A069072 -id:A069072 - cp's OEIS frontend

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

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

A016627 Decimal expansion of log(4).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

This constant (negated) is the 1-dimensional analog of Madelung's constant. - Jean-François Alcover, May 20 2014
This constant is the sum over the reciprocals of the hexagonal numbers A000384(n), n >= 1. See the Downey et al. link, and the formula by Robert G. Wilson v below. - Wolfdieter Lang, Sep 12 2016
log(4) - 1 is the mean ratio between the smaller length and the larger length of the two parts of a stick that is being broken at a point that is uniformly chosen at random (Mosteller, 1965). - Amiram Eldar, Jul 25 2020
From Bernard Schott, Sep 11 2020: (Start)
This constant was the subject of the problem B5 during the 42nd Putnam competition in 1981 (see formula Sep 11 2020 and Putnam link).
Jeffrey Shallit generalizes this result obtained for base 2 to any base b (see Amer. Math. Month. link): Sum_{k>=1} digsum(k)_b / (k*(k+1)) = (b/(b-1)) * log(b) where digsum(k)_b is the sum of the digits of k when expressed in base b (for base 10 see A334388). (End)

			1.38629436111989061883446424291635313615100026872051050824136...

Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 2.
Frederick Mosteller, Fifty challenging problems of probability, Dover, New York, 1965. See problem 42, pp. 10 and 63.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 2, equation 2:13:8 at page 23.

Harry J. Smith, Table of n, a(n) for n = 1..20000
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
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.
I. S. Gradsteyn and I. M. Ryzhik, Table of integrals, series and products (6th ed.), 2000, (eq. 0.236.1 and 0.236.6).
D. H. Lehmer, Interesting series involving the Central Binomial Coefficient, Am. Math. Monthly, Vol. 92, No. 7 (1985) pp. 449-457.
Allon G. Percus, Gabriel Istrate, Bruno Goncalves, Robert Z. Sumi, and Stefan Boettcher, The Peculiar Phase Structure of Random Graph Bisection, arXiv:0808.1549 [cond-mat.stat-mech], 2008.
H.-J. Seiffert, Problem B-771, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 32, No. 4 (1994), p. 374; More Sums, Solution to Problem B-771 by Don Redmond, ibid., Vol. 33, No. 5 (1995), pp. 470-471.
J. O. Shallit, Solutions of Advanced Problems, 6450, The American Mathematical Monthly, Vol. 92, No. 7, Aug.-Sep., 1985, pp. 513-514; DOI: 10.2307/2322523.
42nd Putnam Competition, Problem B5, 1981.
Index entries for sequences related to Olympiads and other Mathematical competitions.
Index entries for transcendental numbers.

Cf. A016732 (continued fraction).
Cf. A002162 (half), A133362 (reciprocal).
Cf. A000384, A001787, A002378, A007531, A066066, A069072, A191907.
Cf. A000045, A000120, A334388.

RealDigits[Log@ 4, 10, 111][[1]] (* Robert G. Wilson v, Aug 31 2014 *)

default(realprecision, 20080); x=log(4); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b016627.txt", n, " ", d)); \\ Harry J. Smith, May 16 2009, corrected May 19 2009

A016627_vec(N)=digits(floor(log(precision(4.,N))*10^(N-1))) \\ Or: default(realprecision,N);digits(log(4)\.1^N) \\ M. F. Hasler, Oct 20 2013

