A282513 -id:A282513 - cp's OEIS frontend

A032528 Concentric hexagonal numbers: floor(3*n^2/2).

0, 1, 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, 3750

Offset: 0

N. J. A. Sloane

From Omar E. Pol, Aug 20 2011: (Start)
Cellular automaton on the hexagonal net. The sequence gives the number of "ON" cells in the structure after n-th stage. A007310 gives the first differences. For a definition without words see the illustration of initial terms in the example section. Note that the cells become intermittent. A083577 gives the primes of this sequences.
A033581 and A003154 interleaved.
Row sums of an infinite square array T(n,k) in which column k lists 2*k-1 zeros followed by the numbers A008458 (see example).  (End)
Sequence found by reading the line from 0, in the direction 0, 1, ... and the same line from 0, in the direction 0, 6, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. Main axis perpendicular to A045943 in the same spiral. - Omar E. Pol, Sep 08 2011

			From _Omar E. Pol_, Aug 20 2011: (Start)
Using the numbers A008458 we can write:
  0, 1, 6, 12, 18, 24, 30, 36, 42,  48,  54, ...
  0, 0, 0,  1,  6, 12, 18, 24, 30,  36,  42, ...
  0, 0, 0,  0,  0,  1,  6, 12, 18,  24,  30, ...
  0, 0, 0,  0,  0,  0,  0,  1,  6,  12,  18, ...
  0, 0, 0,  0,  0,  0,  0,  0,  0,   1,   6, ...
And so on.
===========================================
The sums of the columns give this sequence:
0, 1, 6, 13, 24, 37, 54, 73, 96, 121, 150, ...
...
Illustration of initial terms as concentric hexagons:
.
.                                         o o o o o
.                         o o o o        o         o
.             o o o      o       o      o   o o o   o
.     o o    o     o    o   o o   o    o   o     o   o
. o  o   o  o   o   o  o   o   o   o  o   o   o   o   o
.     o o    o     o    o   o o   o    o   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    6        13           24               37
.
(End)

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

Cf. A003154, A007310, A008458, A033581, A083577, A000326, A001318, A005449, A045943, A032527, A195041. Column 6 of A195040.
Cf. A004524, A004525, A033436, A212959, A238410, A227017, A282513.
Cf. A033581, A003154, A011934, A184533.

a032528 n = a032528_list !! n
a032528_list = scanl (+) 0 a007310_list
-- Reinhard Zumkeller, Jan 07 2012

[Floor(3*n^2/2): n in [0..50]]; // Vincenzo Librandi, Aug 21 2011

f[n_, m_] := Sum[Floor[n^2/k], {k, 1, m}]; t = Table[f[n, 2], {n, 1, 90}] (* Clark Kimberling, Apr 20 2012 *)

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

From Joerg Arndt, Aug 22 2011: (Start)
G.f.: (x+4*x^2+x^3)/(1-2*x+2*x^3-x^4) = x*(1+4*x+x^2)/((1+x)*(1-x)^3).
a(n) = +2*a(n-1) -2*a(n-3) +1*a(n-4). (End)
a(n) = (6*n^2+(-1)^n-1)/4. - Bruno Berselli, Aug 22 2011
a(n) = A184533(n), n >= 2. - Clark Kimberling, Apr 20 2012
First differences of A011934: a(n) = A011934(n) - A011934(n-1) for n>0. - Franz Vrabec, Feb 17 2013
From Paul Curtz, Mar 31 2019: (Start)
a(-n) = a(n).
a(n) = a(n-2) + 6*(n-1) for n > 1.
a(2*n) = A033581(n).
a(2*n+1) = A003154(n+1). (End)
E.g.f.: (3*x*(x + 1)*cosh(x) + (3*x^2 + 3*x - 1)*sinh(x))/2. - Stefano Spezia, Aug 19 2022
Sum_{n>=1} 1/a(n) = Pi^2/36 + tan(Pi/(2*sqrt(3)))*Pi/(2*sqrt(3)). - Amiram Eldar, Jan 16 2023

New name and more terms a(41)-a(50) from Omar E. Pol, Aug 20 2011

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

A211520 Number of ordered triples (w,x,y) with all terms in {1,...,n} and w + 4y = 2x.

Original entry on oeis.org

0, 0, 0, 1, 2, 3, 5, 7, 10, 12, 16, 19, 24, 27, 33, 37, 44, 48, 56, 61, 70, 75, 85, 91, 102, 108, 120, 127, 140, 147, 161, 169, 184, 192, 208, 217, 234, 243, 261, 271, 290, 300, 320, 331, 352, 363, 385, 397, 420, 432, 456, 469, 494, 507, 533, 547, 574

Offset: 0

Clark Kimberling, Apr 14 2012

For a guide to related sequences, see A211422.

