A146325 -id:A146325 - cp's OEIS frontend

A005448 Centered triangular numbers: a(n) = 3n(n-1)/2 + 1.

1, 4, 10, 19, 31, 46, 64, 85, 109, 136, 166, 199, 235, 274, 316, 361, 409, 460, 514, 571, 631, 694, 760, 829, 901, 976, 1054, 1135, 1219, 1306, 1396, 1489, 1585, 1684, 1786, 1891, 1999, 2110, 2224, 2341, 2461, 2584, 2710, 2839, 2971, 3106, 3244, 3385, 3529

Offset: 1

N. J. A. Sloane, R. K. Guy, Dec 12 1974

These are Hogben's central polygonal numbers
2
.P
3 n
Also the sum of three consecutive triangular numbers (A000217); i.e., a(4) = 19 = T4 + T3 + T2 = 10 + 6 + 3. - Robert G. Wilson v, Apr 27 2001
For k>2, Sum_{n=1..k} a(n) gives the sum pertaining to the magic square of order k. E.g., Sum_{n=1..5} a(n) = 1 + 4 + 10 + 19 + 31 = 65. In general, Sum_{n=1..k} a(n) = k*(k^2 + 1)/2. - Amarnath Murthy, Dec 22 2001
Binomial transform of (1,3,3,0,0,0,...). - Paul Barry, Jul 01 2003
a(n) is the difference of two tetrahedral (or pyramidal) numbers: C(n+3,3) = (n+1)(n+2)(n+3)/6. a(n) = A000292(n) - A000292(n-3) = (n+1)(n+2)(n+3)/6 - (n-2)(n-1)(n)/6. - Alexander Adamchuk, May 20 2006
Partial sums are A006003(n) = n(n^2+1)/2. Finite differences are a(n+1) - a(n) = A008585(n) = 3n. - Alexander Adamchuk, Jun 03 2006
If X is an n-set and Y a fixed 3-subset of X then a(n-2) is equal to the number of 3-subsets of X intersecting Y. - Milan Janjic, Jul 30 2007
Equals (1, 2, 3, ...) convolved with (1, 2, 3, 3, 3, ...). a(4) = 19 = (1, 2, 3, 4) dot (3, 3, 2, 1) = (3 + 6 + 6 + 4). - Gary W. Adamson, May 01 2009
Equals the triangular numbers convolved with [1, 1, 1, 0, 0, 0, ...]. - Gary W. Adamson and Alexander R. Povolotsky, May 29 2009
a(n) is the number of triples (w,x,y) having all terms in {0,...,n} and min(w+x,x+y,y+w) = max(w,x,y). - Clark Kimberling, Jun 14 2012
a(n) = number of atoms at graph distance <= n from an atom in the graphite or graphene network (cf. A008486). - N. J. A. Sloane, Jan 06 2013
In 1826, Shiraishi gave a solution to the Diophantine equation a^3 + b^3 + c^3 = d^3 with b = a(n) for n > 1; see A226903. - Jonathan Sondow, Jun 22 2013
For n > 1, a(n) is the remainder of n^2 * (n-1)^2 mod (n^2 + (n-1)^2). - J. M. Bergot, Jun 27 2013
The equation A000578(x) - A000578(x-1) = A000217(y) - A000217(y-2) is satisfied by y=a(x). - Bruno Berselli, Feb 19 2014
A242357(a(n)) = n. - Reinhard Zumkeller, May 11 2014
A255437(a(n)) = 1. - Reinhard Zumkeller, Mar 23 2015
The first differences give A008486. a(n) seems to give the total number of triangles in the n-th generation of the six patterns of triangle expansion shown in the link. - Kival Ngaokrajang, Sep 12 2015
Number of binary shuffle squares of length 2n which contains exactly two 1's. - Bartlomiej Pawlik, Sep 07 2023
The digital root has period 3 (1, 4, 1) (A146325), the same digital root as the centered 12-gonal numbers, or centered dodecagonal numbers A003154(n). - Peter M. Chema, Dec 20 2023