log(4) = Sum_{k >= 1} H(k)/2^k where H(k) is the k-th harmonic number. - Benoit Cloitre, Jun 15 2003
Equals 1 - Sum_{k >= 1} (-1)^k/A002378(k) = 1 + 2*Sum_{k >= 0} 1/A069072(k) = 5/4 - Sum_{k >= 1} (-1)^k/A007531(k+2). - R. J. Mathar, Jan 23 2009
Equals 2*A002162 = Sum_{n >= 1} binomial(2*n, n)/(n*4^n) [D. H. Lehmer, Am. Math. Monthly 92 (1985) 449 and Jolley eq. 262]. - R. J. Mathar, Mar 04 2009
log(4) = Sum_{k >= 1} A191907(4, k)/k, (conjecture). - Mats Granvik, Jun 19 2011
log(4) = lim_{n -> infinity} A066066(n)/n. - M. F. Hasler, Oct 20 2013
Equals Sum_{k >= 1} 1/( 2*k^2 - k ). - Robert G. Wilson v, Aug 31 2014
Equals gamma(0, 1/2) - gamma(0, 1) = -(EulerGamma + polygamma(0, 1/2)), where gamma(n,x) denotes the generalized Stieltjes constants, see A020759. - Peter Luschny, May 16 2018
From Amiram Eldar, Jul 25 2020: (Start)
Equals Sum_{k>=1} (3/4)^k/k.
Equals Sum_{k>=1} 1/(k*2^(k-1)) = Sum_{k>=1} 1/A001787(k).
Equals Integral_{x=0..1} log(1+1/x) dx. (End)
Equals Sum_{k>=1} A000120(k) / (k*(k+1)). - Bernard Schott, Sep 11 2020
Equals 1 + Sum_{k>=1} zeta(2*k+1)/4^k. - Amiram Eldar, May 27 2021
Equals Sum_{k>=1} (2*k+1)*Fibonacci(k)/(k*(k+1)*2^k) (Seiffert, 1994). - Amiram Eldar, Jan 15 2022
Continued fraction: log(4) = 1 + 1/(2 + (1*2)/(2 + (2*3)/(2 + (3*4)/(2 + (4*5)/(2 + ... ))))) due to Euler. - Peter Bala, Mar 05 2024
log(4) = 2*Sum_{k>=1} 1/(k*P(k, 5/3)*P(k-1, 5/3)), where P(k, x) denotes the k-th Legendre polynomial. The first 20 terms of the series gives log(4) correct to 18 decimal places. - Peter Bala, Mar 18 2024
Equals Sum_{k>=1} (2*k - 1)!!/(k*(2*k)!!) [Ross] (see Spanier at p. 23). - Stefano Spezia, Dec 27 2024
Equals 1 + Sum_{k>=1} 1/(k*(4*k^2-1)). - Sean A. Irvine, Apr 05 2025
Equals Sum_{k>=1} (12*k^2-1)/(k*(4*k^2-1)^2). - Sean A. Irvine, Apr 06 2025
Equals Integral_{x=0..1} arctanh(sqrt(x))/sqrt(x) dx. - Kritsada Moomuang, Jun 06 2025
From Kritsada Moomuang, Jun 18 2025: (Start)
Equals Integral_{x=0..1} (x^(n - 1)*(x^(3*n) - 1))/log(x) dx, for n > 0.
Equals Integral_{x=0..Pi} sin(x)/(1 + abs(cos(x))) dx. (End)

A097321 a(n) = (3n-1) 3n (3*n+1).

Original entry on oeis.org

24, 210, 720, 1716, 3360, 5814, 9240, 13800, 19656, 26970, 35904, 46620, 59280, 74046, 91080, 110544, 132600, 157410, 185136, 215940, 249984, 287430, 328440, 373176, 421800, 474474, 531360, 592620, 658416, 728910, 804264, 884640, 970200, 1061106, 1157520

Offset: 1

Ralf Stephan, Aug 07 2004

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
S. Ramanujan, Notebook entry.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A069072, A069140, A002391, A016767.

[27*n^3-3*n: n in [1..40]]; // Vincenzo Librandi, Sep 07 2011

Table[27n^3-3n,{n,40}]  (* Harvey P. Dale, Mar 30 2011 *)

G.f.: 6x * (4x^2 + 19x + 4) / (1-x)^4.
Sum_{n>=1} 1/a(n) = (log(3) - 1)/2. - Amiram Eldar, Jul 04 2020
Sum_{n>=1} (-1)^(n+1)/a(n) = 1/2 - 2*log(2)/3. - Amiram Eldar, May 15 2022
E.g.f.: 3*exp(x)*x*(8 + 27*x + 9*x^2). - Stefano Spezia, Feb 20 2025

A046142 Haüy rhombic dodecahedral numbers.

