A033581 -id:A033581 - cp's OEIS frontend

A085250 4 times hexagonal numbers: a(n) = 4n(2*n-1).

0, 4, 24, 60, 112, 180, 264, 364, 480, 612, 760, 924, 1104, 1300, 1512, 1740, 1984, 2244, 2520, 2812, 3120, 3444, 3784, 4140, 4512, 4900, 5304, 5724, 6160, 6612, 7080, 7564, 8064, 8580, 9112, 9660, 10224, 10804, 11400, 12012, 12640, 13284

Offset: 0

Gary W. Adamson, Jun 23 2003

a(n) also can represented as n concentric squares (see example). - Omar E. Pol, Aug 21 2011

Sequence found by reading the line from 0, in the direction 0, 4, ..., in the square spiral whose vertices are the triangular numbers A000217. - Omar E. Pol, Sep 03 2011

			From _Omar E. Pol_, Aug 21 2011: (Start)
Illustration of initial terms as concentric squares:
.
.                           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   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
.
.      4          24                 60
.
(End)

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A067239, A000384, A033581, A046092, A152734, A152751, A194274.

Table[8*n^2 - 4*n, {n, 0, 50}] (* G. C. Greubel, Jul 14 2017 *)
4 PolygonalNumber[6,Range[0,50]] (* Harvey P. Dale, Oct 19 2022 *)

a(n)=4*n*(2*n-1) \\ Charles R Greathouse IV, Sep 24 2015

a(n) = A067239(n)/2, for n>0.
Sum_{n>0} 1/a(n) = log(2)/2.
a(n) = A000384(n)*4. - Omar E. Pol, Dec 11 2008
a(n) = 16*n + a(n-1) - 12 (with a(0)=0). - Vincenzo Librandi, Aug 08 2010
G.f.: 4*x*(1 + 3*x)/(1 - 3*x + 3*x^2 - x^3). - Colin Barker, Jan 04 2012
E.g.f.: 4*x*(2*x + 1)*exp(x). - G. C. Greubel, Jul 14 2017
a(n) = A046092(2n-1), for n > 0. - Bruce J. Nicholson, Sep 04 2017
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/8 - log(2)/4. - Amiram Eldar, Mar 17 2022

Edited by Don Reble, Nov 13 2005

Added zero, better definition, corrected offset and edited original formula. - Omar E. Pol, Dec 11 2008

A144555 a(n) = 14*n^2.

Original entry on oeis.org

0, 14, 56, 126, 224, 350, 504, 686, 896, 1134, 1400, 1694, 2016, 2366, 2744, 3150, 3584, 4046, 4536, 5054, 5600, 6174, 6776, 7406, 8064, 8750, 9464, 10206, 10976, 11774, 12600, 13454, 14336, 15246, 16184, 17150, 18144, 19166, 20216, 21294, 22400, 23534, 24696

Offset: 0

N. J. A. Sloane, Jan 01 2009

Sequence found by reading the line from 0, in the direction 0, 14, ..., in the square spiral whose vertices are the generalized enneagonal numbers A118277. Also sequence found by reading the same line and direction in the square spiral whose edges have length A195019 and whose vertices are the numbers A195020. - Omar E. Pol, Sep 10 2011

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

See also A033428, A033429, A033581, A033582, A033583, A033584, ... and A249327 for the whole table.
Cf. A000290, A001105, A064761, A118277, A152742, A195019, A195020.
Cf. A008596, A195145.

Table[14*n^2, {n, 0, 45}] (* Amiram Eldar, Feb 03 2021 *)

A144555(n)=14*n^2 \\ M. F. Hasler, Oct 31 2014

a(n) = 14*A000290(n) = 7*A001105(n) = 2*A033582(n). - Omar E. Pol, Jan 01 2009
a(n) = a(n-1) + 14*(2*n-1), with a(0) = 0. - Vincenzo Librandi, Nov 25 2010
From Amiram Eldar, Feb 03 2021: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/84.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/168.
Product_{n>=1} (1 + 1/a(n)) = sqrt(14)*sinh(Pi/sqrt(14))/Pi.
Product_{n>=1} (1 - 1/a(n)) = sqrt(14)*sin(Pi/sqrt(14))/Pi. (End)
From Elmo R. Oliveira, Nov 30 2024: (Start)
G.f.: 14*x*(1 + x)/(1-x)^3.
E.g.f.: 14*x*(1 + x)*exp(x).
a(n) = n*A008596(n) = A195145(2*n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A152751 3 times octagonal numbers: a(n) = 3n(3*n-2).

