, Aug 24 2011
This sequence is related to A070893 by A070893(2*n-1) = n*a(n)-sum(i=0..n-1, a(i)). - Bruno Berselli, Aug 26 2011
Number of integer solutions to 1-n <= x <= y <= z <= n-1. - Michael Somos, Dec 27 2011
Partial sums of A016754. - Reinhard Zumkeller, Apr 02 2012
Also the number of cubes in the n-th Haüy square pyramid. - Eric W. Weisstein, Sep 27 2017

			G.f. = x + 10*x^2 + 35*x^3 + 84*x^4 + 165*x^5 + 286*x^6 + 455*x^7 + 680*x^8 + ...
a(2) = 10 since (-1, -1, -1), (-1, -1, 0), (-1, -1, 1), (-1, 0, 0), (-1, 0, 1), (-1, 1, 1), (0, 0, 0), (0, 0, 1), (0, 1, 1), (1, 1, 1) are the 10 solutions (x, y, z) of -1 <= x <= y <= z <= 1.
a(0) = 0, which corresponds to the empty sum.

G. Chrystal, Textbook of Algebra, Vol. 1, A. & C. Black, 1886, Chap. XX, Sect. 10, Example 2.
F. E. Croxton and D. J. Cowden, Applied General Statistics. 2nd ed., Prentice-Hall, Englewood Cliffs, NJ, 1955, p. 742.
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 140.
C. V. Durell, Advanced Algebra, Volume 1, G. Bell & Son, 1932, Exercise IIIe, No. 4.
L. B. W. Jolley, Summation of Series. 2nd ed., Dover, NY, 1961, p. 7.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
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).

T. D. Noe, Table of n, a(n) for n = 0..1000
J. L. Bailey, A table to facilitate the fitting of certain logistic curves, Annals Math. Stat., Vol. 2 (1931), pp. 355-359. [Annotated scanned copy]
Valentin Bakoev, Algorithmic approach to counting of certain types m-ary partitions, Discrete Mathematics, Vol. 275 (2004), pp. 17-41. - _Valentin Bakoev_, Mar 03 2009
F. E. Croxton and D. J. Cowden, Applied General Statistics, 2nd Ed., Prentice-Hall, Englewood Cliffs, NJ, 1955 [Annotated scans of just pages 742-743]
Milan Janjic, Two Enumerative Functions.
T. P. Martin, Shells of atoms, Phys. Reports, Vol. 273 (1996), pp. 199-241, eq. (11).
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Eric Weisstein's World of Mathematics, Haüy Construction.
Eric Weisstein's World of Mathematics, Square Pyramid.
Index entries for two-way infinite sequences.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

(1/12)*t*(n^3-n)+n for t = 2, 4, 6, ... gives A004006, A006527, A006003, A005900, A004068, A000578, A004126, A000447, A004188, A004466, A004467, A007588, A062025, A063521, A063522, A063523.
Column 1 in triangles A008956 and A008958.
Cf. A035328, A069072, A190152.
A000447 is related to partitions of 2^n into powers of 2, as it is shown in the formula, example and cross-references of A002577. - Valentin Bakoev, Mar 03 2009
Cf. A002412, A016061.

a000447 n = a000447_list !! n
a000447_list = scanl1 (+) a016754_list
-- Reinhard Zumkeller, Apr 02 2012

[n*(4*n^2-1)/3: n in [0..50]]; // Vincenzo Librandi, Jan 12 2016

A000447:=z*(1+6*z+z**2)/(z-1)**4; # Simon Plouffe, 1992 dissertation.
A000447:=n->n*(4*n^2 - 1)/3; seq(A000447(n), n=0..50); # Wesley Ivan Hurt, Mar 30 2014

