A032528 -id:A032528 - cp's OEIS frontend

A195158 Concentric 24-gonal numbers.

0, 1, 24, 49, 96, 145, 216, 289, 384, 481, 600, 721, 864, 1009, 1176, 1345, 1536, 1729, 1944, 2161, 2400, 2641, 2904, 3169, 3456, 3745, 4056, 4369, 4704, 5041, 5400, 5761, 6144, 6529, 6936, 7345, 7776, 8209, 8664, 9121, 9600, 10081, 10584, 11089

Offset: 0

Omar E. Pol, Sep 28 2011

Sequence found by reading the line from 0, in the direction 0, 24, ..., and the same line from 1, in the direction 1, 49, ..., in the square spiral whose vertices are the generalized tetradecagonal numbers A195818. Main axis, perpendicular to A049598 in the same spiral.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Column 24 of A195040.

Cf. A032527, A032528, A195143, A195149, A195058.

[(12*n^2+5*(-1)^n-5)/2: n in [0..50]]; // Vincenzo Librandi, Sep 30 2011

LinearRecurrence[{2,0,-2,1},{0,1,24,49},50] (* Harvey P. Dale, Jan 28 2021 *)

a(n)=6*n^2+5*((-1)^n-1)/2 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 6*n^2 + 5*((-1)^n-1)/2.
a(n) = -a(n-1) + A069190(n). - Vincenzo Librandi, Sep 30 2011
From Colin Barker, Sep 16 2012: (Start)
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
G.f.: x*(1+22*x+x^2)/((1-x)^3*(1+x)). (End)
Sum_{n>=1} 1/a(n) = Pi^2/144 + tan(sqrt(5/6)*Pi/2)*Pi/(4*sqrt(30)). - Amiram Eldar, Jan 17 2023

A122652 a(0) = 0, a(1) = 4; for n > 1, a(n) = 10*a(n-1) - a(n-2).

Original entry on oeis.org

0, 4, 40, 396, 3920, 38804, 384120, 3802396, 37639840, 372596004, 3688320200, 36510605996, 361417739760, 3577666791604, 35415250176280, 350574834971196, 3470333099535680, 34352756160385604, 340057228504320360, 3366219528882817996, 33322138060323859600

Offset: 0

N. J. A. Sloane, Sep 21 2006

Kekulé numbers for the benzenoids P_2(n).
a(n) are the values of m where A032528(m) - 1 has integer square roots. The roots are given by A001079. - Richard R. Forberg, Aug 05 2013
Numbers n such that 6*n^2 + 4 is a square. - Colin Barker, Mar 17 2014

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 283, K{P_2(n)}).

Michael De Vlieger, Table of n, a(n) for n = 0..1004
Andersen, K., Carbone, L. and Penta, D., Kac-Moody Fibonacci sequences, hyperbolic golden ratios, and real quadratic fields, Journal of Number Theory and Combinatorics, Vol 2, No. 3 pp 245-278, 2011. See Section 9.
Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
John M. Campbell, An Integral Representation of Kekulé Numbers, and Double Integrals Related to Smarandache Sequences, arXiv preprint arXiv:1105.3399 [math.GM], 2011.
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (10,-1).

Cf. A001078, A001079, A032528.

CoefficientList[Series[(4 z)/(z^2 - 10 z + 1), {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 11 2011 *)
LinearRecurrence[{10, -1}, {0, 4}, 21] (* Jean-François Alcover, Jan 07 2019 *)

a(n)=if(n<2,(n%2)*4,10*a(n-1)-a(n-2)) \\ Benoit Cloitre, Sep 23 2006

G.f.: 4*x/(1 - 10*x + x^2). - Philippe Deléham, Nov 17 2008
3*a(n)^2 + 2 = 2*A001079(n)^2. - Charlie Marion, Feb 01 2013
a(n) = (2*arcsinh(sqrt(2))*sinh(2*n*arcsinh(sqrt(2)))/log(sqrt(2) + sqrt(3)))/sqrt(6). - Artur Jasinski, Aug 09 2016
a(n) = 2*A001078(n). - Bruno Berselli, Nov 25 2016
E.g.f.: sqrt(6)*exp(5*x)*sinh(2*sqrt(6)*x)/3. - Franck Maminirina Ramaharo, Jan 07 2019

More terms and better definition from Benoit Cloitre, Sep 23 2006

A238410 a(n) = floor((3(n-1)^2 + 1)/2).