			From _Seiichi Manyama_, Aug 12 2017: (Start)
a(1) = 1:
      *
     / \
    /   \
   /     \
  *-------*
.................................................
a(2) = 4:
            *
           / \
          /   \
         /     \
        *---*---*
           / \
      *   /   \   *
     / \ /     \ / \
    /   *-------*   \
   /     \     /     \
  *-------*   *-------*
.................................................
a(3) = 10:
                  *
                 / \
                /   \
               /     \
              *---*---*
                 / \
            *   /   \   *
           / \ /     \ / \
          /   *---*---*   \
         /     \ / \ /     \
        *---*---*   *---*---*
           / \ /     \ / \
      *   /   *---*---*   \   *
     / \ /     \ / \ /     \ / \
    /   *-------*   *-------*   \
   /     \     /     \     /     \
  *-------*   *-------*   *-------*
.................................................
a(4) = 19:
                        *
                       / \
                      /   \
                     /     \
                    *---*---*
                       / \
                  *   /   \   *
                 / \ /     \ / \
                /   *---*---*   \
               /     \ / \ /     \
              *---*---*   *---*---*
                 / \ /     \ / \
            *   /   \---*---*   \   *
           / \ /     \ / \ /     \ / \
          /   *---*---*   *---*---*   \
         /     \ / \ /     \ / \ /     \
        *---*---*   *---*---*   *---*---*
           / \ /     \ / \ /     \ / \
      *   /   *---*---*   *---*---*   \   *
     / \ /     \ / \ /     \ / \ /     \ / \
    /   *-------*   *-------*   *-------*   \
   /     \     /     \     /     \     /     \
  *-------*   *-------*   *-------*   *-------*
(End)

R. Reed, The Lemming Simulation Problem, Mathematics in School, 3 (#6, Nov. 1974), front cover and pp. 5-6.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
D. Bevan, D. Levin, P. Nugent, J. Pantone, and L. Pudwell, Pattern avoidance in forests of binary shrubs, arXiv:1510.08036 [math.CO], 2015.
Jarosław Grytczuk, Bartłomiej Pawlik, and Mariusz Pleszczyński, Variations on shuffle squares, arXiv:2308.13882 [math.CO], 2023. See p. 11.
F. Javier de Vega, On the parabolic partitions of a number, J. Alg., Num. Theor., and Appl. (2023) Vol. 61, No. 2, 135-169.
Guo-Niu Han, Enumeration of Standard Puzzles [Cached copy]
L. Hogben, Choice and Chance by Cardpack and Chessboard, Vol. 1, Max Parrish and Co, London, 1950, p. 22.
Milan Janjic, Two Enumerative Functions
Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.
Kival Ngaokrajang, Illustration of triangles expansion
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. Reed, The Lemming Simulation Problem, Mathematics in School, 3 (#6, Nov. 1974), front cover and pp. 5-6. [Scanned photocopy of pages 5, 6 only, with annotations by R. K. Guy and N. J. A. Sloane]
Leo Tavares, Illustration: Triple Triangles
B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.
Eric Weisstein's World of Mathematics, Centered Triangular Number
Julie Zhang, Noah A. Rosenberg, and Julia A. Palacios, The space of multifurcating ranked tree shapes: enumeration, lattice structure, and Markov chains, arXiv:2506.10856 [math.PR], 2025. See p. 33.
Index entries for sequences related to centered polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A000292, A001263, A001844, A002061, A003154, A006003 (partial sums), A008486, A008585 = first differences, A045943, A134482, A146325, A226903, A242357, A255437.

a005448 n = 3 * n * (n - 1) `div` 2 + 1
a005448_list = 1 : zipWith (+) a005448_list [3, 6 ..]
-- Reinhard Zumkeller, Jun 20 2013

I:=[1,4,10]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..60]]; // Vincenzo Librandi, Sep 13 2015

A005448 := n->(3*(n-1)^2+3*(n-1)+2)/2: seq(A005448(n), n=1..100);
A005448 := -(1+z+z**2)/(z-1)^3; # Simon Plouffe in his 1992 dissertation for offset 0

FoldList[#1 + #2 &, 1, 3 Range@ 50] (* Robert G. Wilson v, Feb 02 2011 *)
Join[{1,4},Total/@Partition[Accumulate[Range[50]],3,1]] (* Harvey P. Dale, Aug 17 2012 *)
LinearRecurrence[{3, -3, 1}, {1, 4, 10}, 50] (* Vincenzo Librandi, Sep 13 2015 *)
Table[ j! Coefficient[Series[Exp[x]*(1 + 3 x^2/2)-1, {x, 0, 20}], x, j], {j, 0, 20}] (* Nikolaos Pantelidis, Feb 07 2023 *)
3#+1&/@Accumulate[Range[0,50]] (* Harvey P. Dale, Nov 20 2024 *)