Original entry on oeis.org

0, 3, 24, 63, 120, 195, 288, 399, 528, 675, 840, 1023, 1224, 1443, 1680, 1935, 2208, 2499, 2808, 3135, 3480, 3843, 4224, 4623, 5040, 5475, 5928, 6399, 6888, 7395, 7920, 8463, 9024, 9603, 10200, 10815, 11448, 12099, 12768, 13455, 14160, 14883, 15624, 16383, 17160

Offset: 0

Omar E. Pol, Dec 12 2008

a(n) also can be represented as n concentric triangles (see example). - Omar E. Pol, Aug 21 2011

			From _Omar E. Pol_, Aug 21 2011: (Start)
Illustration of initial terms as concentric triangles:
.
.                                          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           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
.
.    3            24                       63
(End)

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000567, A064201, A139267, A152995.
3 times n-gonal numbers: A045943, A033428, A062741, A094159, A152773, A152759, A152767, A153783, A153448, A153875.
Cf. A033581, A085250, A152734, A194273. - Omar E. Pol, Aug 21 2011
Cf. numbers of the form n*(n*k - k + 6)/2, this sequence is the case k=18: see Comments lines of A226492.

s=0;lst={s};Do[s+=n;AppendTo[lst,s],{n,3,6!,18}];lst (* Vladimir Joseph Stephan Orlovsky, Apr 02 2009 *)
3*PolygonalNumber[8,Range[0,40]] (* Harvey P. Dale, May 08 2022 *)

a(n)=3*n*(3*n-2) \\ Charles R Greathouse IV, Sep 24 2015

a(n) = 9*n^2 - 6*n = 3*A000567(n) = A064201(n)/3.
a(n) = a(n-1) + 18*n - 15 with n > 0, a(0)=0. - Vincenzo Librandi, Nov 26 2010
G.f.: 3*x*(1+5*x)/(1-x)^3. - Bruno Berselli, Jan 21 2011
From Elmo R. Oliveira, Dec 25 2024: (Start)
E.g.f.: 3*exp(x)*x*(1 + 3*x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3.
a(n) = n + A152995(n). (End)

A055112 a(n) = n(n+1)(2*n+1).

Original entry on oeis.org

0, 6, 30, 84, 180, 330, 546, 840, 1224, 1710, 2310, 3036, 3900, 4914, 6090, 7440, 8976, 10710, 12654, 14820, 17220, 19866, 22770, 25944, 29400, 33150, 37206, 41580, 46284, 51330, 56730, 62496, 68640, 75174, 82110, 89460, 97236, 105450

Offset: 0

Henry Bottomley, Jun 15 2000

Original name: Areas of Pythagorean triangles (X, Y, Z = Y + 1) with X^2 + Y^2 = Z^2.
a(n) is the set of possible y values for 4*x^3 + x^2 = y^2 with the x values being A002378(n). - Gary Detlefs, Feb 22 2010
This sequence is related to A028896 by a(n) = n*A028896(n) - Sum_{i = 0..n-1} A028896(i) and this is the case d = 3 in the identity n*(d*(d+1)*n*(n+1)/4) - Sum_{i = 0..n-1} d*(d+1)*i*(i+1)/4 = d*(d+1)*n*(n+1)*(2*n+1)/12. - Bruno Berselli, Mar 31 2012
Also sums of rows of natural numbers (cf. A001477) seen as triangle with an odd numbers of terms per row, see example. - Reinhard Zumkeller, Jan 24 2013
Without mentioning the connection to Pythagorean triangles, Bolker (1967) gives it as an exercise to prove that these numbers are always divisible by 6. This is easy to prove from the formula that he gives, n(n - 1)(2n - 1): obviously either n or (n - 1) must be even; then, if n is congruent to 2 mod 3 it means that (2n - 1) is a multiple of 3, otherwise either n or (n - 1) is a multiple of 3; thus both prime divisors of 6 are accounted for in a(n). - Alonso del Arte, Oct 13 2013
a(n) = n*(n+1)*(n+(n+1)) is the product of two consecutive integers multiplied by the sum of those two consecutive integers. - Charles Kusniec, Sep 04 2022