Original entry on oeis.org

0, 2, 6, 14, 24, 38, 54, 74, 96, 122, 150, 182, 216, 254, 294, 338, 384, 434, 486, 542, 600, 662, 726, 794, 864, 938, 1014, 1094, 1176, 1262, 1350, 1442, 1536, 1634, 1734, 1838, 1944, 2054, 2166, 2282, 2400, 2522, 2646, 2774, 2904, 3038, 3174, 3314, 3456, 3602, 3750, 3902, 4056, 4214, 4374, 4538, 4704

Offset: 1

Emeric Deutsch, Feb 27 2014

a(n) = the eccentric connectivity index of the path P[n] on n vertices. The eccentric connectivity index of a simple connected graph G is defined to be the sum over all vertices i of G of the product E(i)D(i), where E(i) is the eccentricity and D(i) is the degree of vertex i. For example, a(4)=14 because the vertices of P[4] have degrees 1,2,2,1 and eccentricities 3,2,2,3; we have 1*3 + 2*2 + 2*2 + 1*3 = 14.
From Paul Curtz, Feb 23 2023: (Start)
East spoke of the hexagonal spiral using A004526 with a single 0:
.
            43  42  42  41  41  40
          43  28  28  27  27  26  40
        44  29  17  16  16  15  26  39
      44  29  17   8   8   7  15  25  39
    45  30  18   9   3   2   7  14  25  38
  45  30  18   9   3   0---2---6--14--24--38-->
    31  19  10   4   1   1   6  13  24  37
      31  19  10   4   5   5  13  23  37
        32  20  11  11  12  12  23  36
          32  20  21  21  22  22  36
            33  33  34  34  35  35
.
Other spokes are A032528, A005449, A227017, A045943, A005448, A212959, A000326, A282513, A143689, and A104249. (End)

Matthew House, Table of n, a(n) for n = 1..10000
M. J. Morgan, S. Mukwembi and H. C. Swart, On the eccentric connectivity index of a graph, Discrete Math., 311, 2011, 1229-1234.
B. Zhou and Zh. Du, On eccentric connectivity index, Comm. Math. Comp. Chem. (MATCH), 63, 2010, 181-198.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A004526, A077043.

(Cf. A056105, A056107.)

a := proc (n) options operator, arrow: floor((3/2)*(n-1)^2+1/2) end proc: seq(a(n), n = 1 .. 70);

Table[Floor[(3(n-1)^2+1)/2],{n,80}]  (* or *) LinearRecurrence[{2,0,-2,1},{0,2,6,14},80] (* Harvey P. Dale, Apr 30 2022 *)

a(n)=(3*(n-1)^2 + 1)\2 \\ Charles R Greathouse IV, Feb 15 2017

a(n) = (3*n)^2/6 for n even and a(n) = ((3*n)^2 + 3)/6 for n odd. - Miquel Cerda, Jun 17 2016
From Ilya Gutkovskiy, Jun 17 2016: (Start)
G.f.: 2*x^2*(1 + x + x^2)/((1 - x)^3*(1 + x)).
a(n) = (6*n^2 - 12*n + 7 + (-1)^n)/4.
a(n) = 2* A077043(n-1). (End)
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4). - Matthew House, Feb 15 2017
Sum_{n>=2} 1/a(n) = Pi^2/36 + tanh(Pi/(2*sqrt(3)))*Pi/(2*sqrt(3)). - Amiram Eldar, Mar 12 2023

A122653 a(n) = 10*a(n-1) - a(n-2) with a(0)=0, a(1)=6.

Original entry on oeis.org

0, 6, 60, 594, 5880, 58206, 576180, 5703594, 56459760, 558894006, 5532480300, 54765908994, 542126609640, 5366500187406, 53122875264420, 525862252456794, 5205499649303520, 51529134240578406, 510085842756480540, 5049329293324226994, 49983207090485789400

Offset: 0

N. J. A. Sloane, Sep 21 2006

Kekulé numbers for the benzenoids P''(n).
a(n) are the integer square roots of A032528(m) - 1. A001079 gives the value of m where these roots occur. Also see A122652. - Richard R. Forberg, Aug 05 2013
Numbers n such that 6*n^2 + 9 is a square. - Colin Barker, Mar 17 2014

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 301).

