A244647 -id:A244647 - cp's OEIS frontend

A001107 10-gonal (or decagonal) numbers: a(n) = n(4n-3).

0, 1, 10, 27, 52, 85, 126, 175, 232, 297, 370, 451, 540, 637, 742, 855, 976, 1105, 1242, 1387, 1540, 1701, 1870, 2047, 2232, 2425, 2626, 2835, 3052, 3277, 3510, 3751, 4000, 4257, 4522, 4795, 5076, 5365, 5662, 5967, 6280, 6601, 6930, 7267, 7612, 7965, 8326

Offset: 0

N. J. A. Sloane

Write 0, 1, 2, ... in a square spiral, with 0 at the origin and 1 immediately below it; sequence gives numbers on the negative y-axis (see Example section).
Number of divisors of 48^(n-1) for n > 0. - J. Lowell, Aug 30 2008
a(n) is the Wiener index of the graph obtained by connecting two copies of the complete graph K_n by an edge (for n = 3, approximately: |>-<|). The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices in the graph. - Emeric Deutsch, Sep 20 2010
This sequence does not contain any squares other than 0 and 1. See A188896. - T. D. Noe, Apr 13 2011
For n > 0: right edge of the triangle A033293. - Reinhard Zumkeller, Jan 18 2012
Sequence found by reading the line from 0, in the direction 0, 10, ... and the parallel line from 1, in the direction 1, 27, ..., in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Jul 18 2012
Partial sums give A007585. - Omar E. Pol, Jan 15 2013
This is also a star pentagonal number: a(n) = A000326(n) + 5*A000217(n-1). - Luciano Ancora, Mar 28 2015
Also the number of undirected paths in the n-sunlet graph. - Eric W. Weisstein, Sep 07 2017
After 0, a(n) is the sum of 2*n consecutive integers starting from n-1. - Bruno Berselli, Jan 16 2018
Number of corona of an H0 hexagon with a T(n) triangle. - Craig Knecht, Dec 13 2024

			On a square lattice, place the nonnegative integers at lattice points forming a spiral as follows: place "0" at the origin; then move one step downward (i.e., in the negative y direction) and place "1" at the lattice point reached; then turn 90 degrees in either direction and place a "2" at the next lattice point; then make another 90-degree turn in the same direction and place a "3" at the lattice point; etc. The terms of the sequence will lie along the negative y-axis, as seen in the example below:
  99  64--65--66--67--68--69--70--71--72
   |   |                               |
  98  63  36--37--38--39--40--41--42  73
   |   |   |                       |   |
  97  62  35  16--17--18--19--20  43  74
   |   |   |   |               |   |   |
  96  61  34  15   4---5---6  21  44  75
   |   |   |   |   |       |   |   |   |
  95  60  33  14   3  *0*  7  22  45  76
   |   |   |   |   |   |   |   |   |   |
  94  59  32  13   2--*1*  8  23  46  77
   |   |   |   |           |   |   |   |
  93  58  31  12--11-*10*--9  24  47  78
   |   |   |                   |   |   |
  92  57  30--29--28-*27*-26--25  48  79
   |   |                           |   |
  91  56--55--54--53-*52*-51--50--49  80
   |                                   |
  90--89--88--87--86-*85*-84--83--82--81
[Edited by _Jon E. Schoenfield_, Jan 02 2017]