Original entry on oeis.org

1, 33, 185, 553, 1233, 2321, 3913, 6105, 8993, 12673, 17241, 22793, 29425, 37233, 46313, 56761, 68673, 82145, 97273, 114153, 132881, 153553, 176265, 201113, 228193, 257601, 289433, 323785, 360753, 400433, 442921, 488313, 536705, 588193, 642873, 700841

Offset: 1

Eric W. Weisstein

The Haüy rhombic dodecahedral formula is remarkably similar to that of A254473, the 24-hedral numbers: a(n) = (2*n+1)*(8*n^2+14*n+7). Compare with (2*n-1)*(8*n^2-14*n+7); the differences are simple: (1) the first factor of the dodecahedral formula has "+1" and the 24-hedral formula has "-1"; (2) the second factor of the former has "-14n" and the latter has "+14n". Note that the rhombic dodecahedron has 24 edges. The difference between these sequences is diff(n) = 72*n^2 + 14. - Peter M. Chema, Jan 09 2016

Named after the French priest and mineralogist René Just Haüy (1743-1822). - Amiram Eldar, Jun 22 2021

H. Steinhaus, Mathematical Snapshots, 3rd ed. New York: Dover, pp. 185-186, 1999.

G. C. Greubel, Table of n, a(n) for n = 1..5000
R.-J. Haüy, Essai d'une théorie sur la structure des crystaux appliquée à plusieurs genres de substances crystallisées, 1784.
Jonathan Vos Post, Table of Polytope Numbers, Sorted, Through 1,000,000 which lists Haüy rhombic dodecahedral numbers as "RhoDod(n)."
Eric Weisstein's World of Mathematics, Haüy Construction.
Eric Weisstein's World of Mathematics, Rhombic Dodecahedral Number.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000447, A001845, A016755, A069072, A254473.

[(2*n-1)*(8*n^2-14*n+7): n in [1..40]]; // Vincenzo Librandi, Mar 29 2015

A046142:=n->(2*n-1)*(8*n^2-14*n+7): seq(A046142(n), n=1..50); # Wesley Ivan Hurt, Mar 02 2016

Table[(2 n - 1) (8 n^2 - 14 n + 7), {n, 40}] (* Vincenzo Librandi, Mar 29 2015 *)
LinearRecurrence[{4, -6, 4, -1}, {1, 33, 185, 553}, 20] (* Eric W. Weisstein, Sep 27 2017 *)
CoefficientList[Series[(1 + 29 x + 59 x^2 + 7 x^3)/(-1 + x)^4, {x, 0, 20}], x] (* Eric W. Weisstein, Sep 27 2017 *)

Vec(x*(7*x^3+59*x^2+29*x+1)/(x-1)^4 + O(x^50)) \\ Michel Marcus, Mar 24 2015

a(n) = (2*n - 1)*(8*n^2 - 14*n + 7).
G.f.: x*(7*x^3 +59*x^2 +29*x +1)/(1-x)^4. - Colin Barker, Oct 26 2012
a(n) = A016755(n) + A069072(n-1). - Luciano Ancora, Mar 23 2015
a(n) = A016755(n) + 6*A000447(n-1). - Luciano Ancora, Mar 23 2015
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>4. - Wesley Ivan Hurt, Mar 02 2016
E.g.f.: (-7 +8*x +12*x^2 +16*x^3)*exp(x) + 7. - G. C. Greubel, Nov 04 2017

A069140 a(n) = (4n-1)4n(4n+1).

Original entry on oeis.org

0, 60, 504, 1716, 4080, 7980, 13800, 21924, 32736, 46620, 63960, 85140, 110544, 140556, 175560, 215940, 262080, 314364, 373176, 438900, 511920, 592620, 681384, 778596, 884640, 999900, 1124760, 1259604, 1404816, 1560780, 1727880, 1906500

Offset: 0

Henry Bottomley, Apr 08 2002

			a(10) = 39*40*41 = 63960.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Henry Bottomley, Medians and area bisectors of a triangle.
S. Ramanujan, Notebook entry.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A007531, A069072, A097321.

Table[64n^3-4n,{n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{0,60,504,1716},40] (* Harvey P. Dale, Dec 24 2012 *)

a(n)=(4*n-1)*4*n*(4*n+1) \\ Charles R Greathouse IV, Oct 07 2015

Sum_{i>0} 1/a(i) = log(2)*3/4 - 1/2 = 0.019860..., which is the ratio of the area of the deltoid envelope formed by area bisectors of a triangle to the area of the triangle.
a(n) = 64n^3 - 4n = A007531(4n) = A069072(2n-1).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Harvey P. Dale, Dec 24 2012
Sum_{n>=1} (-1)^(n+1)/a(n) = 1/2 - log(2)/4 + log(tan(Pi/8))/(2*sqrt(2)). Amiram Eldar, Mar 20 2022

A322170 Triangle T(n, k) read by rows, n > 0 and 0 < k <= 3^(n-1): T(n, k) = A321768(n, k) * A321769(n, k) / 2.