			.  n   A001477(n) as triangle with row lengths = 2*n+1   Row sums = a(n)
.  0                         0                                  0
.  1                      1  2  3                               6
.  2                   4  5  6  7  8                           30
.  3                9 10 11 12 13 14 15                        84
.  4            16 17 18 19 20 21 22 23 24                    180
.  5         25 26 27 28 29 30 31 32 33 34 35                 330
.  6      36 37 38 39 40 41 42 43 44 45 46 47 48              546
.  7   49 50 51 52 53 54 55 56 57 58 59 60 61 62 63           840 .
- _Reinhard Zumkeller_, Jan 24 2013

Ethan D. Bolker, Elementary Number Theory: An Algebraic Approach. Mineola, New York: Dover Publications (1969, reprinted 2007): p. 7, Problem 6.5.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
R. Roy, The discovery of the series formula for π by Leibniz, Gregory and Nilakantha, Mathematics Magazine, 63 (5) 1990, 291-306.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A005408 (X values), A046092 (Y values), A001844 (Z values), A002939 (perimeter), A033581.
Similar sequences are listed in A316224.
Cf. A000217, A000290, A005898, A083374.

Table[n(n + 1)(2n + 1), {n, 0, 39}] (* Vladimir Joseph Stephan Orlovsky, Nov 21 2010 *)
LinearRecurrence[{4,-6,4,-1},{0,6,30,84},50] (* Harvey P. Dale, Oct 02 2024 *)

PARI
```
a(n)=n*(n+1)*(2*n+1);
```

def A055112(n): return n*(n*((n<<1) + 3) + 1) # Chai Wah Wu, Nov 14 2022

a(n) = n*(n+1)*(2*n+1).
G.f.: 6*x*(1+x)/(1-x)^4. - Bruno Berselli, Mar 31 2012
From Benoit Cloitre, Apr 30 2002: (Start)
a(n) = 6*A000330(n) = A007531(2*n)/4 = 3*A000292(2*n-1)/2 = A005408(n)*A046092(n)/2 = A005408(n)*(A001844(n)-1)/2.
Sum_{n > 0} 1/a(n) = 3 - 4*log(2). (End)
a(n) = Sum_{i = 1..n} A033581(i). - Jonathan Vos Post, Mar 15 2006
a(n) = A000217(2*n)*A000217(2*n+1)/(2*n+1). - Charlie Marion, Feb 17 2012
a(n) = Sum_{i = 1..2*n + 1} (n^2 + (i-1)). - Charlie Marion, Sep 14 2012
Sum_{n >= 1} (-1)^(n+1)/a(n) = Pi - 3, due to Nilakantha, circa 1500. See Roy p. 304. - Peter Bala, Feb 19 2015
a(n) = A002378(n) * (2n+1). - Bruce J. Nicholson, Aug 31 2017
a(n) = Sum_{k=n^2..(n+1)^2-1} k. - Darío Clavijo, Jan 31 2025
E.g.f.: exp(x)*x*(6 + 9*x + 2*x^2). - Stefano Spezia, Feb 02 2025
a(n) = A005898(n) - A005408(n) = A083374(n+1) - A083374(n). - J.S. Seneschal, Jul 08 2025

A000459 Number of multiset permutations of {1, 1, 2, 2, ..., n, n} with no fixed points.

Original entry on oeis.org

1, 0, 1, 10, 297, 13756, 925705, 85394646, 10351036465, 1596005408152, 305104214112561, 70830194649795010, 19629681235869138841, 6401745422388206166420, 2427004973632598297444857, 1058435896607583305978409166, 526149167104704966948064477665

Offset: 0

N. J. A. Sloane

Original definition: Number of permutations with no hits on 2 main diagonals. (Identical to definition of A000316.) - M. F. Hasler, Sep 27 2015
Card-matching numbers (Dinner-Diner matching numbers): A deck has n kinds of cards, 2 of each kind. The deck is shuffled and dealt in to n hands with 2 cards each. A match occurs for every card in the j-th hand of kind j. A(n) is the number of ways of achieving no matches. The probability of no matches is A(n)/((2n)!/2!^n).
Also, Penrice's Christmas gift numbers (see Penrice 1991).
a(n) is the maximal number of totally mixed Nash equilibria in games of n players, each with 3 pure options. - Raimundas Vidunas, Jan 22 2014

			There are 297 ways of achieving zero matches when there are 2 cards of each kind and 4 kinds of card so a(4)=297.
From _Peter Bala_, Jul 08 2014: (Start)
a(3) = 10: the 10 permutations of the multiset {1,1,2,2,3,3} that have no fixed points are
{2,2,3,3,1,1}, {3,3,1,1,2,2}
{2,3,1,3,1,2}, {2,3,1,3,2,1}
{2,3,3,1,1,2}, {2,3,3,1,2,1}
{3,2,1,3,1,2}, {3,2,1,3,2,1}
{3,2,3,1,1,2}, {3,2,3,1,2,1}
(End)