Table[n (4 n^2 - 1)/3, {n, 0, 80}] (* Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)
LinearRecurrence[{4, -6, 4, -1}, {0, 1, 10, 35}, 80] (* Harvey P. Dale, May 25 2012 *)
Join[{0}, Accumulate[Range[1, 81, 2]^2]] (* Harvey P. Dale, Jul 18 2013 *)
CoefficientList[Series[x (1 + 6 x + x^2)/(-1 + x)^4, {x, 0, 20}], x] (* Eric W. Weisstein, Sep 27 2017 *)

A000447(n):=n*(4*n^2 - 1)/3$ makelist(A000447(n),n,0,20); /* Martin Ettl, Jan 07 2013 */

PARI
```
{a(n) = n * (4*n^2 - 1) / 3};
```

concat(0, Vec(x*(1+6*x+x^2)/(1-x)^4 + O(x^100))) \\ Altug Alkan, Jan 11 2016

def A000447(n): return n*((n**2<<2)-1)//3 # Chai Wah Wu, Feb 12 2023

a(n) = binomial(2*n+1, 3) = A000292(2*n-1).
G.f.: x*(1+6*x+x^2)/(1-x)^4.
a(n) = -a(-n) for all n in Z.
a(n) = A000330(2*n)-4*A000330(n) = A000466(n)*n/3 = A000578(n)+A007290(n-2) = A000583(n)-2*A024196(n-1) = A035328(n)/3. - Henry Bottomley, Jul 14 2003
a(n+1) = (2*n+1)*(2*n+2)(2*n+3)/6. - Valentin Bakoev, Mar 03 2009
a(0)=0, a(1)=1, a(2)=10, a(3)=35, a(n)=4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Harvey P. Dale, May 25 2012
a(n) = v(n,n-1), where v(n,k) is the central factorial numbers of the first kind with odd indices. - Mircea Merca, Jan 25 2014
a(n) = A005917(n+1) - A100157(n+1), where A005917 are the rhombic dodecahedral numbers and A100157 are the structured rhombic dodecahedral numbers (vertex structure 9). - Peter M. Chema, Jan 09 2016
For any nonnegative integers m and n, 8*(n^3)*a(m) + 2*m*a(n) = a(2*m*n). - Ivan N. Ianakiev, Mar 04 2017
E.g.f.: exp(x)*x*(1 + 4*x  + (4/3)*x^2). - Wolfdieter Lang, Mar 11 2017
a(n) = A002412(n) + A016061(n-1), for n>0. - Bruce J. Nicholson, Nov 12 2017
From Amiram Eldar, Jan 04 2022: (Start)
Sum_{n>=1} 1/a(n) = 6*log(2) - 3.
Sum_{n>=1} (-1)^(n+1)/a(n) = 3 - 3*log(2). (End)

Chrystal and Durell references from R. K. Guy, Apr 02 2004

A212959 Number of (w,x,y) such that w,x,y are all in {0,...,n} and |w-x| = |x-y|.

Original entry on oeis.org

1, 4, 11, 20, 33, 48, 67, 88, 113, 140, 171, 204, 241, 280, 323, 368, 417, 468, 523, 580, 641, 704, 771, 840, 913, 988, 1067, 1148, 1233, 1320, 1411, 1504, 1601, 1700, 1803, 1908, 2017, 2128, 2243, 2360, 2481, 2604, 2731, 2860, 2993, 3128, 3267

