A035328 -id:A035328 - 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

A069072 a(n) = (2n+1)(2n+2)(2n+3).

Original entry on oeis.org

6, 60, 210, 504, 990, 1716, 2730, 4080, 5814, 7980, 10626, 13800, 17550, 21924, 26970, 32736, 39270, 46620, 54834, 63960, 74046, 85140, 97290, 110544, 124950, 140556, 157410, 175560, 195054, 215940, 238266, 262080, 287430, 314364, 342930

Offset: 0

Benoit Cloitre, Apr 05 2002

Terms are areas of primitive Pythagorean triangles whose odd sides differ by 2; e.g., the triangle with sides 8,15,17 has area 60. - Lekraj Beedassy, Apr 18 2003

Using (n, n+1), (n, n+2), and (n+1, n+2) to generate three unreduced Pythagorean triangles gives a sum of the areas for all three to be (2*n+1)*(2*n+2)*(2*n+3), which are three consecutive numbers. - J. M. Bergot, Aug 22 2011

T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 190.
Jolley, Summation of Series, Oxford (1961).
Konrad Knopp, Theory and application of infinite series, Dover, p. 269.

M. Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
Konrad Knopp, Theorie und Anwendung der unendlichen Reihen, Berlin, J. Springer, 1922. (Original german edition of "Theory and Application of Infinite Series")
S. Ramanujan, Notebook entry
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A097321, A069140, A000447, A035328.

Array[Times@@(2#+{1,2,3})&,40,0] (* or *) LinearRecurrence[{4,-6,4,-1},{6,60,210,504},40] (* Harvey P. Dale, Dec 08 2013 *)

a(n)=(2*n+1)*(2*n+2)*(2*n+3) \\ Charles R Greathouse IV, Oct 07 2015

log(2) - 1/2 = Sum_{n>=0} 1/a(n); (1/2)*(1-log(2)) = Sum_{n>=0} (-1)^n/a(n). [Jolley eq 236 and 237]

Sum_{n>=0} x^n/a(n) = ((1+x)/sqrt(x)*log((1+sqrt x)/(1-sqrt x)) + 2*log(1-x)-2)/(4x). [Jolley eq 280 for 0

Sum_{n>=0} (-x)^n/a(n) = (1-log(1+x) -(1-x)/sqrt(x)*arctan(x))/(2x). [Jolley eq 281 for 0

a(n) = 6*A000447(n+1). - Lekraj Beedassy, Apr 18 2003

G.f.: 6*(1 + 6*x + x^2) / (x-1)^4 . - R. J. Mathar, Jun 09 2013

a(0)=6, a(1)=60, a(2)=210, a(3)=504, a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Harvey P. Dale, Dec 08 2013

a(n) = 2*A035328(n+1). - J. M. Bergot, Jan 02 2015

A162917 Numbers n which are not in A161983.

Original entry on oeis.org

1, 2, 6, 7, 8, 9, 15, 16, 17, 18, 19, 20, 28, 29, 30, 31, 32, 33, 34, 35, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 120, 121, 122, 123, 124, 125, 126, 127, 128

Offset: 1

Juri-Stepan Gerasimov, Jul 17 2009

They have a block structure: A000384(n) is followed by the next n integers.
Numbers k with the property that the largest Dyck path of the symmetric representation of sigma(k) has a central peak. (Cf. A237593.) - Omar E. Pol, Aug 28 2018
Union of nonzero terms of A000384 and A317303. - Omar E. Pol, Aug 29 2018

			From _Omar E. Pol_, Aug 28 2018: (Start)
Also, written as an irregular triangle in which the row lengths are the positive even numbers, the sequence begins:
   1,  2;
   6,  7,  8,  9;
  15, 16, 17, 18, 19, 20;
  28, 29, 30, 31, 32, 33, 34, 35;
  45, 46, 47, 48, 49, 50, 51, 52, 53,  54;
  66, 67, 68, 69, 70, 71, 72, 73, 74,  75,  76,  77;
  91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104;
...
Row sums give A035328, n >= 1.
Column 1 gives A000384, n >= 1.
Column 2 gives A130883, n >= 1.
Right border gives the positive terms of A014107, also the odd-indexed terms of A000096.
(End)

Complement of A161983.

Cf. A000096, A000384, A014107, A035328, A130883, A237593, A317303.

A000027 \ A161983.

119 removed by R. J. Mathar, Jul 19 2009

A204558 Row sums of the triangle A045975.

Original entry on oeis.org

1, 6, 45, 120, 325, 630, 1225, 2016, 3321, 4950, 7381, 10296, 14365, 19110, 25425, 32640, 41905, 52326, 65341, 79800, 97461, 116886, 140185, 165600, 195625, 228150, 266085, 306936, 354061, 404550, 462241, 523776, 593505, 667590, 750925, 839160, 937765

Offset: 1

Reinhard Zumkeller, Jan 18 2012

nonn

a(n) = sum(A045975(n,k): k = 1 .. n);

a(n) = n * (if n even then A035328(n/2) else A005817((n+1)/2)). [Reinhard Zumkeller, Jan 18 2012]

Haskell
```
a204558 = sum . a045975_row
```

Conjecture: a(n) = (n^2*(-(-1)^n+n^2))/2. G.f.: -x*(x^6 +4*x^5 +31*x^4 +24*x^3 +31*x^2 +4*x +1) / ((x -1)^5*(x +1)^3). - Colin Barker, Apr 02 2013

A365372 Array read by ascending antidiagonals: A(n, k) = n(kn^2 - 1) with k > 0.

Original entry on oeis.org

0, 6, 1, 24, 14, 2, 60, 51, 22, 3, 120, 124, 78, 30, 4, 210, 245, 188, 105, 38, 5, 336, 426, 370, 252, 132, 46, 6, 504, 679, 642, 495, 316, 159, 54, 7, 720, 1016, 1022, 858, 620, 380, 186, 62, 8, 990, 1449, 1528, 1365, 1074, 745, 444, 213, 70, 9, 1320, 1990, 2178, 2040, 1708, 1290, 870, 508, 240, 78, 10

Offset: 1

Stefano Spezia, Sep 02 2023

			The array begins:
    0,   1,   2,   3,    4,    5, ...
    6,  14,  22,  30,   38,   46, ...
   24,  51,  78, 105,  132,  159, ...
   60, 124, 188, 252,  316,  380, ...
  120, 245, 370, 495,  620,  745, ...
  210, 426, 642, 858, 1074, 1290, ...
  ...

Cf. A007531, A017137, A035328 (k=4), A058895 (main diagonal), A365373 (antidiagonal sums).

A[n_,k_]:=n(k n^2-1); Table[A[n-k+1,k],{n,11},{k,n}]//Flatten

G.f.: x*y*(x^2*y + y - 2*x*(y - 3))/((1 - x)^4*(1 - y)^2).
1st column: A(n, 1) = A007531(n+1).
2nd row: A(2, n) = A017137(n-1).

Showing 1-5 of 5 results.

cp's OEIS Frontend

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

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A069072 a(n) = (2n+1)*(2n+2)*(2n+3).

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A162917 Numbers n which are not in A161983.

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A204558 Row sums of the triangle A045975.

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A365372 Array read by ascending antidiagonals: A(n, k) = n*(k*n^2 - 1) with k > 0.

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Programs

Mathematica

Formula

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

A069072 a(n) = (2n+1)(2n+2)(2n+3).

A365372 Array read by ascending antidiagonals: A(n, k) = n(kn^2 - 1) with k > 0.