F. N. David and D. E. Barton, Combinatorial Chance, Hafner, NY, 1962, Ch. 7 and Ch. 12.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 174-178.
R. P. Stanley, Enumerative Combinatorics Volume I, Cambridge University Press, 1997, p. 71.
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..100
Per Alexandersson and Frether Getachew, An involution on derangements, arXiv:2105.08455 [math.CO], 2021.
Shalosh B. Ekhad, Christoph Koutschan, and Doron Zeilberger, There are EXACTLY 1493804444499093354916284290188948031229880469556 Ways to Derange a Standard Deck of Cards (ignoring suits) [and many other such useful facts], arXiv:2101.10147 [math.CO], 2021.
F. F. Knudsen and I. Skau, On the Asymptotic Solution of a Card-Matching Problem, Mathematics Magazine 69 (1996), 190-197.
P. A. MacMahon, Combinatory Analysis Cambridge: The University Press 1915-1916
B. H. Margolius, The Dinner-Diner Matching Problem, Mathematics Magazine, 76 (2003), 107-118.
B. H. Margolius, Dinner-Diner Matching Probabilities
R. D. McKelvey and A. McLennan, The maximal number of regular totally mixed Nash equilibria, J. Economic Theory, 72:411--425, 1997.
L. I. Nicolaescu, Derangements and asymptotics of the Laplace transforms of large powers of a polynomial, New York J. Math. 10 (2004) 117-131.
S. G. Penrice, Derangements, permanents and Christmas presents, The American Mathematical Monthly 98 (1991), 617-620.
John Riordan and N. J. A. Sloane, Correspondence, 1974
R. Vidunas, Counting derangements and Nash equilibria, arXiv preprint arXiv:1401.5400 [math.CO], 2014-2016.
Raimundas Vidunas, Counting derangements and Nash equilibria Ann. Comb. 21, No. 1, 131-152 (2017).
Wikipedia, Permutations of multisets
Index entries for sequences related to card matching

Cf. A000166, A008290, A059056-A059071, A033581, A059073.

Row n=2 of A372307.

I:=[0,1]; [n le 2 select I[n] else n*(2*n-1)*Self(n-1)+2*n*(n-1)*Self(n-2)-(2*n-1): n in [1..30]]; // Vincenzo Librandi, Sep 28 2015

p := (x,k)->k!^2*sum(x^j/((k-j)!^2*j!),j=0..k); R := (x,n,k)->p(x,k)^n; f := (t,n,k)->sum(coeff(R(x,n,k),x,j)*(t-1)^j*(n*k-j)!,j=0..n*k); seq(f(0,n,2)/2!^n,n=0..18);

RecurrenceTable[{(2*n+3)*a[n+3]==(2*n+5)^2*(n+2)*a[n+2]+(2*n+3)*(n+2)*a[n+1]-2*(2*n+5)*(n+1)*(n+2)*a[n],a[1]==0,a[2]==1,a[3]==10},a,{n,1,25}] (* Vaclav Kotesovec, Aug 31 2012 *)
a[n_] := a[n] = n*(2*n-1)*a[n-1] + 2*n*(n-1)*a[n-2] - (2*n-1); a[0] = 1; a[1] = 0; a[2] = 1; Table[a[n], {n, 0, 14}] (* Jean-François Alcover, Mar 04 2013 *)
a[n_] := Sum[(2*(n-m))! / 2^(n-m) Binomial[n, m] Hypergeometric1F1[m-n, 2*(m - n), -4], {m, 0, n}]; Table[a[n], {n, 0, 16}] (* Peter Luschny, Nov 15 2023 *)

a(n) = (2^n*round(2^(n/2+3/4)*Pi^(-1/2)*exp(-2)*n!*besselk(1/2+n,2^(1/2))))/2^n;
vector(15, n, a(n))\\ Altug Alkan, Sep 28 2015

