A014641 -id:A014641 - cp's OEIS frontend

A000567 Octagonal numbers: n(3n-2). Also called star numbers.

0, 1, 8, 21, 40, 65, 96, 133, 176, 225, 280, 341, 408, 481, 560, 645, 736, 833, 936, 1045, 1160, 1281, 1408, 1541, 1680, 1825, 1976, 2133, 2296, 2465, 2640, 2821, 3008, 3201, 3400, 3605, 3816, 4033, 4256, 4485, 4720, 4961, 5208, 5461

Offset: 0

N. J. A. Sloane

From Floor van Lamoen, Jul 21 2001: (Start)
Write 1,2,3,4,... in a hexagonal spiral around 0; then a(n) is the sequence found by reading the line from 0 in the direction 0,1,....
The spiral begins:
.
                 85--84--83--82--81--80
                 /                     \
               86  56--55--54--53--52  79
               /   /                 \   \
             87  57  33--32--31--30  51  78
             /   /   /             \   \   \
           88  58  34  16--15--14  29  50  77
           /   /   /   /         \   \   \   \
         89  59  35  17   5---4  13  28  49  76
         /   /   /   /   /     \   \   \   \   \
       90  60  36  18   6   0   3  12  27  48  75
       /   /   /   /   /   /   /   /   /   /   /
     91  61  37  19   7   1---2  11  26  47  74
       \   \   \   \   \ .       /   /   /   /
       92  62  38  20   8---9--10  25  46  73
         \   \   \   \ .           /   /   /
         93  63  39  21--22--23--24  45  72
           \   \   \ .               /   /
           94  64  40--41--42--43--44  71
             \   \ .                   /
             95  65--66--67--68--69--70
               \ .
               96
.
(End)
From Lekraj Beedassy, Oct 02 2003: (Start)
Also the number of distinct three-cell blocks that may be removed out of A000217(n+1) square cells arranged in a stepping triangular array of side (n+1). A 5-layer triangular array of square cells, for instance, has vertices outlined thus:
  x x
  x x x
  x x x x
  x x x x x
  x x x x x x
  x x x x x x  (End)
First derivative at n of A045991. - Ross La Haye, Oct 23 2004
Starting from n=1, the sequence corresponds to the Wiener index of K_{n,n} (the complete bipartite graph wherein each independent set has n vertices). - Kailasam Viswanathan Iyer, Mar 11 2009
Number of divisors of 24^(n-1) for n > 0 (cf A009968). - J. Lowell, Aug 30 2008
a(n) = A001399(6n-5), number of partitions of 6*n - 5 into parts < 4. For example a(2)=8 and partitions of 6*2 - 5 = 7 into parts < 4 are: [1,1,1,1,1,1,1], [1,1,1,1,1,2],[1,1,1,1,3], [1,1,1,2,2], [1,1,2,3], [1,2,2,2], [1,3,3], [2,2,3]. - Adi Dani, Jun 07 2011
Also, sequence found by reading the line from 0 in the direction 0, 8, ..., and the parallel line from 1 in the direction 1, 21, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. - Omar E. Pol, Sep 10 2011
Partial sums give A002414. - Omar E. Pol, Jan 12 2013
Generate a Pythagorean triple using Euclid's formula with (n, n-1) to give A,B,C. a(n) = B + (A + C)/2. - J. M. Bergot, Jul 13 2013
The number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 773", based on the 5-celled von Neumann neighborhood. - Robert Price, May 23 2016
For n >= 1, the continued fraction expansion of sqrt(27*a(n)) is [9n-4; {1, 2n-2, 3, 2n-2, 1, 18n-8}]. For n=1, this collapses to [5; {5, 10}]. - Magus K. Chu, Oct 10 2022
a(n)*a(n+1) + 1 = (3n^2 + n - 1)^2. In general, a(n)*a(n+k) + k^2 = (3n^2 + (3k-2)n - k)^2. - Charlie Marion, May 23 2023

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 38.
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6.
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. 1.
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).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 19-20.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 123.