{a(n)=3*(n^2-n)/2+1} /* Michael Somos, Sep 23 2006 */

isok(n) = my(k=(2*n-2)/3, m); (n==1) || ((denominator(k)==1) && (m=sqrtint(k)) && (m*(m+1)==k)); \\ Michel Marcus, May 20 2020

Expansion of x*(1-x^3)/(1-x)^4.
a(n) = C(n+3, 3)-C(n, 3) = C(n, 0)+3*C(n, 1)+3*C(n, 2). - Paul Barry, Jul 01 2003
a(n) = 1 + Sum_{j=0..n-1} (3*j). - Xavier Acloque, Oct 25 2003
a(n) = A000217(n) + A000290(n-1) = (3*A016754(n) + 5)/8. - Lekraj Beedassy, Nov 05 2005
Euler transform of length 3 sequence [4, 0, -1]. - Michael Somos, Sep 23 2006
a(1-n) = a(n). - Michael Somos, Sep 23 2006
a(n) = binomial(n+1,n-1) + binomial(n,n-2) + binomial(n-1,n-3). - Zerinvary Lajos, Sep 03 2006
Row sums of triangle A134482. - Gary W. Adamson, Oct 27 2007
Narayana transform (A001263) * [1, 3, 0, 0, 0, ...]. - Gary W. Adamson, Dec 29 2007
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), a(1)=1, a(2)=4, a(3)=10. - Jaume Oliver Lafont, Dec 02 2008
a(n) = A000217(n-1)*3 + 1 = A045943(n-1) + 1. - Omar E. Pol, Dec 27 2008
a(n) = a(n-1) + 3*n-3. - Vincenzo Librandi, Nov 18 2010
Sum_{n>=1} 1/a(n) = A306324. - Ant King, Jun 12 2012
a(n) = 2*a(n-1) - a(n-2) + 3. - Ant King, Jun 12 2012
a(n) = A101321(3,n-1). - R. J. Mathar, Jul 28 2016
E.g.f.: -1 + (2 + 3*x^2)*exp(x)/2. - Ilya Gutkovskiy, Jul 28 2016
a(n) = A002061(n) + A000217(n-1). - Bruce J. Nicholson, Apr 20 2017
From Amiram Eldar, Jun 20 2020: (Start)
Sum_{n>=1} a(n)/n! = 5*e/2 - 1.
Sum_{n>=1} (-1)^n * a(n)/n! = 5/(2*e) - 1. (End)
a(n) = A000326(n) - n + 1. - Charlie Marion, Nov 21 2020

A003154 Centered 12-gonal numbers, or centered dodecagonal numbers: numbers of the form 6k(k-1) + 1.

Original entry on oeis.org

1, 13, 37, 73, 121, 181, 253, 337, 433, 541, 661, 793, 937, 1093, 1261, 1441, 1633, 1837, 2053, 2281, 2521, 2773, 3037, 3313, 3601, 3901, 4213, 4537, 4873, 5221, 5581, 5953, 6337, 6733, 7141, 7561, 7993, 8437, 8893, 9361, 9841, 10333, 10837, 11353, 11881, 12421

Offset: 1

N. J. A. Sloane

Binomial transform of [1, 12, 12, 0, 0, 0, ...]. Narayana transform (A001263) of [1, 12, 0, 0, 0, ...]. - Gary W. Adamson, Dec 29 2007
Numbers k such that 6*k+3 is a square, these squares are given in A016946. - Gary Detlefs and Vincenzo Librandi, Aug 08 2010
Odd numbers of the form floor(n^2/6). - Juri-Stepan Gerasimov, Jul 27 2011
Bisection of A032528. - Omar E. Pol, Aug 20 2011
Sequence found by reading the line from 1, in the direction 1, 13, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. Opposite numbers to the members of A033581 in the same spiral. - Omar E. Pol, Sep 08 2011
The digital root has period 3 (1, 4, 1) (A146325), the same digital root as the centered triangular numbers A005448(n). - Peter M. Chema, Dec 20 2023

