A022144 -id:A022144 - cp's OEIS frontend

A108998 Square array, read by antidiagonals, where row n equals the coordination sequence of B_n lattice, for n >= 0.

1, 1, 0, 1, 2, 0, 1, 8, 2, 0, 1, 18, 16, 2, 0, 1, 32, 74, 24, 2, 0, 1, 50, 224, 170, 32, 2, 0, 1, 72, 530, 768, 306, 40, 2, 0, 1, 98, 1072, 2562, 1856, 482, 48, 2, 0, 1, 128, 1946, 6968, 8130, 3680, 698, 56, 2, 0, 1, 162, 3264, 16394, 28320, 20082, 6432, 954, 64, 2, 0

Offset: 0

Paul D. Hanna, Jun 17 2005

Compare with A108553, where row n equals the crystal ball sequence for D_n lattice.

			Square array begins:
  1,  0,    0,     0,     0,      0,      0,      0, ...
  1,  2,    2,     2,     2,      2,      2,      2, ...
  1,  8,   16,    24,    32,     40,     48,     56, ...
  1, 18,   74,   170,   306,    482,    698,    954, ...
  1, 32,  224,   768,  1856,   3680,   6432,  10304, ...
  1, 50,  530,  2562,  8130,  20082,  42130,  78850, ...
  1, 72, 1072,  6968, 28320,  85992, 214864, 467544, ...
  1, 98, 1946, 16394, 83442, 307314, 907018, ...
Product of the g.f. of row n and (1-x)^n generates the rows of triangle A109001:
  1;
  1,  1;
  1,  6,   1;
  1, 15,  23,    1;
  1, 28, 102,   60,    1;
  1, 45, 290,  402,  125,   1;
  1, 66, 655, 1596, 1167, 226, 1; ...

Muniru A Asiru, Rows n=0..110 of antidiagonals, flattened
R. Bacher, P. de la Harpe and B. Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.

Cf. A108999 (main diagonal), A109000 (antidiagonal sums), A109001, A022144 (row 2), A022145 (row 3), A022146 (row 4), A022147 (row 5), A022148 (row 6), A022149 (row 7), A022150 (row 8), A022151 (row 9), A022152 (row 10), A022153 (row 11), A022154 (row 12).

T(n,k)=if(n<0 || k<0,0,sum(j=0,k, binomial(n+k-j-1,k-j)*(binomial(2*n+1,2*j)-2*n*binomial(n-1,j-1))))

T(n, k) = Sum_{j=0..k} C(n+k-j-1, k-j)*(C(2*n+1, 2*j)-2*n*C(n-1, j-1)) for n >= k >= 0.

G.f. for coordination sequence of B_n lattice: ((Sum_{i=0..n} binomial(2*n+1, 2*i)*z^i)-2*n*z*(1+z)^(n-1))/(1-z)^n. [Bacher et al.]

A201279 a(n) = 6n^2 + 10n + 5.

Original entry on oeis.org

5, 21, 49, 89, 141, 205, 281, 369, 469, 581, 705, 841, 989, 1149, 1321, 1505, 1701, 1909, 2129, 2361, 2605, 2861, 3129, 3409, 3701, 4005, 4321, 4649, 4989, 5341, 5705, 6081, 6469, 6869, 7281, 7705, 8141, 8589, 9049, 9521, 10005, 10501, 11009, 11529, 12061

Offset: 0

Eleonora Echeverri-Toro, Nov 29 2011

Numbers n where 6n-5 is a square of a number type 6n-1.
Also sequence found by reading the line from 5, in the direction 5, 21,..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. - Omar E. Pol, Jul 18 2012
The spiral mentioned above naturally appears on a "graphene" like lattice (planar net 6^3). The opposite diagonal is A080859. - Yuriy Sibirmovsky, Oct 04 2016
First differences of A048395. - Leo Tavares, Nov 24 2021 [Corrected by Omar E. Pol, Dec 26 2021]

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Yuriy Sibirmovsky, Diagonals of a 'graphene' number spiral.
Leo Tavares, Illustration: Diamond Frame Stars
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A136392, A080859.

Cf. A003154, A022144, A016754, A046092, A048395.

[6*n^2 + 10*n + 5: n in [0..60]]; // Vincenzo Librandi, Dec 01 2011

LinearRecurrence[{3,-3,1},{5,21,49},50] (* Vincenzo Librandi, Dec 01 2011 *)
Table[6 n^2 + 10 n + 5, {n, 0, 44}] (* or *)
CoefficientList[Series[(1 + x) (5 + x)/(1 - x)^3, {x, 0, 44}], x] (* Michael De Vlieger, Oct 04 2016 *)

a(n)=6*n^2+10*n+5 \\ Charles R Greathouse IV, Nov 29 2011

G.f.: (1+x)*(5+x)/(1-x)^3. - Colin Barker, Jan 09 2012
a(n) = 1 + A033579(n+1). - Omar E. Pol, Jul 18 2012
a(n) = (n+1)*A001844(n+1)-n*A001844(n). [Bruno Berselli, Jan 15 2013]
From Leo Tavares, Nov 24 2021: (Start)
a(n) = A003154(n+2) - A022144(n+1). See Diamond Frame Stars illustration.
a(n) = A016754(n) + A046092(n+1). (End)

A217873 a(n) = 4n(n^2 + 2)/3.