T. D. Noe, Table of n, a(n) for n = 0..1000
Raghavendra N. Bhat, Cristian Cobeli, and Alexandru Zaharescu, A lozenge triangulation of the plane with integers, arXiv:2403.10500 [math.NT], 2024.
Francesco Brenti and Paolo Sentinelli, Wachs permutations, Bruhat order and weak order, arXiv:2212.04932 [math.CO], 2022.
Cesar Ceballos and Viviane Pons, The s-weak order and s-permutahedra II: The combinatorial complex of pure intervals, arXiv:2309.14261 [math.CO], 2023. See p. 42.
C. K. Cook and M. R. Bacon, Some polygonal number summation formulas, Fib. Q., 52 (2014), 336-343.
John Elias, Illustration: Hexagonal spiral grids based on generalized octagonal numbers; Illustration: Generalized Square-Octagonal Grids.
John Elias, Illustration: Nesting Cube-frames; Nesting Cube Animation; Nesting-frames Decomposition; Factorial Compartmentalization.
Ghislain R. Franssens, On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling number triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1.
Lancelot Hogben, Choice and Chance by Cardpack and Chessboard, Vol. 1, Max Parrish and Co, London, 1950, p. 36.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 342.
Milan Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
R. Kemp, On the number of words in the language {w in Sigma* | w = w^R }^2, Discrete Math., 40 (1982), 225-234. See Table 1.
Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2002), 65-75.
Kaie Kubjas, Luca Sodomaco, and Elias Tsigaridas, Exact solutions in low-rank approximation with zeros, arXiv:2010.15636 [math.AG], 2020.
Viktor Levandovskyy, Christoph Koutschan, and Oleksandr Motsak, On Two-generated Non-commutative Algebras Subject to the Affine Relation, arXiv:1108.1108 [cs.SC], 2011.
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
Omar E. Pol, Illustration of initial terms of A000217, A000290, A000326, A000384, A000566, A000567.
Leo Tavares, Illustration: Square Rays.
Leo Tavares, Illustration: Twin Rectangular Rays.
Leo Tavares, Illustration: Star Rows.
Leo Tavares, Illustration: Split Stars.
Eric Weisstein's World of Mathematics, Complete Bipartite Graph.
Eric Weisstein's World of Mathematics, Octagonal Number.
Eric Weisstein's World of Mathematics, Wiener Index.
Index to sequences related to polygonal numbers.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A014641, A014642, A014793, A014794, A001835, A016777, A045944, A093563 ((6, 1) Pascal, column m=2). A016921 (differences).
Cf. A005408 (the odd numbers).
Cf. A000290, A000217, A046092, A000384, A002378, A003154.

List([0..50], n -> n*(3*n-2)); # G. C. Greubel, Nov 15 2018

a000567 n = n * (3 * n - 2)  -- Reinhard Zumkeller, Dec 20 2012

[n*(3*n-2) : n in [0..50]]; // Wesley Ivan Hurt, Oct 10 2021

A000567 := proc(n)
    n*(3*n-2) ;
end proc:
seq(A000567(n), n=1..50) ;

Table[n (3 n - 2), {n, 0, 50}] (* Harvey P. Dale, May 06 2012 *)
Table[PolygonalNumber[RegularPolygon[8], n], {n, 0, 43}] (* Arkadiusz Wesolowski, Aug 27 2016 *)
PolygonalNumber[8, Range[0, 20]] (* Eric W. Weisstein, Sep 07 2017 *)
LinearRecurrence[{3, -3, 1}, {1, 8, 21}, {0, 20}] (* Eric W. Weisstein, Sep 07 2017 *)

a(n)=n*(3*n-2) \\ Charles R Greathouse IV, Jun 10 2011

vector(50, n, n--; n*(3*n-2)) \\ G. C. Greubel, Nov 15 2018

# Intended to compute the initial segment of the sequence, not isolated terms.
def aList():
     x, y = 1, 1
     yield 0
     while True:
         yield x
         x, y = x + y + 6, y + 6
A000567 = aList()
print([next(A000567) for i in range(49)]) # Peter Luschny, Aug 04 2019