Offset: 0

Clark Kimberling, Jun 01 2012

In the following guide to related sequences: M=max(x,y,z), m=min(x,y,z), and R=range=M-m. In some cases, it is an offset of the listed sequence which fits the conditions shown for w,x,y. Each sequence satisfies a linear recurrence relation, some of which are identified in the list by the following code (signature):
  A: 2,  0, -2,  1, i.e., a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4);
  B: 3, -2, -2,  3, -1;
  C: 4, -6,  4, -1;
  D: 1,  2, -2, -1,  1;
  E: 2,  1, -4,  1,  2, -1;
  F: 2, -1,  1, -2,  1;
  G: 2, -1,  0,  1, -2,  1;
  H: 2, -1,  2, -4,  2, -1,  2, -1;
  I: 3, -3,  2, -3,  3, -1;
  J: 4, -7,  8, -7,  4, -1.
  ...
A212959 ... |w-x|=|x-y| ...... recurrence type A
A212960 ... |w-x| != |x-y| ................... B
A212683 ... |w-x| < |x-y| .................... B
A212684 ... |w-x| >= |x-y| ................... B
A212963 ... see entry for definition ......... B
A212964 ... |w-x| < |x-y| < |y-w| ............ B
A006331 ... |w-x| < y ........................ C
A005900 ... |w-x| <= y ....................... C
A212965 ... w = R ............................ D
A212966 ... 2*w = R
A212967 ... w < R ............................ E
A212968 ... w >= R ........................... E
A077043 ... w = x > R ........................ A
A212969 ... w != x and x > R ................. E
A212970 ... w != x and x < R ................. E
A055998 ... w = x + y - 1
A011934 ... w < floor((x+y)/2) ............... B
A182260 ... w > floor((x+y)/2) ............... B
A055232 ... w <= floor((x+y)/2) .............. B
A011934 ... w >= floor((x+y)/2) .............. B
A212971 ... w < floor((x+y)/3) ............... B
A212972 ... w >= floor((x+y)/3) .............. B
A212973 ... w <= floor((x+y)/3) .............. B
A212974 ... w > floor((x+y)/3) ............... B
A212975 ... R is even ........................ E
A212976 ... R is odd ......................... E
A212978 ... R = 2*n - w - x
A212979 ... R = average{w,x,y}
A212980 ... w < x + y and x < y .............. B
A212981 ... w <= x+y and x < y ............... B
A212982 ... w < x + y and x <= y ............. B
A212983 ... w <= x + y and x <= y ............ B
A002623 ... w >= x + y and x <= y ............ B
A087811 ... w = 2*x + y ...................... A
A008805 ... w = 2*x + 2*y .................... D
A000982 ... 2*w = x + y ...................... F
A001318 ... 2*w = 2*x + y .................... F
A001840 ... w = 3*x + y
A212984 ... 3*w = x + y
A212985 ... 3*w = 3*x + y
A001399 ... w = 2*x + 3*y
A212986 ... 2*w = 3*x + y
A008810 ... 3*x = 2*x + y .................... F
A212987 ... 3*w = 2*x + 2*y
A001972 ... w = 4*x + y ...................... G
A212988 ... 4*w = x + y ...................... G
A212989 ... 4*w = 4*x + y
A008812 ... 5*w = 2*x + 3*y
A016061 ... n < w + x + y <= 2*n ............. C
A000292 ... w + x + y <=n .................... C
A000292 ... 2*n < w + x + y <= 3*n ........... C
A212977 ... n/2 < w + x + y <= n
A143785 ... w < R < x ........................ E
A005996 ... w < R <= x ....................... E
A128624 ... w <= R <= x ...................... E
A213041 ... R = 2*|w - x| .................... A
A213045 ... R < 2*|w - x| .................... B
A087035 ... R >= 2*|w - x| ................... B
A213388 ... R <= 2*|w - x| ................... B
A171218 ... M < 2*m .......................... B
A213389 ... R < 2|w - x| ..................... E
A213390 ... M >= 2*m ......................... E
A213391 ... 2*M < 3*m ........................ H
A213392 ... 2*M >= 3*m ....................... H
A213393 ... 2*M > 3*m ........................ H
A213391 ... 2*M <= 3*m ....................... H
A047838 ... w = |x + y - w| .................. A
A213396 ... 2*w < |x + y - w| ................ I
A213397 ... 2*w >= |x + y - w| ............... I
A213400 ... w < R < 2*w
A069894 ... min(|w-x|,|x-y|) = 1
A000384 ... max(|w-x|,|x-y|) = |w-y|
A213395 ... max(|w-x|,|x-y|) = w
A213398 ... min(|w-x|,|x-y|) = x ............. A
A213399 ... max(|w-x|,|x-y|) = x ............. D
A213479 ... max(|w-x|,|x-y|) = w+x+y ......... D
A213480 ... max(|w-x|,|x-y|) != w+x+y ........ E
A006918 ... |w-x| + |x-y| > w+x+y ............ E
A213481 ... |w-x| + |x-y| <= w+x+y ........... E
A213482 ... |w-x| + |x-y| < w+x+y ............ E
A213483 ... |w-x| + |x-y| >= w+x+y ........... E
A213484 ... |w-x|+|x-y|+|y-w| = w+x+y
A213485 ... |w-x|+|x-y|+|y-w| != w+x+y ....... J
A213486 ... |w-x|+|x-y|+|y-w| > w+x+y ........ J
A213487 ... |w-x|+|x-y|+|y-w| >= w+x+y ....... J
A213488 ... |w-x|+|x-y|+|y-w| < w+x+y ........ J
A213489 ... |w-x|+|x-y|+|y-w| <= w+x+y ....... J
A213490 ... w,x,y,|w-x|,|x-y| distinct
A213491 ... w,x,y,|w-x|,|x-y| not distinct
A213493 ... w,x,y,|w-x|,|x-y|,|w-y| distinct
A213495 ... w = min(|w-x|,|x-y|,|w-y|)
A213492 ... w != min(|w-x|,|x-y|,|w-y|)
A213496 ... x != max(|w-x|,|x-y|)
A213498 ... w != max(|w-x|,|x-y|,|w-y|)
A213497 ... w = min(|w-x|,|x-y|)
A213499 ... w != min(|w-x|,|x-y|)
A213501 ... w != max(|w-x|,|x-y|)
A213502 ... x != min(|w-x|,|x-y|)
...
A211795 includes a guide for sequences that count 4-tuples (w,x,y,z) having all terms in {0,...,n} and satisfying selected properties.  Some of the sequences indexed at A211795 satisfy recurrences that are represented in the above list.
Partial sums of the numbers congruent to {1,3} mod 6 (see A047241). - Philippe Deléham, Mar 16 2014