			From _Omar E. Pol_, Aug 21 2011: (Start)
1. Classic illustration of initial terms of the star numbers:
.
.                                     o
.                                    o o
.                  o            o o o o o o o
.               o o o o          o o o o o o
.     o          o o o            o o o o o
.               o o o o          o o o o o o
.                  o            o o o o o o o
.                                    o o
.                                     o
.
.     1            13                 37
.
2. Alternative illustration of initial terms using n-1 concentric hexagons around a central element:
.
.                                 o o o o o
.                                o         o
.                o o o          o   o o o   o
.               o     o        o   o     o   o
.     o        o   o   o      o   o   o   o   o
.               o     o        o   o     o   o
.                o o o          o   o o o   o
.                                o         o
.                                 o o o o o
(End)

Martin Gardner, Time Travel and Other Mathematical Bewilderments. Freeman, NY, 1988, p. 20.
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 = 1..1000
John Elias, Illustration: Star Configurations on the Zero-Centered Hexagonal Number Spiral.
John Elias, Illustration: Star Configurations on the Zero-Centered Square and Hexagonal Number Spirals.
John Elias, Illustration: Generalized Pentagonal and Octagonal Numbers in the Star-Crossed Configurations.
John Elias, Illustration: Generalized Pentagonal and Octagonal Integration in Centered 9-gonal Triangles.
Martin Gardner and N. J. A. Sloane, Correspondence, 1973-74.
Marco Matone and Roberto Volpato, Vector-Valued Modular Forms from the Mumford Form, Schottky-Igusa Form, Product of Thetanullwerte and the Amazing Klein Formula, arXiv:1102.0006 [math.AG], 2011-2012, c_n.
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
Amelia C. Sparavigna, Groupoid of OEIS A003154 numbers (star numbers or centered dodecagonal numbers), Politecnico di Torino, Repository istituzionale (2019).
Amelia Carolina Sparavigna, Groupoid of OEIS A003154 Numbers (star numbers or centered dodecagonal numbers), Department of Applied Science and Technology, Politecnico di Torino (Italy, 2019).
Amelia Carolina Sparavigna, Generalized Sum of Stella Octangula Numbers, Politecnico di Torino (Italy, 2021).
Leo Tavares, Illustration: Twin Hexagons.
Leo Tavares, Illustration: Diamond Rays.
Eric Weisstein's World of Mathematics, Star Number.
R. Yin, J. Mu, and T. Komatsu, The p-Frobenius Number for the Triple of the Generalized Star Numbers, Preprints 2024, 2024072280. See p. 1.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
Index entries for sequences related to centered polygonal numbers.

Cf. A001263, A001318, A003215, A007588, A016946, A032528, A033581, A049598, A056827, A306980.
Row 4 of A257565.
Cf. A000217, A005448, A016754, A146325.

List([1..50], n-> 12*Binomial(n,2)+1 ); # G. C. Greubel, Jul 23 2019

([: >: 6 * ] * <:) i.1000 NB. Stephen Makdisi, May 06 2018

[12*Binomial(n,2)+1: n in [1..50]]; // G. C. Greubel, Jul 23 2019

A003154:=n->6*n*(n-1) + 1: seq(A003154(n), n=1..100); # Wesley Ivan Hurt, Oct 23 2017

FoldList[#1 + #2 &, 1, 12 Range@50] (* Robert G. Wilson v *)
LinearRecurrence[{3,-3,1},{1,13,37},50] (* Harvey P. Dale, Jul 18 2016 *)
12*Binomial[Range[50], 2] + 1 (* G. C. Greubel, Jul 23 2019 *)

a(n)=6*n*(n-1)+1 \\ Charles R Greathouse IV, Nov 20 2012

print([6*n*(n-1)+1 for n in range(1, 47)]) # Michael S. Branicky, Jan 13 2021

G.f.: x*(1+10*x+x^2)/(1-x)^3. Simon Plouffe in his 1992 dissertation
a(n) = 1 + Sum_{j=0..n} (12*j). E.g., a(2)=37 because 1 + 12*0 + 12*1 + 12*2 = 37. - Xavier Acloque, Oct 06 2003
a(n) = numerator in B_2(x) = (1/2)x^2 - (1/2)x + 1/12 = Bernoulli polynomial of degree 2. - Gary W. Adamson, May 30 2005
a(n) = 12*(n-1) + a(n-1), with n>1, a(1)=1. - Vincenzo Librandi, Aug 08 2010
a(n) = A049598(n-1) + 1. - Omar E. Pol, Oct 03 2011
Sum_{n>=1} 1/a(n) = A306980 = Pi * tan(Pi/(2*sqrt(3))) / (2*sqrt(3)). - Vaclav Kotesovec, Jul 23 2019
From Amiram Eldar, Jun 21 2020: (Start)
Sum_{n>=1} a(n)/n! = 7*e - 1.
Sum_{n>=1} (-1)^n * a(n)/n! = 7/e - 1. (End)
a(n) = 2*A003215(n-1) - 1. - Leo Tavares, Jul 30 2021
E.g.f.: exp(x)*(1 + 6*x^2) - 1. - Stefano Spezia, Aug 19 2022