[n*(3*n-2) for n in range(50)] # Gennady Eremin, Mar 10 2022

[n*(3*n-2) for n in range(50)] # G. C. Greubel, Nov 15 2018

a(n) = n*(3*n-2).
a(n) = (3n-2)*(3n-1)*(3n)/((3n-1) + (3n-2) + (3n)), i.e., (the product of three consecutive numbers)/(their sum). a(1) = 1*2*3/(1+2+3), a(2) = 4*5*6/(4+5+6), etc. - Amarnath Murthy, Aug 29 2002
E.g.f.: exp(x)*(x+3*x^2). - Paul Barry, Jul 23 2003
G.f.: x*(1+5*x)/(1-x)^3. Simon Plouffe in his 1992 dissertation
a(n) = Sum_{k=1..n} (5*n - 4*k). - Paul Barry, Sep 06 2005
a(n) = n + 6*A000217(n-1). - Floor van Lamoen, Oct 14 2005
a(n) = C(n+1,2) + 5*C(n,2).
Starting (1, 8, 21, 40, 65, ...) = binomial transform of [1, 7, 6, 0, 0, 0, ...]. - Gary W. Adamson, Apr 30 2008
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), a(0)=0, a(1)=1, a(2)=8. - Jaume Oliver Lafont, Dec 02 2008
a(n) = A000578(n) - A007531(n). - Reinhard Zumkeller, Sep 18 2009
a(n) = a(n-1) + 6*n - 5 (with a(0)=0). - Vincenzo Librandi, Nov 20 2010
a(n) = 2*a(n-1) - a(n-2) + 6. - Ant King, Sep 01 2011
a(n) = A000217(n) + 5*A000217(n-1). - Vincenzo Librandi, Nov 20 2010
a(n) = (A185212(n) - 1) / 4. - Reinhard Zumkeller, Dec 20 2012
a(n) = A174709(6n). - Philippe Deléham, Mar 26 2013
a(n) = (2*n-1)^2 - (n-1)^2. - Ivan N. Ianakiev, Apr 10 2013
a(6*a(n) + 16*n + 1) = a(6*a(n) + 16*n) + a(6*n + 1). - Vladimir Shevelev, Jan 24 2014
a(0) = 0, a(n) = Sum_{k=0..n-1} A005408(A051162(n-1,k)), n >= 1. - L. Edson Jeffery, Jul 28 2014
Sum_{n>=1} 1/a(n) = (sqrt(3)*Pi + 9*log(3))/12 = 1.2774090575596367311949534921... . - Vaclav Kotesovec, Apr 27 2016
From Ilya Gutkovskiy, Jul 29 2016: (Start)
Inverse binomial transform of A084857.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(2*sqrt(3)) = A093766. (End)
a(n) = n * A016777(n-1) = A053755(n) - A000290(n+1). - Bruce J. Nicholson, Aug 10 2017
Product_{n>=2} (1 - 1/a(n)) = 3/4. - Amiram Eldar, Jan 21 2021
P(4k+4,n) = ((k+1)*n - k)^2 - (k*n - k)^2 where P(m,n) is the n-th m-gonal number (a generalization of the Apr 10 2013 formula, a(n) = (2*n-1)^2 - (n-1)^2). - Charlie Marion, Oct 07 2021
From Leo Tavares, Oct 31 2021: (Start)
a(n) = A000290(n) + 4*A000217(n-1). See Square Rays illustration.
a(n) = A000290(n) + A046092(n-1)
a(n) = A000384(n) + 2*A000217(n-1). See Twin Rectangular Rays illustration.
a(n) = A000384(n) + A002378(n-1)
a(n) = A003154(n) - A045944(n-1). See Star Rows illustration. (End)

Incorrect example removed by Joerg Arndt, Mar 11 2010

A008598 Multiples of 16.

Original entry on oeis.org

0, 16, 32, 48, 64, 80, 96, 112, 128, 144, 160, 176, 192, 208, 224, 240, 256, 272, 288, 304, 320, 336, 352, 368, 384, 400, 416, 432, 448, 464, 480, 496, 512, 528, 544, 560, 576, 592, 608, 624, 640, 656, 672, 688, 704, 720, 736, 752, 768, 784, 800, 816, 832

Offset: 0

N. J. A. Sloane

If X is an n-set and Y_i (i=1,2,3,4) mutually disjoint 2-subsets of X then a(n-6) is equal to the number of 5-subsets of X intersecting each Y_i (i=1,2,3,4). - Milan Janjic, Aug 26 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 328.
Milan Janjic, Two Enumerative Functions.
Tanya Khovanova, Recursive Sequences.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Leo Tavares, Illustration: Square Block Star Frames
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008590, A008596, A008597, A014641, A174312, A185212.