Original entry on oeis.org

6, 30, 210, 60, 84, 1320, 630, 1560, 7140, 1386, 924, 2340, 210, 180, 4620, 2730, 10920, 45144, 7854, 7980, 23184, 2574, 5016, 63336, 26910, 49476, 242556, 50490, 25200, 57420, 4290, 3570, 34650, 12540, 14490, 79794, 18564, 5610, 10374, 504, 330, 11970, 7956

Offset: 1

Rémy Sigrist, Nov 29 2018

This sequence gives the areas of the primitive Pythagorean triangles corresponding to the primitive Pythagorean triples in the tree described in A321768.

If we order the terms in this sequence and keep duplicates then we obtain A024406.

			The first rows are:
   6
   30, 210, 60
   84, 1320, 630, 1560, 7140, 1386, 924, 2340, 210
T(1,1) corresponds to the area of the triangle with sides 3, 4, 5; hence T(1, 1) = 3 * 4 / 2 = 6.

Rémy Sigrist, Rows n = 1..9, flattened
Index entries related to Pythagorean Triples

Cf. A024406, A029549, A055112, A069072, A321768, A321769.

M = [[1, -2, 2; 2, -1, 2; 2, -2, 3], [1, 2, 2; 2, 1, 2; 2, 2, 3], [-1, 2, 2; -2, 1, 2; -2, 2, 3]];
T(n, k) = my (t=[3; 4; 5], d=digits(3^(n-1)+k-1, 3)); for (i=2, #d, t = M[d[i]+1] * t); return (t[1, 1] * t[2, 1] / 2)

Empirically:
- T(n, 1) = A055112(n),
- T(n, (3^(n-1) + 1)/2) = A029549(n),
- T(n, 3^(n-1)) = A069072(n-1).

A342313 T(n, k) = (n + k - 1)(n + k)binomial(2*n + 1, n - k + 1) with T(0, 0) = T(1, 0) = 1. Triangle read by rows, T(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 1, 6, 20, 60, 60, 210, 420, 420, 210, 1512, 2520, 2520, 1512, 504, 9240, 13860, 13860, 9240, 3960, 990, 51480, 72072, 72072, 51480, 25740, 8580, 1716, 270270, 360360, 360360, 270270, 150150, 60060, 16380, 2730, 1361360, 1750320, 1750320, 1361360, 816816, 371280, 123760, 28560, 4080

Offset: 0

Peter Luschny, Mar 08 2021

The triangle can be seen as representing the denominators of a sequence of rational polynomials. Let p_{n}(x) = Sum_{k=0..n} (A342312(n, k)/T(n, k))*x^k. Then p_{n}(1) = B_{n}(1), where B_{n}(x) are the Bernoulli polynomials. See A342312 for a formula for the polynomials.

			Triangle starts:
[0] 1
[1] 1,       6
[2] 20,      60,      60
[3] 210,     420,     420,     210
[4] 1512,    2520,    2520,    1512,    504
[5] 9240,    13860,   13860,   9240,    3960,   990
[6] 51480,   72072,   72072,   51480,   25740,  8580,   1716
[7] 270270,  360360,  360360,  270270,  150150, 60060,  16380,  2730
[8] 1361360, 1750320, 1750320, 1361360, 816816, 371280, 123760, 28560, 4080

Cf. A069072 (main diagonal), A342312 (numerators).

T := (n, k) -> `if`(n=0, 1,`if`(n=1 and k=0, 1,
(n + k - 1)*(n + k)*binomial(2*n + 1, n - k + 1))):
seq(print(seq(T(n, k), k = 0..n)), n = 0..8);

T[0, 0] := 1; T[1, 0] := 1;
T[n_, k_] := (n - 1 + k) (n + k) Binomial[2n + 1, n - k + 1];
Table[T[n, k], {n, 0, 8}, {k, 0, n}]

A237576 Smallest integer areas of integer-sided triangles such that the perimeter equals n times the smallest side.