{ A000459(n) = sum(m=0,n, sum(k=0,n-m, (-1)^k * binomial(n,k) * binomial(n-k,m) * 2^(2*k+m-n) * (2*n-2*m-k)! )); } \\ Max Alekseyev, Oct 06 2016

a(n) = A000316(n)/2^n.
a(n) = Sum_{k=0..n} Sum_{m=0..n-k} (-1)^k * n!/(k!*m!*(n-k-m)!) * 2^(2*k+m-n) * (2*n-2*m-k)!. - Max Alekseyev, Oct 06 2016
G.f.: Sum_{j=0..n*k} coeff(R(x, n, k), x, j)*(t-1)^j*(n*k-j)! where n is the number of kinds of cards, k is the number of cards of each kind (2 in this case) and coeff(R(x, n, k), x, j) is the coefficient of x^j of the rook polynomial R(x, n, k) = (k!^2*sum(x^j/((k-j)!^2*j!))^n (see Riordan or Stanley).
D-finite with recurrence a(n) = n*(2*n-1)*a(n-1)+2*n*(n-1)*a(n-2)-(2*n-1), a(1) = 0, a(2) = 1.
a(n) = round(2^(n/2 + 3/4)*Pi^(-1/2)*exp(-2)*n!*BesselK(1/2+n,2^(1/2))). - Mark van Hoeij, Oct 30 2011
(2*n+3)*a(n+3)=(2*n+5)^2*(n+2)*a(n+2)+(2*n+3)*(n+2)*a(n+1)-2*(2*n+5)*(n+1)*(n+2)*a(n). - Vaclav Kotesovec, Aug 31 2012
Asymptotic: a(n) ~ n^(2*n)*2^(n+1)*sqrt(Pi*n)/exp(2*n+2), Vaclav Kotesovec, Aug 31 2012
a(n) = (1/2^n)*A000316(n) = int_{0..inf} exp(-x)*(1/2*x^2 - 2*x + 1)^n dx. Asymptotic: a(n) ~ ((2*n)!/2^n)*exp(-2)*( 1 - 1/(2*n) - 23/(96*n^2) + O(1/n^3) ). See Nicolaescu. - Peter Bala, Jul 07 2014
Let S = x_1 + ... + x_n. a(n) equals the coefficient of (x_1*...*x_n)^2 in the expansion of product {i = 1..n} (S - x_i)^2 (MacMahon, Chapter III). - Peter Bala, Jul 08 2014
Conjecture: a(n+k) - a(n) is divisible by k. - Mark van Hoeij, Nov 15 2023

More terms and edited by Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Dec 22 2000
Edited by M. F. Hasler, Sep 27 2015
a(0)=1 prepended by Max Alekseyev, Oct 06 2016

A135453 a(n) = 12*n^2.

Original entry on oeis.org

0, 12, 48, 108, 192, 300, 432, 588, 768, 972, 1200, 1452, 1728, 2028, 2352, 2700, 3072, 3468, 3888, 4332, 4800, 5292, 5808, 6348, 6912, 7500, 8112, 8748, 9408, 10092, 10800, 11532, 12288, 13068, 13872, 14700, 15552, 16428, 17328, 18252, 19200, 20172, 21168, 22188

Offset: 0

Ben Paul Thurston, Dec 14 2007

Areas of perfect 4:3 rectangles (for n > 0).
Sequence found by reading the line from 0, in the direction 0, 12, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. Semi-axis opposite to A069190 in the same spiral. - Omar E. Pol, Sep 16 2011
(x,y,z) = (-a(n), 1 + n*a(n), 1 - n*a(n)) are solutions of the Diophantine equation x^3 + 2*y^3 + 2*z^3 = 4. - XU Pingya, Apr 30 2022

			192 is on the list since 16*12 is a 4:3 rectangle with integer sides and an area of 192.

Ivan Panchenko, Table of n, a(n) for n = 0..10000
John Elias, Illustration: Even Ordered Star Perimeters.
Leo Tavares, Illustration.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000290, A001105, A016742, A033428, A033581, A069190, A086729.

Cf. A008594, A195143.