A008598:=n->16*n; seq(A008598(n), n=0..100); # Wesley Ivan Hurt, Nov 13 2013

Range[0, 1000, 16] (* Vladimir Joseph Stephan Orlovsky, May 31 2011 *)
CoefficientList[Series[16 x / (x - 1)^2, {x, 0, 60}], x] (* Vincenzo Librandi Jun 10 2013 *)

a(n)=16*n \\ Charles R Greathouse IV, Sep 24 2015

a(n) = Sum_{k=1..8n} (i^k+1)*(i^(8n-k)+1), where i=sqrt(-1). - Bruno Berselli, Mar 19 2012
G.f.: 16*x/(x-1)^2. - Vincenzo Librandi, Jun 10 2013
a(n) = A014641(n) - A185212(n). - Leo Tavares, May 24 2022
From Elmo R. Oliveira, Apr 07 2025: (Start)
E.g.f.: 16*x*exp(x).
a(n) = 16*n = 2*A008590(n) = A174312(n)/2.
a(n) = 2*a(n-1) - a(n-2). (End)

A014642 Even octagonal numbers: a(n) = 4n(3*n-1).

Original entry on oeis.org

0, 8, 40, 96, 176, 280, 408, 560, 736, 936, 1160, 1408, 1680, 1976, 2296, 2640, 3008, 3400, 3816, 4256, 4720, 5208, 5720, 6256, 6816, 7400, 8008, 8640, 9296, 9976, 10680, 11408, 12160, 12936, 13736, 14560, 15408, 16280, 17176, 18096, 19040, 20008, 21000, 22016

Offset: 0

Mohammad K. Azarian

8 times pentagonal numbers. - Omar E. Pol, Dec 11 2008
Sequence found by reading the line from 0, in the direction 0, 8, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. - Omar E. Pol, Jul 18 2012
The sequence forms the even nesting cube-frames (see illustrations in A000567), which separate and appear according to formula along the axes on the zero-centered and one-centered hexagonal number spirals, as well as the axes of the zero-centered and one-centered square number spirals. See illustrations in links. - John Elias, Jul 20 2022

G. C. Greubel, Table of n, a(n) for n = 0..1000
John Elias, Octagonal Nesting Cubes on the Hexagonal Number Spiral Octagonal Nesting Cubes on the Square Number Spiral
Craig Knecht, Number of positions the remaining tiles can occupy in a 4*n length polyiamond bilayer when one tile is missing.
Yaohui Zhu, Kaiming Sun, Zhengdong Luo, and Lingfeng Wang, Progressive Self-Learning for Domain Adaptation on Symbolic Regression of Integer Sequences, Proc. 39th AAAI Conf. Artif. Intel. (2025) Vol. 39, No. 1, 1692-1699. See p. 1698.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000567, A000326, A001082, A014641, A014793, A014794, A033579, A049450.

List([0..50], n-> 8*Binomial(3*n,2)/3); # G. C. Greubel, Oct 09 2019

[8*Binomial(3*n,2)/3: n in [0..50]]; // G. C. Greubel, Oct 09 2019

seq(8*binomial(3*n,2)/3, n=0..50); # G. C. Greubel, Oct 09 2019

LinearRecurrence[{3,-3,1},{0,8,40}, 50] (* G. C. Greubel, Jun 07 2017 *)
PolygonalNumber[8,Range[0,90,2]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 19 2020 *)

vector(51, n, 8*binomial(3*(n-1),2)/3 ) \\ G. C. Greubel, Jun 07 2017

[8*binomial(3*n,2)/3 for n in (0..50)] # G. C. Greubel, Oct 09 2019

a(n) = A000326(n)*8. - Omar E. Pol, Dec 11 2008
a(n) = A049450(n)*4 = A033579(n)*2. - Omar E. Pol, Dec 13 2008
a(n) = a(n-1) + 24*n - 16 (with a(0)=0). - Vincenzo Librandi, Nov 20 2010
G.f.: x*(8+16*x)/(1-3*x+3*x^2-x^3). - Colin Barker, Jan 06 2012
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - G. C. Greubel, Jun 07 2017
E.g.f.: 4*x*(2 + 3*x)*exp(x). - G. C. Greubel, Oct 09 2019
From Amiram Eldar, Mar 24 2021: (Start)
Sum_{n>=1} 1/a(n) = 3*log(3)/8 - Pi/(8*sqrt(3)).
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2)/2 - Pi/(4*sqrt(3)). (End)

More terms from Patrick De Geest

A243201 Odd octagonal numbers indexed by triangular numbers.