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

Cf. A178804, A211422, A282513.

a211520 n = a211520_list !! n
a211520_list = 0 : 0 : 0 : scanl1 (+) a178804_list
-- Reinhard Zumkeller, Nov 15 2014

seq(floor((n-1)^2/4)-floor((n-1)/4)*floor((n+1)/4), n=0..60); # Ridouane Oudra, Nov 21 2024

t[n_] := t[n] = Flatten[Table[w - 2 x + 4 y, {w, 1, n}, {x, 1, n}, {y, 1, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 70}]  (* this sequence *)
FindLinearRecurrence[t]
LinearRecurrence[{1,1,-1,1,-1,-1,1},{0,0,0,1,2,3,5},57] (* Ray Chandler, Aug 02 2015 *)

{ my(x='x+O('x^66)); concat([0,0,0],Vec( x^3*(1+x+x^3) / ( (1-x)^3*(1+x)^2*(1+x^2) ) ) ) } \\ Joerg Arndt, Apr 02 2017

a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4) - a(n-5) - a(n-6) + a(n-7).
a(n) - a(n-1) = A178804(n-2). - Reinhard Zumkeller, Nov 15 2014
a(n) = (6*n^2-10*n+3+(2*n-7)*(-1)^n-4*(-1)^((2*n-3-(-1)^n)/4))/32. - Luce ETIENNE, Dec 31 2015
a(n) = Sum_{k=1..floor(n/2)} floor((n-k)/2). - Wesley Ivan Hurt, Apr 01 2017
G.f.: x^3 * (1+x+x^3) / ( (1-x)^3*(1+x)^2*(1+x^2) ). - Joerg Arndt, Apr 02 2017
a(n)+a(n-1) = A282513(n-2). - R. J. Mathar, Jun 23 2021
a(n) = floor((n-1)^2/4) - floor((n-1)/4)*floor((n+1)/4). - Ridouane Oudra, Nov 21 2024

A336234 Edge length of 'Prime squares': sum the four numbers at the corners of a square drawn on a diagonally numbered 2D board, with 1 at the corner of the square. The sequence gives the size of the square such that the sum is a prime number.

Original entry on oeis.org

1, 3, 7, 9, 13, 15, 19, 25, 31, 37, 39, 51, 61, 63, 69, 81, 87, 97, 99, 109, 117, 135, 145, 147, 151, 153, 163, 165, 171, 183, 189, 195, 201, 207, 213, 219, 223, 229, 235, 241, 249, 253, 267, 271, 273, 277, 297, 307, 319, 325, 337, 343, 345, 355, 373, 381, 387, 391, 393, 409, 435, 447, 451, 457

Offset: 1

Eric Angelini and Scott R. Shannon, Jul 13 2020

nonn

			The board is numbered as follows:
.
   1  2  4  7 11 16  .
   3  5  8 12 17  .
   6  9 13 18  .
  10 14 19  .
  15 20  .
  21  .
  .
a(1) = 1 as the four numbers {1,2,5,3} form the corners of a square of size 1, and the sum of these number is 11, a prime number.
a(2) = 3 as the four numbers {1,7,25,10} form the corners of a square of size 3, and the sum of these number is 43, a prime number.
a(3) = 7 as the four numbers {1,29,113,36} form the corners of a square of size 7, and the sum of these number is 179, a prime number.

Harvey P. Dale, Table of n, a(n) for n = 1..2000
Eric Angelini, Prime squares and square squares, personal blog "Cinquante signes", Jun. 29, 2020.
Eric Angelini, Prime squares and square squares, personal blog "Cinquante signes", Jun. 29, 2020. [Cached copy]

Cf. A000040, A185505, A000124, A000217, A282513.

Select[Range[1,501,2],PrimeQ[3#^2+4#+4]&] (* Harvey P. Dale, May 26 2022 *)

The sequence is the values of d where 3*d^2+4*d+4, the sum of the four numbers for a square of size d, is prime. For even d this sum will always be even, thus all terms are odd.

cp's OEIS Frontend

A032528 Concentric hexagonal numbers: floor(3*n^2/2).

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

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A211520 Number of ordered triples (w,x,y) with all terms in {1,...,n} and w + 4y = 2x.

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Haskell

Maple

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A336234 Edge length of 'Prime squares': sum the four numbers at the corners of a square drawn on a diagonally numbered 2D board, with 1 at the corner of the square. The sequence gives the size of the square such that the sum is a prime number.

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Programs

Mathematica

Formula