List([0..100],n->12*n^2); # Muniru A Asiru, Jan 29 2018

seq(12*h^2,n=0..100); # Muniru A Asiru, Jan 29 2018

Table[12*n^2, {n, 0, 60}] (* Stefan Steinerberger, Dec 17 2007 *)
LinearRecurrence[{3,-3,1},{0,12,48},50] (* Harvey P. Dale, Jan 19 2020 *)

a(n)=12*n^2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 12*A000290(n) = 6*A001105(n) = 4*A033428(n) = 3*A016742(n) = 2*A033581(n). - Omar E. Pol, Dec 13 2008
From Amiram Eldar, Feb 03 2021: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/72 (A086729).
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/144.
Product_{n>=1} (1 + 1/a(n)) = 2*sqrt(3)*sinh(Pi/(2*sqrt(3)))/Pi.
Product_{n>=1} (1 - 1/a(n)) = 2*sqrt(3)*sin(Pi/(2*sqrt(3)))/Pi. (End)
From Elmo R. Oliveira, Nov 30 2024: (Start)
G.f.: 12*x*(1 + x)/(1-x)^3.
E.g.f.: 12*x*(1 + x)*exp(x).
a(n) = n*A008594(n) = A195143(2*n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

More terms from Stefan Steinerberger, Dec 17 2007

Minor edits from Omar E. Pol, Dec 15 2008

A244630 a(n) = 17*n^2.

Original entry on oeis.org

0, 17, 68, 153, 272, 425, 612, 833, 1088, 1377, 1700, 2057, 2448, 2873, 3332, 3825, 4352, 4913, 5508, 6137, 6800, 7497, 8228, 8993, 9792, 10625, 11492, 12393, 13328, 14297, 15300, 16337, 17408, 18513, 19652, 20825, 22032, 23273, 24548, 25857, 27200, 28577, 29988

Offset: 0

Vincenzo Librandi, Jul 03 2014

First bisection of A195047. - Bruno Berselli, Jul 03 2014

Norms of purely imaginary numbers in Z[sqrt(-17)] (for example, 3*sqrt(-17) has norm 153). - Alonso del Arte, Jun 23 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A008599, A195047.

Cf. similar sequences of the type k*n^2: A000290 (k = 1), A001105 (k = 2), A033428 (k = 3), A016742 (k = 4), A033429 (k = 5), A033581 (k = 6), A033582 (k = 7), A139098 (k = 8), A016766 (k = 9), A033583 (k = 10), A033584 (k = 11), A135453 (k = 12), A152742 (k = 13), A144555 (k = 14), A064761 (k = 15), A016802 (k = 16), this sequence (k = 17), A195321 (k = 18), A244631 (k = 19), A195322 (k = 20), A064762 (k = 21), A195323 (k = 22), A244632 (k = 23), A195824 (k = 24), A016850 (k = 25), A244633 (k = 26), A244634 (k = 27), A064763 (k = 28), A244635 (k = 29), A244636 (k = 30).

List([0..45],n->17*n^2); # Muniru A Asiru, Jun 29 2018

Magma
```
[17*n^2: n in [0..40]];
```

seq(17*n^2,n=0..45); # Muniru A Asiru, Jun 29 2018

Mathematica
```
Table[17 n^2, {n, 0, 40}]
```

a(n)=17*n^2 \\ Charles R Greathouse IV, Oct 07 2015

for (i <- 0 to 50) yield 17 * (i * i) // Alonso del Arte, Jun 29 2018

G.f.: 17*x*(1 + x)/(1 - x)^3. [corrected by Bruno Berselli, Jul 03 2014]
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.
a(n) = 17*A000290(n). - Omar E. Pol, Jul 03 2014
a(n) = a(-n). - Muniru A Asiru, Jun 29 2018
From Elmo R. Oliveira, Dec 02 2024: (Start)
E.g.f.: 17*x*(1 + x)*exp(x).
a(n) = n*A008599(n) = A195047(2*n). (End)

A140811 a(n) = 6*n^2 - 1.

Original entry on oeis.org

-1, 5, 23, 53, 95, 149, 215, 293, 383, 485, 599, 725, 863, 1013, 1175, 1349, 1535, 1733, 1943, 2165, 2399, 2645, 2903, 3173, 3455, 3749, 4055, 4373, 4703, 5045, 5399, 5765, 6143, 6533, 6935, 7349, 7775, 8213, 8663, 9125, 9599, 10085, 10583, 11093, 11615