Original entry on oeis.org

1, 21, 133, 481, 1281, 2821, 5461, 9633, 15841, 24661, 36741, 52801, 73633, 100101, 133141, 173761, 223041, 282133, 352261, 434721, 530881, 642181, 770133, 916321, 1082401, 1270101, 1481221, 1717633, 1981281, 2274181, 2598421, 2956161, 3349633, 3781141, 4253061, 4767841, 5328001

Offset: 0

Mathew Englander, Jun 01 2014

			a(2) = 133 because the second triangular number is 3 and third odd octagonal number is 133.
a(3) = 481 because the third triangular number is 6 and the sixth odd octagonal number is 481.
a(4) = 1281 because the fourth triangular number is 10 and the tenth odd octagonal number is 1281.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Row 5 of A059259 (coefficients of 1 + 4*n + 7*n^2 + 6*n^3 + 3*n^4 + 0*n^5 which is a formula for the within sequence).
Column 5 of A081297.
Column 6 of A072024.
Diagonal T(n + 1, n) of A219069, n > 0.
Cf. A000217, A000567, A002061, A003215, A005408, A014641, A022522, A140676, A187156.

[3*n^4+6*n^3+7*n^2+4*n+1: n in [0..40]]; // Bruno Berselli, Jun 03 2014

Table[((3 n^2 + 3 n + 2)^2 - 1)/3, {n, 0, 39}] (* Alonso del Arte, Jun 01 2014 *)

[3*n^4+6*n^3+7*n^2+4*n+1 for n in (0..40)] # Bruno Berselli, Jun 03 2014

a(n) = 3*n^4 + 6*n^3 + 7*n^2 + 4*n + 1.
a(n) = (n^2 + n + 1)*(3*n^2 + 3*n + 1).
a(n) = ((3*n^2 + 3*n + 2)^2 - 1)/3.
a(n) = A003215(n) * A002061(n + 1).
a(n) = A022522(n) / A005408(n).
a(n) = A000567(n^2 + n + 1).
a(n) = A014641((n^2 + n)/2).
a(n) = 1 + A140676(n^2 + n).
a(n) = 1 + A187156((n^2 + n + 4)/2) (empirical).
G.f.: (1 + 16*x + 38*x^2 + 16*x^3 + x^4)/(1 - x)^5. - Bruno Berselli, Jun 03 2014
E.g.f.: exp(x)*(1 + 20*x + 46*x^2 + 24*x^3 + 3*x^4). - Stefano Spezia, Apr 16 2022

A350470 Array read by ascending antidiagonals. T(n, k) = J(k, n) where J are the Jacobsthal polynomials.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 5, 1, 1, 1, 7, 9, 11, 1, 1, 1, 9, 13, 29, 21, 1, 1, 1, 11, 17, 55, 65, 43, 1, 1, 1, 13, 21, 89, 133, 181, 85, 1, 1, 1, 15, 25, 131, 225, 463, 441, 171, 1, 1, 1, 17, 29, 181, 341, 937, 1261, 1165, 341, 1

Offset: 0

Peter Luschny, Mar 19 2022

			Array starts:
n\k 0, 1,  2,  3,   4,    5,    6,     7,      8,      9, ...
---------------------------------------------------------------------
[0] 1, 1,  1,  1,   1,    1,    1,     1,      1,      1, ... A000012
[1] 1, 1,  3,  5,  11,   21,   43,    85,    171,    341, ... A001045
[2] 1, 1,  5,  9,  29,   65,  181,   441,   1165,   2929, ... A006131
[3] 1, 1,  7, 13,  55,  133,  463,  1261,   4039,  11605, ... A015441
[4] 1, 1,  9, 17,  89,  225,  937,  2737,  10233,  32129, ... A015443
[5] 1, 1, 11, 21, 131,  341, 1651,  5061,  21571,  72181, ... A015446
[6] 1, 1, 13, 25, 181,  481, 2653,  8425,  40261, 141361, ... A053404
[7] 1, 1, 15, 29, 239,  645, 3991, 13021,  68895, 251189, ... A350468
[8] 1, 1, 17, 33, 305,  833, 5713, 19041, 110449, 415105, ... A168579
[9] 1, 1, 19, 37, 379, 1045, 7867, 26677, 168283, 648469, ... A350469
      A005408 | A082108 |
           A016813   A014641