More terms from Michael Somos

A357638 Triangle read by rows where T(n,k) is the number of integer partitions of n with skew-alternating sum k, where k ranges from -n to n in steps of 2.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 3, 1, 1, 0, 0, 1, 4, 1, 1, 0, 0, 1, 4, 4, 1, 1, 0, 0, 0, 4, 5, 4, 1, 1, 0, 0, 0, 1, 10, 5, 4, 1, 1, 0, 0, 0, 1, 5, 13, 5, 4, 1, 1, 0, 0, 0, 0, 4, 13, 14, 5, 4, 1, 1, 0, 0, 0, 0, 1, 13, 17, 14, 5, 4, 1, 1

Offset: 0

Gus Wiseman, Oct 10 2022

We define the skew-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A - B - C + D + E - F - G + ....

			Triangle begins:
  1
  0  1
  0  1  1
  0  1  1  1
  0  0  3  1  1
  0  0  1  4  1  1
  0  0  1  4  4  1  1
  0  0  0  4  5  4  1  1
  0  0  0  1 10  5  4  1  1
  0  0  0  1  5 13  5  4  1  1
  0  0  0  0  4 13 14  5  4  1  1
  0  0  0  0  1 13 17 14  5  4  1  1
  0  0  0  0  1  5 28 18 14  5  4  1  1
Row n = 7 counts the following partitions:
  .  .  .  (322)      (43)      (52)     (61)  (7)
           (331)      (421)     (511)
           (2221)     (3211)    (4111)
           (1111111)  (22111)   (31111)
                      (211111)

Row sums are A000041.
Number of nonzero entries in row n appears to be A004396(n+1).
First nonzero entry of each row appears to converge to A146325.
The central column is A035544, half A035363.
Column sums appear to be A298311.
For original alternating sum we have A344651, ordered A097805.
The half-alternating version is A357637.
The ordered version (compositions) is A357646, half A357645.
The reverse version is A357705, half A357704.
A351005 = alternately equal and unequal partitions, compositions A357643.
A351006 = alternately unequal and equal partitions, compositions A357644.
A357621 gives half-alternating sum of standard compositions, skew A357623.
A357629 gives half-alternating sum of prime indices, skew A357630.
A357633 gives half-alternating sum of Heinz partition, skew  A357634.
Cf. A035594, A053251, A357136, A357189, A357486, A357487, A357488, A357624, A357631, A357632, A357636.

skats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[(i+1)/2]),{i,Length[f]}];
Table[Length[Select[IntegerPartitions[n],skats[#]==k&]],{n,0,12},{k,-n,n,2}]

Conjecture: The columns are palindromes with sums A298311.

A188659 Decimal expansion of (1+sqrt(26))/5.

Original entry on oeis.org

1, 2, 1, 9, 8, 0, 3, 9, 0, 2, 7, 1, 8, 5, 5, 6, 9, 6, 6, 0, 0, 5, 6, 4, 4, 8, 2, 1, 8, 0, 4, 5, 5, 6, 3, 9, 7, 9, 1, 2, 7, 5, 4, 1, 8, 9, 2, 1, 9, 9, 1, 9, 2, 8, 1, 5, 1, 6, 9, 9, 4, 1, 6, 0, 8, 8, 5, 1, 8, 6, 7, 2, 6, 4, 1, 2, 4, 4, 4, 8, 3, 9, 1, 1, 7, 6, 6, 9, 7, 7, 0, 2, 1, 8, 7, 8, 6, 4, 0, 1, 6, 7, 2, 2, 3, 8, 0, 5, 1, 6, 4, 8, 2, 0, 7, 9, 8, 2, 1, 3, 2, 2, 8, 6, 3, 5, 5, 8, 6, 8

Offset: 1

Clark Kimberling, Apr 09 2011

Decimal expansion of the shape of a (2/5)-extension rectangle; see A188640 for definitions of shape and r-extension rectangle. Briefly, shape=length/width, and an r-extension rectangle is composed of two rectangles of shape 1/r when r<1.