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189.
Bruce C. Berndt, Ramanujan's Notebooks, Part II, Springer; see p. 23.
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6.
S. M. Ellerstein, The square spiral, J. Recreational Mathematics 29 (#3, 1998) 188; 30 (#4, 1999-2000), 246-250.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd ed., 1994, p. 99.
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
Soren Laing Aletheia-Zomlefer, Lenny Fukshansky, and Stephan Ramon Garcia, The Bateman-Horn Conjecture: Heuristics, History, and Applications, arXiv:1807.08899 [math.NT], 2018-2019. See 6.6.3 p. 33.
Emilio Apricena, A version of the Ulam spiral.
Yin Choi Cheng, Greedy Sidon sets for linear forms, J. Num. Theor. (2024).
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 344.
Craig Knecht, Corona of the H0 hexagon with a T(n) triangle.
Minh Nguyen, 2-adic Valuations of Square Spiral Sequences, Honors Thesis, Univ. of Southern Mississippi (2021).
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
Leo Tavares, Illustration: Conjoined Hexagon/Square Pairs
Eric Weisstein's World of Mathematics, Barbell Graph.
Eric Weisstein's World of Mathematics, Decagonal Number.
Eric Weisstein's World of Mathematics, Graph Path.
Eric Weisstein's World of Mathematics, Sunlet Graph.
Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A007585, A028994.
Cf. A093565 ((8, 1) Pascal, column m = 2). Partial sums of A017077.
Sequences from spirals: A001107 (this), A002939, A007742, A033951, A033952, A033953, A033954, A033989, A033990, A033991, A002943, A033996, A033988.
Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.
Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.
Cf. A003215.

[4*n^2-3*n : n in [0..50] ]; // Wesley Ivan Hurt, Jun 05 2014

A001107:=-(1+7*z)/(z-1)**3; # Simon Plouffe in his 1992 dissertation

LinearRecurrence[{3, -3, 1}, {0, 1, 10}, 60] (* Harvey P. Dale, May 08 2012 *)
Table[PolygonalNumber[RegularPolygon[10], n], {n, 0, 46}] (* Arkadiusz Wesolowski, Aug 27 2016 *)
Table[4 n^2 - 3 n, {n, 0, 49}] (* Alonso del Arte, Jan 24 2017 *)
PolygonalNumber[10, Range[0, 20]] (* Eric W. Weisstein, Sep 07 2017 *)
LinearRecurrence[{3, -3, 1}, {1, 10, 27}, {0, 20}] (* Eric W. Weisstein, Sep 07 2017 *)

PARI
```
a(n)=4*n^2-3*n
```

a=lambda n: 4*n**2-3*n # Indranil Ghosh, Jan 01 2017
def aList(): # Intended to compute the initial segment of the sequence, not isolated terms.
     x, y = 1, 1
     yield 0
     while True:
         yield x
         x, y = x + y + 8, y + 8
A001107 = aList()
print([next(A001107) for i in range(49)]) # Peter Luschny, Aug 04 2019

a(n) = A033954(-n) = A074377(2*n-1).
a(n) = n + 8*A000217(n-1). - Floor van Lamoen, Oct 14 2005
G.f.: x*(1 + 7*x)/(1 - x)^3.
Partial sums of odd numbers 1 mod 8, i.e., 1, 1 + 9, 1 + 9 + 17, ... . - Jon Perry, Dec 18 2004
1^3 + 3^3*(n-1)/(n+1) + 5^3*((n-1)*(n-2))/((n+1)*(n+2)) + 7^3*((n-1)*(n-2)*(n-3))/((n+1)*(n+2)*(n+3)) + ... = n*(4*n-3) [Ramanujan]. - Neven Juric, Apr 15 2008
Starting (1, 10, 27, 52, ...), this is the binomial transform of [1, 9, 8, 0, 0, 0, ...]. - Gary W. Adamson, Apr 30 2008
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2, a(0)=0, a(1)=1, a(2)=10. - Jaume Oliver Lafont, Dec 02 2008
a(n) = 8*n + a(n-1) - 7 for n>0, a(0)=0. - Vincenzo Librandi, Jul 10 2010
a(n) = 8 + 2*a(n-1) - a(n-2). - Ant King, Sep 04 2011
a(n) = A118729(8*n). - Philippe Deléham, Mar 26 2013
a(8*a(n) + 29*n+1) = a(8*a(n) + 29*n) + a(8*n + 1). - Vladimir Shevelev, Jan 24 2014
Sum_{n >= 1} 1/a(n) = Pi/6 + log(2) = 1.216745956158244182494339352... = A244647. - Vaclav Kotesovec, Apr 27 2016
From Ilya Gutkovskiy, Aug 28 2016: (Start)
E.g.f.: x*(1 + 4*x)*exp(x).
Sum_{n >= 1} (-1)^(n+1)/a(n) = (sqrt(2)*Pi - 2*log(2) + 2*sqrt(2)*log(1 + sqrt(2)))/6 = 0.92491492293323294695... (End)
a(n) = A000217(3*n-2) - A000217(n-2). In general, if P(k,n) be the n-th k-gonal number and T(n) be the n-th triangular number, A000217(n), then P(T(k),n) = T((k-1)*n - (k-2)) - T(k-3)*T(n-2). - Charlie Marion, Sep 01 2020
Product_{n>=2} (1 - 1/a(n)) = 4/5. - Amiram Eldar, Jan 21 2021
a(n) = A003215(n-1) + A000290(n) - 1. - Leo Tavares, Jul 23 2022