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (10,-1).

CoefficientList[Series[(6 z)/(z^2 - 10 z + 1), {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 11 2011 *)
LinearRecurrence[{10,-1},{0,6},30] (* Harvey P. Dale, Dec 16 2014 *)

a(n)=if(n<2,(n%2)*6,10*a(n-1)-a(n-2)) \\ Benoit Cloitre, Sep 23 2006

G.f.: 6x/(1 - 10x + x^2). - Philippe Deléham, Nov 17 2008
a(n) = 6*A004189(n). - R. J. Mathar, Jun 22 2020
6*a(n)^2+9 = (3*A001079(n))^2  - detail of the Barker comment. - R. J. Mathar, Jun 22 2020

More terms and better definition from Benoit Cloitre, Sep 23 2006

A184533 a(n) = floor(1/{(2+n^3)^(1/3)}), where {}=fractional part.

Original entry on oeis.org

2, 6, 13, 24, 37, 54, 73, 96, 121, 150, 181, 216, 253, 294, 337, 384, 433, 486, 541, 600, 661, 726, 793, 864, 937, 1014, 1093, 1176, 1261, 1350, 1441, 1536, 1633, 1734, 1837, 1944, 2053, 2166, 2281, 2400, 2521, 2646, 2773, 2904, 3037, 3174, 3313, 3456, 3601

Offset: 1

Clark Kimberling, Jan 16 2011

Column 2 of the array at A184532.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A183532, A183534. Essenitally the same as A032528.

p[n_]:=FractionalPart[(n^3+2)^(1/3)]; q[n_]:=Floor[1/p[n]]; Table[q[n],{n,1,120}]
Join[{2},Table[(6*n^2 - (1-(-1)^n))/4,{n,2,50}]] (* or *) Join[{2}, LinearRecurrence[{2,0,-2,1},{6, 13, 24, 37},50]] (* G. C. Greubel, Feb 20 2017 *)

a(n)=my(x=sqrtn(n^3+2,3));x-=n;1/x\1 \\ Charles R Greathouse IV, Aug 23 2011

concat([2], for(n=2,25, print1((6*n^2 - (1-(-1)^n))/4, ", "))) \\ G. C. Greubel, Feb 20 2017

a(n) = floor(1/{(2+n^3)^(1/3)}), where {}=fractional part.
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
a(n) = (6*n^2 - (1-(-1)^n))/4 for n>1.
From Alexander R. Povolotsky, Aug 22 2011: (Start)
a(n+1) +a(n) = 3*n^2 + 3*n + 1.
G.f.: x*(-2 - 2*x - x^2 - 2*x^3 + x^4)/((-1 + x)^3*(1 + x)). (End)
a(n) = floor(1/((n^3+2)^(1/3)-n)). - Charles R Greathouse IV, Aug 23 2011
E.g.f.: (3*x*(x + 1)*cosh(x) + (3*x^2 + 3*x - 1)*sinh(x) + 2*x)/2. - Stefano Spezia, Apr 19 2025

A194273 Concentric triangular numbers (see Comments lines for definition).

Original entry on oeis.org

0, 1, 3, 6, 9, 12, 15, 19, 24, 30, 36, 42, 48, 55, 63, 72, 81, 90, 99, 109, 120, 132, 144, 156, 168, 181, 195, 210, 225, 240, 255, 271, 288, 306, 324, 342, 360, 379, 399, 420, 441, 462, 483, 505, 528, 552, 576, 600, 624, 649, 675, 702, 729, 756, 783, 811

Offset: 0

Omar E. Pol, Aug 20 2011

This can be interpreted as a cellular automaton on the infinite hexagonal net. The sequence gives the number of cells "ON" in the structure after n-th stage. A194272 gives the first differences. For a definition without words see the illustration of initial terms in the example section. For other concentric polygonal numbers see A194274, A194275 and A032528.

Also, row sums of an infinite square array T(n,k) in which column k lists 6*k-1 zeros followed by the numbers A008486 (see example).

			Using the numbers A008486 we can write:
0, 1, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36,...
0, 0, 0, 0, 0,  0,  0,  1,  3,  6,  9, 12, 15, 18,...
0, 0, 0, 0, 0,  0,  0,  0,  0,  0,  0,  0,  0,  1,...
And so on.
=========================================================
The sums of the columns give this sequence:
0, 1, 3, 6, 9, 12, 15, 19, 24, 30, 36, 42, 48, 55,...
...
Illustration of initial terms:
.                                              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
.
. 1    3      6        9          12           15
.
.                                           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
.
.       19               24                 30

Index entries for sequences related to cellular automata

Cf. A000217, A008486, A032528, A069131, A152751, A194272, A194274, A194275.