The continued fraction of the constant is 1, 4, 1, 1, 4, 1, ... = A146325.

			1.219803902718556966005644821804556397912754189219919281516994...

Index entries for algebraic numbers, degree 2.

Cf. A146325, A188640, A188658, A188729, A188730, A010481.

RealDigits[(1 + Sqrt[26])/5, 10, 111][[1]] (* Robert G. Wilson v, Aug 18 2011 *)

Equals exp(arcsinh(1/5)). - Amiram Eldar, Jul 04 2023

A226023 A142705 (numerators of 1/4-1/(4n^2)) sorted to natural order.

Original entry on oeis.org

0, 2, 3, 6, 12, 15, 20, 30, 35, 42, 56, 63, 72, 90, 99, 110, 132, 143, 156, 182, 195, 210, 240, 255, 272, 306, 323, 342, 380, 399, 420, 462, 483, 506, 552, 575, 600, 650, 675, 702, 756, 783, 812, 870, 899

Offset: 0

Paul Curtz, May 23 2013

A198442(n) without indices 4*n+2.
a(n)/A130823(n+1) = 0, 2,3,2, 4,5,4, 6,7,6, 8,9,8, ... (equal to A133310+1, after 0; see also A008611).
-1,   0,   2,   3, is divisible by  1 (for a(-1)=-1),
3,    6,  12,  15,                  3,
15,  20,  30,  35                   5,
35,  42,  56,  63                   7,
63,  72,  90,  99                   9,
99, 110, 132, 143,                 11, etc.
First column:  A000466(n),
second column: A002943(n),
third column:  A002939(n+1),
fourth column: A000466(n+1).
a(n) is also the numerator of 1/4-1/(4*n+2)^2: 0/1, 2/9, 3/16, 6/25, 12/49, 15/64, 20/81, 30/121, 35/144, 42/169, 56/225,...
The n-th denominator is equal to 4*a(n) + A146325(n+2).
Note that the differences of a(n-1): 1, 2, 1, 3, 6, 3, 5, 10, 5, 7, 14, 7, 9, 18, 9, 11, 22,... (from A043547 by pairs and 2*n+1) has the same recurrence.
(Of course every sequence which obeys a linear recurrence with constant coefficients has first differences that obey the same linear recurrence. - R. J. Mathar, Jun 14 2013)

Paolo Xausa, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,2,-2,0,-1,1).

Trisections: A002939, A000466, A002943.

A226023 := proc(n)
    option remember;
    if n <=6 then
        op(n+1,[0,2,3,6,12,15,20]) ;
    else
        procname(n-1)+2*procname(n-3)-2*procname(n-4)-procname(n-6)+procname(n-7) ;
    end if;
end proc: # R. J. Mathar, Jun 28 2013

A226023[n_]:=Floor[(2n+1)/3]Floor[(2n+5)/3];
Array[A226023,100,0] (* Paolo Xausa, Dec 05 2023 *)

a(n) = floor( (2*n + 1)/3 ) * floor( (2*n + 5)/3 ) = A004396(n) * A004396(n+2).
Recurrences: a(n) = 3*a(n-3) -3*a(n-6) +a(n-9) = a(n-1) +2*a(n-3) -2*a(n-4) -a(n-6) +a(n-7).
a(n+15)  - a(n) = 10*A042968(n+8).
a(n+1) - a(n-2) = 2*A042968(n) with a(-2)=0, a(-1)=-1.
G.f.: x*(2+x+3*x^2+2*x^3+x^4-x^5)/((1-x)^3 * (1+x+x^2)^2). [Ralf Stephan, May 24 2013]

cp's OEIS Frontend

A005448 Centered triangular numbers: a(n) = 3*n*(n-1)/2 + 1.

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A003154 Centered 12-gonal numbers, or centered dodecagonal numbers: numbers of the form 6*k*(k-1) + 1.

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A357638 Triangle read by rows where T(n,k) is the number of integer partitions of n with skew-alternating sum k, where k ranges from -n to n in steps of 2.

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A188659 Decimal expansion of (1+sqrt(26))/5.

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A226023 A142705 (numerators of 1/4-1/(4n^2)) sorted to natural order.

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A005448 Centered triangular numbers: a(n) = 3n(n-1)/2 + 1.

A003154 Centered 12-gonal numbers, or centered dodecagonal numbers: numbers of the form 6k(k-1) + 1.