Original entry on oeis.org

0, 0, 0, 6, 60, 30, 210, 24, 84, 60, 198, 330, 1716, 546, 2730, 252, 4080, 36, 5814, 210, 7980, 2310, 10626, 924, 1380, 1248, 90, 4914, 4176, 6090, 26970, 480, 32736, 1224, 39270, 1938, 46620, 2394, 54834, 4560, 63960, 4620, 74046, 19866, 85140, 22770, 97290

Offset: 1

Michel Lagneau, Feb 09 2014

nonn

The area A of a triangle whose sides have lengths a, b, and c is given by Heron's formula: A = sqrt(s*(s-a)*(s-b)*(s-c)), where s = (a+b+c)/2.
The sequence a(n) is the union of four subsequences A, B, C and D where:
A is the subsequence with areas 60, 210, 1716, 2730, 4080, 5814, 7980, 10626, ... where n is odd, and the corresponding sides are of the form (4k, 4k^2-1, 4k^2+1) with areas 2k(4k^2-1) for k = 2, 3, 6, 7, 8, 9, 11, ... These areas are in the sequence A069072 (areas of primitive Pythagorean triangles whose odd sides differ by 2).
B is the subsequence with areas 6, 30, 84, 330, 546, 2310, 4914, 6090, ... where n is even, and the corresponding sides are of the form (2k+1, 2k(k+1), 2k(k+1)+1) with areas k(k+1)(2k+1) for k = 1, 2, 3, 5, 6, 7, 10, 13, 14, ... These areas are in the sequence A055112 (Areas of Pythagorean triangles (a, b, c) with c = b+1).
C is the subsequence with areas 84, 198, 1380, 4176, ... where n is odd but the areas are not Pythagorean triangles.
D is the subsequence with areas 24, 60, 210, 924, 1248, 480, 1224, 1938, ... where n is even but the areas are not Pythagorean triangles.
The triangles with the same areas are not unique; for example:
(8, 15, 17) and (6, 25, 29) => A = 60; the first is a Pythagorean triangle, the second is not.
(12, 35, 37) and (7, 65, 68) => A = 210; the first is a Pythagorean triangle, the second is not.
The following table gives the first values (n, A, p, a, b, c) where A is the area of the triangles, p is the perimeter and a, b, c are the sides.
+----+------+-------------+----+-----+-----+
|  n |   A  |      p      |  a |  b  |  c  |
+----+------+-------------+----+-----+-----+
|  4 |    6 |  12 = 4*3   |  3 |   4 |   5 |
|  5 |   60 |  40 = 5*8   |  8 |  15 |  17 |
|  6 |   30 |  30 = 6*5   |  5 |  12 |  13 |
|  7 |  210 |  84 = 7*12  | 12 |  35 |  37 |
|  8 |   24 |  32 = 8*4   |  4 |  13 |  15 |
|  9 |   84 |  72 = 9*8   |  8 |  29 |  35 |
| 10 |   60 |  60 = 10*6  |  6 |  25 |  29 |
| 11 |  198 | 132 = 11*12 | 12 |  55 |  65 |
| 12 |  330 | 132 = 12*11 | 11 |  60 |  61 |
| 13 | 1716 | 312 = 13*24 | 24 | 143 | 145 |
| 14 |  546 | 182 = 14*13 | 13 |  84 |  85 |
| 15 | 2730 | 420 = 15*28 | 28 | 195 | 197 |
+----+------+-------------+----+-----+-----+

Cf. A188158.

with(numtheory):nn:=600:for n from 4 to 50 do: ii:=0:for a from 1
  to nn while(ii=0) do: for b from a to nn while(ii=0) do: for c from b to nn while(ii=0) do: p:=(a+b+c)/2 : x:=p*(p-a)*(p-b)*(p-c): if x>0 then x0:= sqrt(x):else fi:if x0=floor(x0) and 2*p=n*a then ii:=1:printf ( "%d %d %d %d %d \n",n,x0,a,b,c):else fi:od:od:od:od:

nn=600;lst={};Do[k=0;Do[s=(a+b+c)/2;If[IntegerQ[s],area2=s (s-a) (s-b) (s-c);If[0

cp's OEIS Frontend

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A322170 Triangle T(n, k) read by rows, n > 0 and 0 < k <= 3^(n-1): T(n, k) = A321768(n, k) * A321769(n, k) / 2.

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

A097321 a(n) = (3n-1) 3n (3*n+1).

A069140 a(n) = (4n-1)4n(4n+1).

A342313 T(n, k) = (n + k - 1)(n + k)binomial(2*n + 1, n - k + 1) with T(0, 0) = T(1, 0) = 1. Triangle read by rows, T(n, k) for 0 <= k <= n.