G.f.: x/(1-3*x+3*x^2-3*x^4+3*x^5-x^6) = x/((1-x)^3*(1+x)*(1-x+x^2)).

A277644 Beatty sequence for sqrt(6)/2.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 15, 17, 18, 19, 20, 22, 23, 24, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 39, 40, 41, 42, 44, 45, 46, 47, 48, 50, 51, 52, 53, 55, 56, 57, 58, 60, 61, 62, 63, 64, 66, 67, 68, 69, 71, 72, 73, 74, 75, 77, 78, 79, 80, 82, 83, 84, 85, 86

Offset: 1

Jason Kimberley, Oct 26 2016

Eggleton et al. show that k is in this sequence if and only if A277515(k)=3.

			a(5)=6 because the quotient of 3*5^2 by 2 is 37 which lies between 6^2 and 7^2.

R. B. Eggleton, J. S. Kimberley and J. A. MacDougall, Square-free rank of integers, submitted.

Jason Kimberley, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Beatty Sequence.
Index entries for sequences related to Beatty sequences.

Cf. A000196, A032528, A115754, A277515. Complement of A277645.

Magma
```
[Isqrt(3*n^2 div 2): n in [1..60]];
```

Floor[Range[100]*Sqrt[3/2]] (* Paolo Xausa, Jul 11 2024 *)

a(n)=sqrtint(3*n^2\2) \\ Charles R Greathouse IV, Jul 11 2024

a(n) = floor(n*sqrt(6)/2).

a(n) = A000196(A032528(n)).

A211783 Rectangular array: R(n,k)=n^2+[(n^2)/2]+...+[(n^2)/k], where [ ]=floor, by antidiagonals.

Original entry on oeis.org

1, 4, 1, 9, 6, 1, 16, 13, 7, 1, 25, 24, 16, 8, 1, 36, 37, 29, 18, 8, 1, 49, 54, 45, 33, 19, 8, 1, 64, 73, 66, 51, 36, 20, 8, 1, 81, 96, 89, 75, 56, 38, 21, 8, 1, 100, 121, 117, 101, 82, 60, 40, 22, 8, 1, 121, 150, 148, 133, 110, 88, 63, 42, 23, 8, 1, 144, 181, 183

Offset: 0

Clark Kimberling, Apr 20 2012

For n>=1, row n is a homogeneous linear recurrence sequence with palindromic recurrence coefficients in the sense described at A211701.
Row 1:  A000290
Row 2:  A032528
Row 3:  A211784
R(n,n)=A118014(n,n)
The sequence approached as a limit of the rows is A175346: (1,8,23,50,87,140,...)

			Northwest corner:
1....4....9....16....25....36
1....6....13...24....37....54
1....7....16...29....35....66
1....8....18...33....51....75
1....8....19...36....56....82
1....8....20...38....60....88
1....8....21...40....63....93

Cf. A211701

f[n_, m_] := Sum[Floor[n^2/k], {k, 1, m}]
TableForm[Table[f[n, m], {m, 1, 40}, {n, 1, 16}]]
Flatten[Table[f[n + 1 - m, m], {n, 1, 14}, {m, 1, n}]]

A211786 n^3 + floor(n^3/2).

Original entry on oeis.org

1, 12, 40, 96, 187, 324, 514, 768, 1093, 1500, 1996, 2592, 3295, 4116, 5062, 6144, 7369, 8748, 10288, 12000, 13891, 15972, 18250, 20736, 23437, 26364, 29524, 32928, 36583, 40500, 44686, 49152, 53905, 58956, 64312, 69984, 75979, 82308

Offset: 1

Clark Kimberling, Apr 20 2012

Row 2 of the array A211785.

Bruno Berselli, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3, -2, -2, 3, -1).

Cf. A032766 (n+floor(n/2)), A032528 (n^2+floor(n^2/2)), A211701.