			a(1)=4 counts these (x,y,z): (0,0,0), (1,1,1), (0,1,0), (1,0,1).
Numbers congruent to {1, 3} mod 6: 1, 3, 7, 9, 13, 15, 19, ...
a(0) = 1;
a(1) = 1 + 3 = 4;
a(2) = 1 + 3 + 7 = 11;
a(3) = 1 + 3 + 7 + 9 = 20;
a(4) = 1 + 3 + 7 + 9 + 13 = 33;
a(5) = 1 + 3 + 7 + 9 + 13 + 15 = 48; etc. - _Philippe Deléham_, Mar 16 2014

A. Barvinok, Lattice Points and Lattice Polytopes, Chapter 7 in Handbook of Discrete and Computational Geometry, CRC Press, 1997, 133-152.
P. Gritzmann and J. M. Wills, Lattice Points, Chapter 3.2 in Handbook of Convex Geometry, vol. B, North-Holland, 1993, 765-797.

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A047241, A211795.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[Abs[w - x] == Abs[x - y], s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 50]]   (* A212959 *)

a(n)=(6*n^2+8*n+3)\/4 \\ Charles R Greathouse IV, Jul 28 2015

a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
G.f.: (1+2*x+3*x^2)/((1+x)*(1-x)^3).
a(n) + A212960(n) = (n+1)^3.
a(n) = (6*n^2 + 8*n + 3 + (-1)^n)/4. - Luce ETIENNE, Apr 05 2014
a(n) = 2*A069905(3*(n+1)+2) - 3*(n+1). - Ayoub Saber Rguez, Aug 31 2021

A120070 Triangle of numbers used to compute the frequencies of the spectral lines of the hydrogen atom.