A244645 Decimal expansion of the sum of the reciprocals of the octagonal numbers (A000567).

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Jul 03 2014

			1.2774090575596367311949534921024332115566344803902472326934919840751515151955452...

Lawrence Downey, Boon W. Ong, and James A. Sellers, Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers, Coll. Math. J., 39, no. 5 (2008), 391-394.
Wikipedia, Polygonal number

Cf. A000038, A013661, A244639, A244644, A244646, A244647, A244648, A244649, A000567.

RealDigits[ Sum[1/(3n^2 - 2n), {n, 1 , Infinity}], 10, 111][[1]]

sumpos(n=1, 1/(3*n^2 - 2*n)) \\ Michel Marcus, Sep 12 2016

sumnumrat(1/(3*n-2)/n,1) \\ Charles R Greathouse IV, Feb 08 2023

Equals Sum_{n>=1} 1/(3*n^2 - 2*n).

Equals Pi/(4*sqrt(3)) + 3*log(3)/4. - Vaclav Kotesovec, Jul 05 2014

A244646 Decimal expansion of the sum of the reciprocals of the 9-gonal (or enneagonal or nonagonal) numbers (A001106).

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Jul 03 2014

			1.2433209261537129892066077396310142821358441010300996244152817525...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Ravi P. Agarwal, Pythagoreans Figurative Numbers: The Beginning of Number Theory and Summation of Series, Journal of Applied Mathematics and Physics, Vol.9, No.8 (2021), pp. 2038-2113. See p. 2076.
Wikipedia, Polygonal number.

Cf. A000038, A001106, A013661, A001620, A244639, A244644, A244645, A244647, A244648, A244649.

RealDigits[ Sum[2/(7n^2 - 5n), {n, 1 , Infinity}], 10, 111][[1]]

Equals Sum_{n>=1} 2/(7n^2 - 5n).
Equals (2*log(14) + 4*(cos(Pi/7)*log(cos(3*Pi/14)) + log(sin(Pi/7))*sin(Pi/14) - log(cos(Pi/14)) * sin(3*Pi/14)) + Pi*tan(3*Pi/14))/5. - Vaclav Kotesovec, Jul 04 2014
Equals 14/25 - (2/5)*(gamma + psi(-5/7)), where gamma is Euler's constant (A001620) and psi(x) is the digamma function (Agarwal, 2021), psi(-5/7) = psi(2/7)+7/5 = -2.285517..., see A354628. - Amiram Eldar, Nov 12 2021

A244649 Decimal expansion of the sum of the reciprocals of the Dodecagonal numbers (A051624).

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Jul 03 2014

From Wolfdieter Lang, Nov 09 2017: (Start)
In the Downey et al. link this is the instance k = 5 of the formula given there for S_{2*k+2}. A simpler formula is given in the Koecher reference as (5/4)*v_5(1) on p. 192. See the Kotesovec formula given below.
The partial sums are given in A294520/A294521. (End)

			1.1779560579226638587351739680918874184458572345666798028425228573...

Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, pp. 189 - 193.

Lawrence Downey, Boon W. Ong, and James A. Sellers, Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers, Coll. Math. J., 39, no. 5 (2008), 391-394.
Wikipedia, Polygonal number

Cf. A000038, A013661, A051624, A244639, A244644, A244645, A244646, A244647, A244648, A294520/A294521.