Offset: 0

Paul Curtz, Jul 16 2008

Also: The numerators in the j=2 column of the array a(i,j) defined in A140825, where the columns j=0 and j=1 are represented by A000012 and A005408. This could be extended to column j=3: 1, -1, 9, 55, 161, ... The common feature of these sequences derived from a(i,j) is that their j-th differences are constant sequences defined by A091137(j).
a(n) is the set of all k such that 6*k + 6 is a perfect square. - Gary Detlefs, Mar 04 2010
The identity (6*n^2 - 1)^2 - (9*n^2 - 3)*(2*n)^2 = 1 can be written as a(n+1)^2 - A157872(n)*A005843(n+1)^2 = 1. - Vincenzo Librandi, Feb 05 2012
Apart from first term, sequence found by reading the line from 5, in the direction 5, 23, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. - Omar E. Pol, Jul 18 2012
From Paul Curtz, Sep 17 2018: (Start)
Terms from center to right in the following spiral:
.
                    65--63--61--59
                    /             \
                  67  31--29--27  57
                  /   /         \   \
                69  33   9---7  25  55
                /   /   /     \   \   \
              71  35  11  -1===5==23==53==>
                  /   /   /   /   /   /
                37  13   1---3  21  51
                  \   \         /   /
                  39  15--17--19  49
                    \             /
                    41--43--45--47 (End)

P. Curtz, Intégration numérique des systèmes différentiels à conditions initiales, Note 12, Centre de Calcul Scientifique de l'Armement, Arcueil, 1969, 132 pages, pp. 28-36. CCSA, then CELAR. Now DGA Maitrise de l'Information 35131 Bruz.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Leo Tavares, Illustration: Barred Stars.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005843, A157872, A060747, A103115(n+1), A141417 (array).
Cf. A000012, A001318, A005408, A091137, A140825.
Cf. A003154, A032528, A033581, A016777, A017593.

[6*n^2 - 1: n in [0..50]]; // Vincenzo Librandi, Jun 02 2011

LinearRecurrence[{3,-3,1},{-1,5,23},40] (* Vincenzo Librandi, Feb 05 2012 *)
CoefficientList[Series[(1-8*x-5*x^2)/(x-1)^3 , {x, 0, 40}], x] (* Stefano Spezia, Sep 17 2018 *)

a(n)=6*n^2-1 \\ Charles R Greathouse IV, Jun 01 2011

a(n) = 2*a(n-1) - a(n-2) + 12.
First differences: a(n+1) - a(n) = A017593(n).
Second differences: A071593(n+1) - A071593(n) = 12.
G.f.: (1-8*x-5*x^2)/(x-1)^3. - Jaume Oliver Lafont, Aug 30 2009
From Vincenzo Librandi, Feb 05 2012: (Start)
a(n) = a(n-1) + 12*n - 6.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
a(n) = A033581(n) - 1. - Omar E. Pol, Jul 18 2012
a(n) = A032528(2*n) - 1. - Adriano Caroli, Jul 21 2013
For n > 0, a(n) = floor(3/(cosh(1/n) - 1)) = floor(1/(n*sinh(1/n) - 1)); for similar formulas for cosine and sine, see A033581. - Clark Kimberling, Oct 19 2014, corrected by M. F. Hasler, Oct 21 2014
a(-n) = a(n). - Paul Curtz, Sep 17 2018
From Amiram Eldar, Feb 04 2021: (Start)
Sum_{n>=1} 1/a(n) = (1 - (Pi/sqrt(6))*cot(Pi/sqrt(6)))/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = ((Pi/sqrt(6))*csc(Pi/sqrt(6)) - 1)/2.
Product_{n>=1} (1 + 1/a(n)) = (Pi/sqrt(6))*csc(Pi/sqrt(6)).
Product_{n>=1} (1 - 1/a(n)) = csc(Pi/sqrt(6))*sin(Pi/sqrt(3))/sqrt(2). (End)
a(n) = A003154(n+1) - 2*A016777(n). - Leo Tavares, May 13 2022
E.g.f.: exp(x)*(6*x^2 + 6*x - 1). - Elmo R. Oliveira, Jan 16 2025

Edited and extended by R. J. Mathar, Aug 06 2008

Better description Ray Chandler, Feb 03 2009

A152734 5 times pentagonal numbers: 5n(3*n-1)/2.

Original entry on oeis.org

0, 5, 25, 60, 110, 175, 255, 350, 460, 585, 725, 880, 1050, 1235, 1435, 1650, 1880, 2125, 2385, 2660, 2950, 3255, 3575, 3910, 4260, 4625, 5005, 5400, 5810, 6235, 6675, 7130, 7600, 8085, 8585, 9100, 9630, 10175, 10735, 11310, 11900, 12505, 13125, 13760, 14410

Offset: 0

Omar E. Pol, Dec 11 2008

a(n) can be represented as a figurate number using n concentric pentagons (see example). - Omar E. Pol, Aug 21 2011

			From _Omar E. Pol_, Aug 22 2011 (Start):
Illustration of initial terms as concentric pentagons (in a precise representation the pentagons should be strictly concentric):
.
.                                          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 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
.
.  5             25                        60
(End)

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000326, A033581, A085250, A152751, A194275.