Original entry on oeis.org

3, 8, 5, 15, 12, 7, 24, 21, 16, 9, 35, 32, 27, 20, 11, 48, 45, 40, 33, 24, 13, 63, 60, 55, 48, 39, 28, 15, 80, 77, 72, 65, 56, 45, 32, 17, 99, 96, 91, 84, 75, 64, 51, 36, 19, 120, 117, 112, 105, 96, 85, 72, 57, 40, 21

Offset: 2

Wolfdieter Lang, Jul 20 2006

The rationals r(m,n):=a(m,n)/(m^2*n^2), for m-1 >= n, else 0, are used to compute the frequencies of the spectral lines of the H-atom according to quantum theory: nu(m,n) = r(m,n)*c*R' with c*R'=3.287*10^15 s^(-1) an approximation for the Rydberg frequency. R' indicates, that the correction factor 1/(1+m_e/m_p), approximately 0.9995, with the masses for the electron and proton, has been used for the Rydberg constant R_infinity. c:=299792458 m/s is, per definition, the velocity of light in vacuo (see A003678).
In order to compute the wave length of the spectral lines approximately one uses the reciprocal rationals: lambda(m,n):= c/nu(m,n) = (1/r(m,n))*91.1961 nm. 1 nm = 10^{-9} m. For the corresponding energies one uses approximately E(m,n) = r(m,n)*13.599 eV (electron Volts).
The author was inspired by Dewdney's book to compile this table and related ones.
For the approximate frequencies, energies and wavelengths of the first members of the Lyman (n=1, m>=2), Balmer (n=2, m>=3), Paschen (n=3, m>=4), Brackett (n=4, m>=5) and Pfund (n=5, m>=6) series see the W. Lang link under A120072.
Frenicle wrote this as a(n+1) = A140978(n) - A133819(n-1). - Paul Curtz, Aug 19 2008
This triangle also has an interpretation related to particle spin. For proper offset such that T(0,0) = 3, then, where h-bar = h/(2*Pi) = A003676/A019692 (= The Dirac constant, also known as Planck's reduced constant) and Spin(n/2) = h-bar/2*sqrt(n(n+2)), it follows that: h-bar/2*sqrt(T(r,k)) = h-bar/2*sqrt(T(r,0) - T(k-1,0)) = sqrt((Spin((r+1)/2))^2 - (Spin(k/2))^2). For example, for r = k = 4, then h-bar/2*sqrt(11) = h-bar/2*sqrt(T(4,4)) = h-bar/2*sqrt(T(4,0) - T(3,0)) = sqrt(h-bar^2/4*T(4,0) - h-bar^2/4*T(3,0)) = sqrt(h-bar^2/4*35 - h-bar^2/4*24) = sqrt((Spin((4+1)/2))^2 - (Spin(4/2))^2); 35 = 5*(5+2) & 24 = 4*(4+2). - Raphie Frank, Dec 30 2012

			Triangle begins
  [ 3];
  [ 8, 5];
  [15, 12,  7];
  [24, 21, 16,  9];
  [35, 32, 27, 20, 11];
  ...

A. K. Dewdney, Reise in das Innere der Mathematik, Birkhäuser, Basel, 2000, pp. 148-154; engl.: A Mathematical Mystery Tour, John Wiley & Sons, N.Y., 1999.

Stanislav Sykora, Table of n, a(n) for n = 2..79801
W. Lang: First ten rows and more.
M. de Frenicle, Methode pour trouver la solutions des problemes par les exclusions, in: Divers ouvrages des mathematiques et de physique par messieurs de l'academie royale des sciences, (1693) pp 1-44, page 11.
Wikipedia, Spin (physics)
Wikipedia, Hydrogen spectral series

Row sums give A016061(n-1), n>=2.

Cf. A120072/A120073 numerator and denominator tables for rationals r(m, n).