RealDigits[ Sum[1/(5n^2 - 4n), {n, 1 , Infinity}], 10, 111][[1]]

Equals Sum_{n>=1} 1/(5n^2 - 4n).
Equals Pi/8*sqrt(1+2/sqrt(5)) + (5*log(5) + sqrt(5)*log((3+sqrt(5))/2))/16. - Vaclav Kotesovec, Jul 04 2014
This is the value given in  the Koecher reference (see a comment above), and rewritten with the golden section phi = (1 + sqrt(5))/2 this becomes
  ((5/2)*log(5) + (2*phi - 1)*(log(phi) + (Pi/5)*sqrt(3 + 4*phi)))/8. - Wolfdieter Lang, Nov 09 2017

A244648 Decimal expansion of the sum of the reciprocals of the hendecagonal numbers (A051682).

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Jul 03 2014

			1.195434116529627974352499234698499354884682627084658062386021603017...

Wikipedia, Polygonal number

Cf. A000038, A013661, A244639, A244644, A244645, A244646, A244647, A244649.

RealDigits[ Sum[2/(9n^2 - 7n), {n, 1 , Infinity}], 10, 111][[1]]

Sum_{n=1..infinity} 2/(9n^2 - 7n).

Equals (5*log(3) + Pi*cot(2*Pi/9) - 4*cos(2*Pi/9)*log(cos(Pi/18)) + 4*cos(Pi/9)*log(sin(2*Pi/9)) - 4*log(sin(Pi/9))*sin(Pi/18))/7. - Vaclav Kotesovec, Jul 04 2014

A244641 Decimal expansion of the sum of the reciprocals of the pentagonal numbers (A000326).

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Jul 03 2014

			1.482037501770111223591657453125421381658405425375507779634198065524359698529473...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Hongwei Chen and G. C. Greubel, Sum of the Reciprocals of Polygonal Numbers (Solved), SIAM Problems and solutions.
Hongwei Chen and G. C. Greubel, Siam, Problems and Solutions, problem 07-003 and the solution

Cf. A000326, A016650, A093602, A294514.

Decimal expansion of the sum of the reciprocals of the m-gonal numbers: A000038 (m=3), A013661 (m=4), this sequence (m=5), A016627 (m=6), A244639 (m=7), A244645 (m=8), A244646 (m=9), A244647 (m=10), A244648 (m=11), A244649 (m=12), A275792 (m=14).

SetDefaultRealField(RealField(139)); R:= RealField(); 3*Log(3)-Pi(R)*Sqrt(3)/3; // G. C. Greubel, Mar 24 2024

RealDigits[Sum[2/(3*n^2-n), {n,1,Infinity}], 10, 111][[1]]
RealDigits[3*Log[3] - Pi*Sqrt[3]/3, 10, 140][[1]] (* G. C. Greubel, Mar 24 2024 *)

numerical_approx(3*log(3)-pi*sqrt(3)/3, digits=139) # G. C. Greubel, Mar 24 2024

Sum_{n>=1} 2/(3*n^2 - n).
Equals 3*log(3) - Pi*sqrt(3)/3 = A016650 - A093602. - Michel Marcus, Jul 03 2014
Equals 2*A294514. - Hugo Pfoertner, Apr 24 2025

A294515 Denominators of partial sums of the reciprocals of the decagonal numbers.

Original entry on oeis.org

1, 10, 270, 7020, 119340, 835380, 4176900, 242260200, 888287400, 32866633800, 1347531985800, 4042595957400, 28298171701800, 1499803100195400, 28496258903712600, 3476543586252937200, 3476543586252937200, 26653500827939185200, 1945705560439560519600, 1945705560439560519600, 52534050131868134029200

Offset: 0

Wolfdieter Lang, Nov 02 2017

The corresponding numerators are given by A250551(n+1), n >= 0.
The positive decagonal numbers are A001107(k+1) = (k + 1)*(4*k + 1), k >= 0.
For the general case V(m,r;n) = Sum_{k=0..n} 1/((k + 1)*(m*k + r)) = (1/(m - r))*Sum_{k=0..n} (m/(m*k + r) - 1/(k+1)), for r = 1, ..., m-1 and m = 2, 3, ..., and their limits see a comment in A294512. Here [m,r] = [4,1].
The limit of the series is V(4,1) = lim_{n -> oo} V(4,1;n) = log(2) + Pi/6 = 1.216745956158244182... given in A244647.