Cf. sequences of the form n*(d*n+10-d)/2 indexed in A140090.

[5*n*(3*n-1)/2 : n in [0..50]]; // Wesley Ivan Hurt, Sep 19 2014

A152734:=n->5*n*(3*n-1)/2: seq(A152734(n), n=0..50); # Wesley Ivan Hurt, Sep 19 2014

Table[5 n (3 n - 1)/2, {n, 0, 50}] (* Wesley Ivan Hurt, Sep 19 2014 *)
5*PolygonalNumber[5,Range[0,50]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 13 2020 *)

a(n)=5*n*(3*n-1)/2 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = 5*A000326(n).
a(n) = a(n-1)+15*n-10 (with a(0)=0). - Vincenzo Librandi, Nov 26 2010
G.f.: 5*x*(1+2*x)/(1-x)^3. a(n) = 4*A000217(n)+A051865(n). - Bruno Berselli, Feb 11 2011
E.g.f.: (5/2)*(3*x^2 + 2*x)*exp(x). - G. C. Greubel, Jul 17 2017
From Amiram Eldar, Feb 26 2022: (Start)
Sum_{n>=1} 1/a(n) = (9*log(3) - sqrt(3)*Pi)/15.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*(sqrt(3)*Pi- 6*log(2))/15. (End)

A080859 a(n) = 6n^2 + 4n + 1.

Original entry on oeis.org

1, 11, 33, 67, 113, 171, 241, 323, 417, 523, 641, 771, 913, 1067, 1233, 1411, 1601, 1803, 2017, 2243, 2481, 2731, 2993, 3267, 3553, 3851, 4161, 4483, 4817, 5163, 5521, 5891, 6273, 6667, 7073, 7491, 7921, 8363, 8817, 9283, 9761, 10251, 10753, 11267

Offset: 0

Paul Barry, Feb 23 2003

The old definition of this sequence was "Generalized polygonal numbers".
Column T(n,4) of A080853.
Sequence found by reading the line from 1, in the direction 1, 11, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. - Omar E. Pol, Sep 08 2011

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Subsequence of A186424.
Cf. A000384, A001318, A033579, A033581.
Cf. A220083 for a list of numbers of the form n*P(s,n)-(n-1)*P(s,n-1), where P(s,n) is the n-th polygonal number with s sides.

[6*n^2+4*n+1: n in [0..50]]; // Vincenzo Librandi, Apr 29 2016

Table[6 n^2 + 4 n + 1, {n, 0, 50}] (* Vincenzo Librandi, Apr 29 2016 *)

a(n)=6*n^2+4*n+1 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: (C(3,0) + (C(5,2) - 2)*x + C(3,2)*x^2)/(1-x)^3 = (1 + 8*x + 3*x^2)/(1-x)^3.
E.g.f.: (1 + 10*x + 6*x^2)*exp(x). - Vincenzo Librandi, Apr 29 2016
a(n) = C(4,0) + C(4,1)n + C(4,2)n^2.
a(n) = A186424(2*n).
a(n) = 12*n + a(n-1) - 2 with n > 0, a(0)=1. - Vincenzo Librandi, Aug 08 2010
a(n) = (n+1)*A000384(n+1) - n*A000384(n). - Bruno Berselli, Dec 10 2012
a(n) = (n+1)^4 mod n^3 for n >= 7. - J. M. Bergot, Aug 14 2017
a(n) = (2*n+1)^2 + 2*n^2. - Robert FERREOL, Jan 13 2024

Definition replaced with the closed form by Bruno Berselli, Dec 10 2012

cp's OEIS Frontend

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A144555 a(n) = 14*n^2.

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A152751 3 times octagonal numbers: a(n) = 3*n*(3*n-2).

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A055112 a(n) = n*(n+1)*(2*n+1).

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A000459 Number of multiset permutations of {1, 1, 2, 2, ..., n, n} with no fixed points.

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A135453 a(n) = 12*n^2.

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A085250 4 times hexagonal numbers: a(n) = 4n(2*n-1).

A152751 3 times octagonal numbers: a(n) = 3n(3*n-2).

A055112 a(n) = n(n+1)(2*n+1).

A152734 5 times pentagonal numbers: 5n(3*n-1)/2.

A080859 a(n) = 6n^2 + 4n + 1.