Rows: A000012, A001045, A006131, A015441, A015443, A015446, A053404, A350468, A168579, A350469.
Columns: A000012, A005408, A016813, A082108, A014641.
Cf. A350467 (main diagonal), A352361 (Fibonacci polynomials), A352362 (Lucas polynomials).

J := (n, x) -> add(2^k*binomial(n - k, k)*x^k, k = 0..n):
seq(seq(J(k, n-k), k = 0..n), n = 0..10);

T[n_, k_] := Hypergeometric2F1[(1 - k)/2, -k/2, -k, -8 n];
Table[T[n, k], {n, 0, 9}, {k, 0, 9}] // TableForm
(* or *)
T[n_, k_] := With[{s = Sqrt[8*n+1]}, ((1+s)^(k+1) - (1-s)^(k+1)) / (2^(k+1)*s)];
Table[Simplify[T[n, k]], {n, 0, 9}, {k, 0, 9}] // TableForm

T(n, k) = ([1, 2; k, 0]^n)[1, 1] ;
export(T)
for(k = 0, 9, print(parvector(10, n, T(n - 1, k))))

T(n, k) = Sum_{j=0..k} binomial(k - j, j)*(2*n)^j.
T(n, k) = ((1+s)^(k+1) - (1-s)^(k+1)) / (2^(k+1)*s) where s = sqrt(8*n + 1).
T(n, k) = [x^k] (1 / (1 - x - 2*n*x^2)).
T(n, k) = hypergeom([1/2 - k/2, -k/2], [-k], -8*n).

A014793 Squares of odd octagonal numbers.

Original entry on oeis.org

1, 441, 4225, 17689, 50625, 116281, 231361, 416025, 693889, 1092025, 1640961, 2374681, 3330625, 4549689, 6076225, 7958041, 10246401, 12996025, 16265089, 20115225, 24611521, 29822521, 35820225, 42680089, 50481025, 59305401

Offset: 0

Mohammad K. Azarian

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000567, A014641, A014642, A014794.

G.f.: (-25*x^4 - 964*x^3 - 2030*x^2 - 436*x - 1)/(x-1)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009

a(n) = ((1 + 2*n)*(1 + 6*n))^2. - Stefano Spezia, Apr 16 2022

More terms from Patrick De Geest, Aug 15 2000

A014794 Squares of even octagonal numbers.

Original entry on oeis.org

0, 64, 1600, 9216, 30976, 78400, 166464, 313600, 541696, 876096, 1345600, 1982464, 2822400, 3904576, 5271616, 6969600, 9048064, 11560000, 14561856, 18113536, 22278400, 27123264, 32718400, 39137536, 46457856, 54760000

Offset: 0

Mohammad K. Azarian

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000567, A014641, A014642, A014793.

[16*n^2*(3*n-1)^2: n in [1..50]]; // Vincenzo Librandi, Jan 07 2012

Table[16*n^2*(3*n-1)^2,{n,0,30}] (* Vincenzo Librandi, Jan 07 2012 *)
LinearRecurrence[{5,-10,10,-5,1},{0,64,1600,9216,30976},30] (* Harvey P. Dale, Nov 27 2015 *)
Select[PolygonalNumber[8,Range[0,50]],EvenQ]^2 (* Harvey P. Dale, Aug 03 2025 *)

a(n) = 16*n^2*(3*n-1)^2 \\ Vincenzo Librandi, Jan 07 2012

G.f.: 64*x*(1+20*x+29*x^2+4*x^3)/(1-x)^5. - Colin Barker, Jan 06 2012
a(n) = 16n^2*(3n-1)^2. - Vincenzo Librandi, Jan 07 2012
E.g.f.: 16*exp(x)*x*(4 + 46*x + 48*x^2 + 9*x^3). - Stefano Spezia, Apr 16 2022

More terms from Patrick De Geest, Aug 2000

a(8) corrected by Vincenzo Librandi, Jan 07 2012

A272126 a(n) = 120n^3 + 60n^2 + 2*n + 1.