[n^3+Floor(n^3/2): n in [1..38]]; // Bruno Berselli, May 06 2012

f[n_, m_] := Sum[Floor[n^3/k], {k, 1, m}]
t = Table[f[n, 2], {n, 1, 90}]
FindLinearRecurrence[t]
LinearRecurrence[{3, -2, -2, 3, -1},{1, 12, 40, 96, 187},38] (* Ray Chandler, Aug 02 2015 *)

a(n) = 3*a(n-1)-2*a(n-2)-2*a(n-3)+3*a(n-4)-a(n-5).
G.f.: x*(1+9*x+6*x^2+2*x^3)/((1+x)*(1-x)^4). [Bruno Berselli, May 06 2012]
a(n) = floor(3*n^3/2) = (6*n^3+(-1)^n-1)/4. [Bruno Berselli, May 06 2012]

A229834 Expansion of (1+4x+x^2) / ((1-x)^3(1+x)^4).

Original entry on oeis.org

1, 3, 1, 11, -2, 26, -10, 50, -25, 85, -49, 133, -84, 196, -132, 276, -195, 375, -275, 495, -374, 638, -494, 806, -637, 1001, -805, 1225, -1000, 1480, -1224, 1768, -1479, 2091, -1767, 2451, -2090, 2850, -2450, 3290, -2849, 3773, -3289, 4301, -3772, 4876, -4300, 5500, -4875, 6175, -5499, 6903, -6174, 7686, -6902

Offset: 0

Stefano Maruelli, Dec 19 2013

The sequence can be generated in the following way:
---------------------------       --------------------------
      [0]  [1] [2] [3]  [4]  ...             [i]
---------------------------       --------------------------
[0]    1,   1,  1,  1,   1,  ...  t(0,i) =    1
[1]    7,   6,  5,  4,   3,  ...  t(1,i) = t(1,i-1) - t(0,i)
[2]   19,  13,  8,  4,   1,  ...  t(2,i) = t(2,i-1) - t(1,i)
[3]   37,  24, 16, 12,  11,  ...  t(3,i) = t(3,i-1) - t(2,i)
[4]   61,  37, 21,  9,  -2,  ...  t(4,i) = t(4,i-1) - t(3,i)
[5]   91,  54, 33, 24,  26,  ...             etc.
[6]  127,  73, 40, 16, -10,  ...
[7]  169,  96, 56, 40,  50,  ...
[8]  217, 121, 65, 25, -25,  ...
[9]  271, 150, 85, 60,  85,  ...
...
Column 0 is A003215;
column 1 is A032528;
column 2 is A001082;
column 3 is A241496;
column 4 is this sequence.
The third differences are 16, -35, 64, -105, 160, ..., a signed variant of A077415. - R. J. Mathar, Apr 18 2014

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

Cf. A077415; A058373: a(2k) = -A058373(k); A051925: a(2k+1) = A051925(k+2).

Columns of the table in Comments section: A001082, A003215, A032528.

Table[1 + n (n + 5) (9 - (2 n + 5) (-1)^n)/48, {n, 0, 60}] (* Bruno Berselli, Apr 22 2014 *)
CoefficientList[Series[(1+4x+x^2)/((1-x)^3(1+x)^4),{x,0,60}],x] (* or *) LinearRecurrence[{-1,3,3,-3,-3,1,1},{1,3,1,11,-2,26,-10},60] (* Harvey P. Dale, Jan 27 2022 *)

G.f.: (1 + 4*x + x^2) / ((1 - x)^3*(1 + x)^4). - R. J. Mathar, Apr 18 2014

a(n) = a(-n-5) = 1 + n*(n + 5)*(9 - (2*n + 5)*(-1)^n)/48. [Bruno Berselli, Apr 22 2014]

cp's OEIS Frontend

A195158 Concentric 24-gonal numbers.

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A122652 a(0) = 0, a(1) = 4; for n > 1, a(n) = 10*a(n-1) - a(n-2).

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A238410 a(n) = floor((3(n-1)^2 + 1)/2).

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A122653 a(n) = 10*a(n-1) - a(n-2) with a(0)=0, a(1)=6.

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A184533 a(n) = floor(1/{(2+n^3)^(1/3)}), where {}=fractional part.

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Mathematica

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PARI

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A194273 Concentric triangular numbers (see Comments lines for definition).

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A277644 Beatty sequence for sqrt(6)/2.

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A229834 Expansion of (1+4x+x^2) / ((1-x)^3(1+x)^4).