ColumnForm[Table[n^2 - k^2, {n, 2, 13}, {k, n - 1}], Center] (* Alonso del Arte, Oct 26 2011 *)

nmax=400;a=vector(1+nmax*(nmax-1)\2);idx=1;for(n=2,nmax,for(k=1,n-1,a[idx]=n*n-k*k;idx++)) \\ Stanislav Sykora, Feb 17 2014

T(n,k)=n^2-k^2;
for (n=1,10,for(k=1,n-1, print1(T(n,k),", ")));
\\ Joerg Arndt, Feb 23 2014

a(m,n) = m^2 - n^2 for m-1 >= n, otherwise 0.
G.f. for column n=1,2,...: x^(n+1)*((2*n+1)- (2*n-1)*x)/(1-x)^3.
G.f. for rationals r(m,n), n=1,2,...,10 see W. Lang link.
T(r,k) = T(r,0) - T(k-1,0), T(0,0) = 3. - Raphie Frank, Dec 27 2012

A002492 Sum of the first n even squares: 2n(n+1)(2n+1)/3.

Original entry on oeis.org

0, 4, 20, 56, 120, 220, 364, 560, 816, 1140, 1540, 2024, 2600, 3276, 4060, 4960, 5984, 7140, 8436, 9880, 11480, 13244, 15180, 17296, 19600, 22100, 24804, 27720, 30856, 34220, 37820, 41664, 45760, 50116, 54740, 59640, 64824, 70300, 76076, 82160

Offset: 0

N. J. A. Sloane

Total number of possible bishop moves on an n+1 X n+1 chessboard, if the bishop is placed anywhere. E.g., on a 3 X 3-Board: bishop has 8 X 2 moves and 1 X 4 moves, so a(2)=20. - Ulrich Schimke (ulrschimke(AT)aol.com)
Let M_n denote the n X n matrix M_n(i,j)=(i+j)^2; then the characteristic polynomial of M_n is x^n - a(n)x^(n-1) - .... - Michael Somos, Nov 14 2002
Partial sums of A016742. - Lekraj Beedassy, Jun 19 2004
0,4,20,56,120 gives the number of electrons in closed shells in the double shell periodic system of elements. This is a new interpretation of the periodic system of the elements. The factor 4 in the formula 4*n(n+1)(2n+1)/6 plays a significant role, since it designates the degeneracy of electronic states in this system. Closed shells with more than 120 electrons are not expected to exist. - Karl-Dietrich Neubert (kdn(AT)neubert.net)
Inverse binomial transform of A240434. - Wesley Ivan Hurt, Apr 13 2014
Atomic number of alkaline-earth metals of period 2n. - Natan Arie Consigli, Jul 03 2016
a(n) are the negative cubic coefficients in the expansion of sin(kx) into powers of sin(x) for the odd k: sin(kx) = k sin(x) - c(k) sin^3(x) + O(sin^5(x)); a(n) = c(2n+1) = A000292(2n). - Mathias Zechmeister, Jul 24 2022
Also the number of distinct series-parallel networks under series-parallel reduction on three unlabeled edges of n element kinds. - Michael R. Hayashi, Aug 02 2023

A. O. Barut, Group Structure of the Periodic System, in Wybourne, Ed., The Structure of Matter, University of Canterbury Press, Christchurch, 1972, p. 126.
Edward G. Mazur, Graphic Representation of the Periodic System during One Hundred Years, University of Alabama Press, Alabama, 1974.
W. Permans and J. Kemperman, "Nummeringspribleem van S. Dockx, Mathematisch Centrum. Amsterdam," Rapport ZW; 1949-005, 4 leaves, 19.8 X 34 cm.
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).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Jose Manuel Garcia Calcines, Luis Javier Hernandez Paricio, and Maria Teresa Rivas Rodriguez, Semi-simplicial combinatorics of cyclinders and subdivisions, arXiv:2307.13749 [math.CO], 2023. See p. 32.
Milan Janjic, Two Enumerative Functions
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
Milan Janjic and Boris Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 13 2013
Milan Janjic and Boris Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014), Article #14.3.5.
Ângela Mestre and José Agapito, Square Matrices Generated by Sequences of Riordan Arrays, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
D. Neubert, Double Shell Structure of the Periodic System of the Elements, Z. Naturforschung, 25A (1970), p. 210.
Karl-Dietrich Neubert, Double-Shell PSE: Metals - Nonmetals.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992, arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
D. Suprijanto and Rusliansyah, Observation on Sums of Powers of Integers Divisible by Four, Applied Mathematical Sciences, Vol. 8, No. 45 (2014), pp. 2219-2226.
Index entries for sequences related to Chebyshev polynomials.
Index entries for two-way infinite sequences.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000292, A000330, A005408, A006331, A053120, A081277.
Cf. A033586 (King), A035005 (Queen), A035006 (Rook), A035008 (Knight) and A049450 (Pawn).
Cf. A002412, A016061.