Original entry on oeis.org

1, 183, 1205, 3787, 8649, 16511, 28093, 44115, 65297, 92359, 126021, 167003, 216025, 273807, 341069, 418531, 506913, 606935, 719317, 844779, 984041, 1137823, 1306845, 1491827, 1693489, 1912551, 2149733, 2405755, 2681337, 2977199, 3294061, 3632643, 3993665

Offset: 0

Vincenzo Librandi, Apr 25 2016

This is the polynomial Qbar(3,n) in Brent. See A160485 for the triangle of coefficients (with signs) of the Qbar polynomials. - Peter Bala, Jan 22 2019

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Richard P. Brent, Generalising Tuenter's binomial sums, arXiv:1407.3533 [math.CO], 2014. (page 16).
Richard P. Brent, Generalising Tuenter's binomial sums, Journal of Integer Sequences, 18 (2015), Article 15.3.2.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A014641, A158673, A160485, A245244, A272127, A272129.

[120*n^3 + 60*n^2 + 2*n + 1: n in [0..50]];

Table[120 n^3 + 60 n^2 + 2 n + 1, {n, 0, 40}]
LinearRecurrence[{4,-6,4,-1},{1,183,1205,3787},40] (* Harvey P. Dale, Nov 08 2020 *)

a(n) = 120*n^3 + 60*n^2 + 2*n + 1; \\ Altug Alkan, Apr 30 2016

O.g.f.: (1 + 179*x + 479*x^2 + 61*x^3)/(1-x)^4.
E.g.f.: (1 + 182*x + 420*x^2 + 120*x^3)*exp(x).
a(n) = (2*n+1)*(60*n^2+1).
a(n) = (2*n+1) * A158673(n).
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>3.
See page 7 in Brent's paper:
a(n) = (2*n+1)^2*A014641(n) - 2*n*(2*n+1)*A014641(n-1).
A272127(n) = (2*n+1)^2*a(n) - 2*n*(2*n+1)*a(n-1).
From Peter Bala, Jan 22 2019: (Start)
a(n) = 1/4^n * Sum_{k = 0..n} (2*k + 1)^6 * binomial(2*n + 1, n - k).
a(n-1) = 2/4^n * binomial(2*n,n) * ( 1 + 3^6*(n - 1)/(n + 1) + 5^6*(n - 1)*(n - 2)/((n + 1)*(n + 2)) + 7^6*(n - 1)*(n - 2)*(n - 3)/((n + 1)*(n + 2)*(n + 3)) + ... ). (End)

A272127 a(n) = 1680n^4 - 168n^2 + 128*n + 1.

Original entry on oeis.org

1, 1641, 26465, 134953, 427905, 1046441, 2172001, 4026345, 6871553, 11010025, 16784481, 24577961, 34813825, 47955753, 64507745, 85014121, 110059521, 140268905, 176307553, 218881065, 268735361, 326656681, 393471585, 470046953, 557289985, 656148201, 767609441

Offset: 0

Vincenzo Librandi, Apr 25 2016

This is the polynomial Qbar(4,n) in Brent. See A160485 for the triangle of coefficients (with signs) of the Qbar polynomials. - Peter Bala, Jan 21 2019

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Richard P. Brent, Generalising Tuenter's binomial sums, arXiv:1407.3533 [math.CO], 2014. (page 16).
Richard P. Brent, Generalising Tuenter's binomial sums, Journal of Integer Sequences, 18 (2015), Article 15.3.2.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A014641, A272126, A160485, A245244, A272129.

[1680*n^4-168*n^2+128*n+1: n in [0..50]];