			The rationals V(4,1;n), n >= 0, begin: 1, 11/10, 307/270, 8117/7020, 139393/119340, 982381/835380, 4935773/4176900, 287319059/242260200, 1056494083/888287400, 39179109811/32866633800, ...
V(4,1;10^4) = 1.216720959 (Maple, 10 digits) to be compared with 1.216745956 from V(4,1) from A244647.

Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, pp. 189 - 193.

Robert Israel, Table of n, a(n) for n = 0..866

Cf. A001107, A244647, A250550, A250551.

map(denom,ListTools:-PartialSums([seq(1/((k+1)*(4*k+1)),k=0..50)])); # Robert Israel, Nov 08 2017

Denominator@ Accumulate@ Array[1/PolygonalNumber[10, #] &, 23] (* Michael De Vlieger, Nov 02 2017 *)

a(n) = denominator(V(4,1;n)) with V(4,1;n) = Sum_{k=0..n} 1/((k + 1)*(4*k + 1)) = Sum_{k=0..n} 1/A001107(n+1) = (1/3)*Sum_{k=0..n} (4/(4*k + 1) - 1/(k+1)).
a(n) = A250550(n+1)/(n+1) [conjecture].
In the Koecher reference v_4(1) = (3/4)*V(4,1) = (3/4)*log(2) + Pi/8 = 0.91255946711868313687... .

A365522 Decimal expansion of (Pisqrt(3) + 9log(3))/24.

Original entry on oeis.org

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

Offset: 0

Claude H. R. Dequatre, Sep 08 2023

This sequence is also the decimal expansion of Sum_{k>=1} 1/(f(k) +g(k)), where f(k) and g(k) are respectively the k-th triangular and the 13-gonal numbers (A000217 and A051865).

			0.63870452877981836559747674605121660577831724019512...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Michael Ian Shamos, A catalog of the real numbers (2011), p. 544.
Wikipedia, Polygonal number.

Cf. A000217, A000796, A002194, A002391, A051865.

Cf. A244639, A244641, A244645, A244646, A244647, A244648, A244649.

SetDefaultRealField(RealField(139)); R:= RealField(); (Pi(R)*Sqrt(3)+9*Log(3))/24; // G. C. Greubel, Mar 24 2024

RealDigits[(Pi*Sqrt[3] + 9*Log[3])/24, 10 , 100][[1]] (* Amiram Eldar, Sep 08 2023 *)

PARI
```
(Pi*sqrt(3)+9*log(3))/24
```

numerical_approx((pi*sqrt(3)+9*log(3))/24, digits=139) # G. C. Greubel, Mar 24 2024

Equals Sum_{k>=1} 1/(6*k^2 - 4*k) = A244645/2 [Shamos].

Equals - Integral_{x=0..1} log(1-x^6)/x^5 dx [Shamos].

cp's OEIS Frontend

A001107 10-gonal (or decagonal) numbers: a(n) = n*(4*n-3).

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A244645 Decimal expansion of the sum of the reciprocals of the octagonal numbers (A000567).

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A244646 Decimal expansion of the sum of the reciprocals of the 9-gonal (or enneagonal or nonagonal) numbers (A001106).

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A244649 Decimal expansion of the sum of the reciprocals of the Dodecagonal numbers (A051624).

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A244648 Decimal expansion of the sum of the reciprocals of the hendecagonal numbers (A051682).

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A244641 Decimal expansion of the sum of the reciprocals of the pentagonal numbers (A000326).

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A294515 Denominators of partial sums of the reciprocals of the decagonal numbers.

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A001107 10-gonal (or decagonal) numbers: a(n) = n(4n-3).

A365522 Decimal expansion of (Pisqrt(3) + 9log(3))/24.