[2*n*(n+1)*(2*n+1)/3: n in [0..40]]; // Vincenzo Librandi, Jun 16 2011

A002492:=n->2*n*(n+1)*(2*n+1)/3; seq(A002492(n), n=0..50); # Wesley Ivan Hurt, Apr 04 2014

Table[2n(n+1)(2n+1)/3, {n,0,40}] (* or *) Binomial[2*Range[0,40]+2,3] (* or *) LinearRecurrence[{4,-6,4,-1}, {0,4,20,56},40] (* Harvey P. Dale, Aug 15 2012 *)
Accumulate[(2*Range[0,40])^2] (* Harvey P. Dale, Jun 04 2019 *)

PARI
```
a(n)=2*n*(n+1)*(2*n+1)/3
```

G.f.: 4*x*(1+x)/(1-x)^4. - Simon Plouffe in his 1992 dissertation
a(-1-n) = -a(n).
a(n) = 4*A000330(n) = 2*A006331(n) = A000292(2*n).
a(n) = (-1)^(n+1)*A053120(2*n+1,3) (fourth unsigned column of Chebyshev T-triangle, zeros omitted).
a(n) = binomial(2*n+2, 3). - Lekraj Beedassy, Jun 19 2004
A035005(n+1) = a(n) + A035006(n+1) since Queen = Bishop + Rook. - Johannes W. Meijer, Feb 04 2010
a(n) - a(n-1) = 4*n^2. - Joerg Arndt, Jun 16 2011
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>3. - Harvey P. Dale, Aug 15 2012
a(n) = Sum_{k=0..3} C(n-2+k,n-2)*C(n+3-k,n), for n>2. - J. M. Bergot, Jun 14 2014
a(n) = 2*A006331(n). - R. J. Mathar, May 28 2016
From Natan Arie Consigli Jul 03 2016: (Start)
a(n) = A166464(n) - 1.
a(n) = A168380(2*n). (End)
a(n) = Sum_{i=0..n} A005408(i)*A005408(i-1)+1 with A005408(-1):=-1. - Bruno Berselli, Jan 09 2017
a(n) = A002412(n) + A016061(n). - Bruce J. Nicholson, Nov 12 2017
From Amiram Eldar, Jan 04 2022: (Start)
Sum_{n>=1} 1/a(n) = 9/2 - 6*log(2).
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*Pi/2 - 9/2. (End)
a(n) = A081277(3, n-1) = (1+2*n)*binomial(n+1, n-2)*2^2/(n-1) for n > 0. - Mathias Zechmeister, Jul 26 2022
E.g.f.: 2*exp(x)*x*(6 + 9*x + 2*x^2)/3. - Stefano Spezia, Jul 31 2022

Minor errors corrected and edited by Johannes W. Meijer, Feb 04 2010

Title modified by Charles R Greathouse IV at the suggestion of J. M. Bergot, Apr 05 2014

A159797 Triangle read by rows in which row n lists n+1 terms, starting with n, such that the difference between successive terms is equal to n-1.