Table[1680 n^4 - 168 n^2 + 128 n + 1, {n, 0, 30}]
LinearRecurrence[{5,-10,10,-5,1},{1,1641,26465,134953,427905},30] (* Harvey P. Dale, Nov 27 2017 *)

a(n) = 1680*n^4 - 168*n^2 + 128*n + 1; \\ Altug Alkan, Apr 30 2016

O.g.f.: (1+1636*x+18270*x^2+19028*x^3+1385*x^4)/(1-x)^5.
E.g.f.: (1+1640*x+11592*x^2+10080*x^3+1680*x^4)*exp(x).
a(n) = (2*n+1)*(840*n^3-420*n^2+126*n+1).
a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5), for n>4
See page 7 in Brent's paper:
a(n) = (2*n+1)^2*A272126(n) - 2*n*(2*n+1)*A272126(n-1).
A272128(n) = (2*n+1)^2*a(n) - 2*n*(2*n+1)*a(n-1).
From Peter Bala, Jan 22 2019: (Start)
a(n) = 1/4^n * Sum_{k = 0..n} (2*k + 1)^8 * binomial(2*n + 1, n - k).
a(n-1) = 2/4^n * binomial(2*n,n) * ( 1 + 3^8*(n - 1)/(n + 1) + 5^8*(n - 1)*(n - 2)/((n + 1)*(n + 2)) + 7^8*(n - 1)*(n - 2)*(n - 3)/((n + 1)*(n + 2)*(n + 3)) + ... ). (End)

A272128 a(n) = 30240n^5-25200n^4+5040n^3+7320n^2-2638*n+1.

Original entry on oeis.org

1, 14763, 628805, 5501167, 24943689, 79549811, 203823373, 449807415, 890712977, 1624547899, 2777745621, 4508793983, 7011864025, 10520438787, 15310942109, 21706367431, 30079906593, 40858578635, 54526858597, 71630306319, 92779195241, 118652141203

Offset: 0

Vincenzo Librandi, Apr 25 2016

This is the polynomial Qbar(5,n) in Brent. See A160485 for the triangle of coefficients (with signs) of the Qbar polynomials. - Peter Bala, Feb 01 2019

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Richard P. Brent, Generalising Tuenter's binomial sums, arXiv:1407.3533 [math.CO], 2014. (page 16).
Richard P. Brent, Generalising Tuenter's binomial sums, Journal of Integer Sequences, 18 (2015), Article 15.3.2.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A014641, A272126, A272127.

[30240*n^5-25200*n^4+5040*n^3+7320*n^2-2638*n+1: n in [0..40]];

Table[30240 n^5 - 25200 n^4 + 5040 n^3 + 7320 n^2 - 2638 n + 1, {n, 0, 30}]

a(n)=30240*n^5-25200*n^4+5040*n^3+7320*n^2-2638*n+1 \\ Charles R Greathouse IV, Apr 30 2016

O.g.f.: (1 + 14757*x + 540242*x^2 + 1949762*x^3 + 1073517*x^4 + 50521*x^5)/(1-x)^6.
E.g.f.: (1 + 14762*x + 299640*x^2 + 609840*x^3 + 277200*x^4 + 30240*x^5)*exp(x).
a(n) = (2*n+1)*(15120*n^4-20160*n^3+12600*n^2-2640*n+1).
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).
See page 7 in Brent's paper: a(n) = (2*n+1)^2*A272127(n) - 2*n*(2*n+1)*A272127(n-1).
From Peter Bala, Feb 01 2019: (Start)
a(n) = 1/4^n * Sum_{k = 0..n} (2*k + 1)^10 * binomial(2*n + 1, n - k).
a(n-1) = 2/4^n * binomial(2*n,n) * ( 1 + 3^10*(n - 1)/(n + 1) + 5^10*(n - 1)*(n - 2)/((n + 1)*(n + 2)) + 7^10*(n - 1)*(n - 2)*(n - 3)/((n + 1)*(n + 2)*(n + 3)) + ... ). (End)

cp's OEIS Frontend

A000567 Octagonal numbers: n*(3*n-2). Also called star numbers.

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A008598 Multiples of 16.

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A014642 Even octagonal numbers: a(n) = 4*n*(3*n-1).

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A243201 Odd octagonal numbers indexed by triangular numbers.

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A350470 Array read by ascending antidiagonals. T(n, k) = J(k, n) where J are the Jacobsthal polynomials.

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A014793 Squares of odd octagonal numbers.

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A000567 Octagonal numbers: n(3n-2). Also called star numbers.

A014642 Even octagonal numbers: a(n) = 4n(3*n-1).

A272126 a(n) = 120n^3 + 60n^2 + 2*n + 1.

A272127 a(n) = 1680n^4 - 168n^2 + 128*n + 1.

A272128 a(n) = 30240n^5-25200n^4+5040n^3+7320n^2-2638*n+1.