Original entry on oeis.org

0, 1, 1, 2, 3, 4, 3, 5, 7, 9, 4, 7, 10, 13, 16, 5, 9, 13, 17, 21, 25, 6, 11, 16, 21, 26, 31, 36, 7, 13, 19, 25, 31, 37, 43, 49, 8, 15, 22, 29, 36, 43, 50, 57, 64, 9, 17, 25, 33, 41, 49, 57, 65, 73, 81, 10, 19, 28, 37, 46, 55, 64, 73, 82, 91, 100, 11, 21, 31, 41, 51, 61, 71, 81, 91, 101

Offset: 0

Omar E. Pol, Jul 09 2009

Note that the last term of the n-th row is the n-th square A000290(n).
See also A162611, A162614 and A162622.
The triangle sums, see A180662 for their definitions, link the triangle A159797 with eleven sequences, see the crossrefs. - Johannes W. Meijer, May 20 2011
T(n,k) is the number of distinct sums in the direct sum of {1, 2, ... n} with itself k times for 1 <= k <= n+1, e.g., T(5,3) = the number of distinct sums in the direct sum {1,2,3,4,5} + {1,2,3,4,5} + {1,2,3,4,5}. The sums range from 1+1+1=3 to 5+5+5=15. So there are 13 distinct sums. - Derek Orr, Nov 26 2014

			Triangle begins:
0;
1, 1;
2, 3, 4;
3, 5, 7, 9;
4, 7,10,13,16;
5, 9,13,17,21,25;
6,11,16,21,26,31,36;

Harvey P. Dale, Table of n, a(n) for n = 0..1000
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A000290, A001477, A081493, A159798, A162609, A162610, A162611, A162614, A162622.
Cf.: A006002 (row sums). - R. J. Mathar, Jul 17 2009
Cf. A163282, A163283, A163284, A163285. - Omar E. Pol, Nov 18 2009
From Johannes W. Meijer, May 20 2011: (Start)
Triangle sums (see the comments): A006002 (Row1), A050187 (Row2), A058187 (Related to Kn11, Kn12, Kn13, Fi1 and Ze1), A006918 (Related to Kn21, Kn22, Kn23, Fi2 and Ze2), A000330 (Kn3), A016061 (Kn4), A190717 (Related to Ca1 and Ze3), A144677 (Related to Ca2 and Ze4), A000292 (Related to Ca3, Ca4, Gi3 and Gi4) A190718 (Related to Gi1) and A144678 (Related to Gi2). (End)

A159797:=proc(n) local m: m := floor( (sqrt(8*n+1)-1)/2 ): A159797(n):= m + (n - m*(m+1)/2)*(m-1) end: seq(A159797(n),n=0..75); # Johannes W. Meijer, May 20 2011

Flatten[Table[NestList[#+n-1&,n,n],{n,0,12}]] (* Harvey P. Dale, Aug 04 2014 *)

Given m = floor( (sqrt(8*n+1)-1)/2 ), then a(n) = m + (n - m*(m+1)/2)*(m-1). - Carl R. White, Jul 24 2010

Edited by Omar E. Pol, Jul 18 2009
More terms from Omar E. Pol, Nov 18 2009
More terms from Carl R. White, Jul 24 2010

cp's OEIS Frontend

A014105 Second hexagonal numbers: a(n) = n*(2*n + 1).

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A001845 Centered octahedral numbers (crystal ball sequence for cubic lattice).

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A002412 Hexagonal pyramidal numbers, or greengrocer's numbers.

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A002623 Expansion of 1/((1-x)^4*(1+x)).

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A000447 a(n) = 1^2 + 3^2 + 5^2 + 7^2 + ... + (2*n-1)^2 = n*(4*n^2 - 1)/3.

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A014105 Second hexagonal numbers: a(n) = n(2n + 1).

A000447 a(n) = 1^2 + 3^2 + 5^2 + 7^2 + ... + (2n-1)^2 = n(4*n^2 - 1)/3.

A002492 Sum of the first n even squares: 2n(n+1)(2n+1)/3.

A004188 a(n) = n(3n^2 - 1)/2.