Original entry on oeis.org

0, 4, 16, 44, 96, 180, 304, 476, 704, 996, 1360, 1804, 2336, 2964, 3696, 4540, 5504, 6596, 7824, 9196, 10720, 12404, 14256, 16284, 18496, 20900, 23504, 26316, 29344, 32596, 36080, 39804, 43776, 48004, 52496, 57260, 62304, 67636, 73264, 79196, 85440, 92004

Offset: 0

M. F. Hasler, Oct 13 2012

Occurs as 4th column in the table A142978 of figurate numbers for n-dimensional cross polytope.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A006527, A008412, A008590, A022144, A082138, A112087, A142978, A174794, A292022.

[4*n*(n^2+2)/3: n in [0..45]]; // Vincenzo Librandi, Nov 08 2012

Table[4n(n^2 + 2)/3, {n, 0, 39}] (* Alonso del Arte, Oct 22 2012 *)
LinearRecurrence[{4,-6,4,-1},{0,4,16,44},50] (* Harvey P. Dale, Mar 16 2015 *)

makelist(4*n*(n^2+2)/3, n, 0, 41); /* Martin Ettl, Oct 15 2012 */

PARI
```
a(n)=(n^2+2)*n/3*4
```

a(n) = 4*A006527(n).
From Peter Luschny, Oct 14 2012: (Start)
a(n) = A008412(n)/2.
a(n) = A174794(n+1) - 1.
First differences are in A112087.
Second differences are in A008590 and A022144.
Binomial transformation of (a(n), n > 0) is A082138.  (End)
G.f.: 4*x*(1 + x^2)/(x - 1)^4. - R. J. Mathar, Oct 15 2012
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), a(0)=0, a(1)=4, a(2)=16, a(3)=44. - Harvey P. Dale, Mar 16 2015
From Elmo R. Oliveira, Aug 09 2025: (Start)
E.g.f.: 4*exp(x)*x*(3 + 3*x + x^2)/3.
a(n) = A292022(n)/3. (End)

A109001 Triangle, read by rows, where g.f. of row n equals the product of (1-x)^n and the g.f. of the coordination sequence for root lattice B_n, for n >= 0.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 15, 23, 1, 1, 28, 102, 60, 1, 1, 45, 290, 402, 125, 1, 1, 66, 655, 1596, 1167, 226, 1, 1, 91, 1281, 4795, 6155, 2793, 371, 1, 1, 120, 2268, 12040, 23750, 18888, 5852, 568, 1, 1, 153, 3732, 26628, 74574, 91118, 49380, 11124, 825, 1, 1, 190, 5805, 53544, 201810, 350196, 291410, 114600, 19629, 1150, 1

Offset: 0

Paul D. Hanna, Jun 17 2005

Compare to triangle A108558, where row n equals the (n+1)-th differences of the crystal ball sequence for D_n lattice.

			G.f.s of initial rows of square array A108998 are:
  (1),
  (1 + x)/(1-x),
  (1 + 6*x + x^2)/(1-x)^2;
  (1 + 15*x + 23*x^2 + x^3)/(1-x)^3;
  (1 + 28*x + 102*x^2 + 60*x^3 + x^4)/(1-x)^4.
Triangle begins:
  1;
  1,   1;
  1,   6,    1;
  1,  15,   23,     1;
  1,  28,  102,    60,     1;
  1,  45,  290,   402,   125,     1;
  1,  66,  655,  1596,  1167,   226,     1;
  1,  91, 1281,  4795,  6155,  2793,   371,     1;
  1, 120, 2268, 12040, 23750, 18888,  5852,   568,   1;
  1, 153, 3732, 26628, 74574, 91118, 49380, 11124, 825, 1;

Muniru A Asiru, Rows n=0..100 of triangle, flattened
R. Bacher, P. de la Harpe and B. Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.

Cf. A108998, A108999, A109000, A022144 (row 2), A022145 (row 3), A022146 (row 4), A022147 (row 5), A022148 (row 6), A022149 (row 7), A022150 (row 8), A022151 (row 9), A022152 (row 10), A022153 (row 11), A022154 (row 12).

Flat(List([0..10],n->List([0..n],k->Binomial(2*n+1,2*k)-2*n*Binomial(n-1,k-1)))); # Muniru A Asiru, Dec 14 2018

T[n_, k_] := Binomial[2n+1, 2k] - 2n * Binomial[n-1, k-1]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Dec 14 2018 *)

T(n,k)=binomial(2*n+1,2*k)-2*n*binomial(n-1,k-1)

T(n, k) = C(2*n+1, 2*k) - 2*n*C(n-1, k-1).
Row sums are 2^n*(2^n - n) for n >= 0.
G.f. for coordination sequence of B_n lattice: ((Sum_{i=0..n} binomial(2*n+1, 2*i)*z^i) - 2*n*z*(1+z)^(n-1))/(1-z)^n. [Bacher et al.]

A019561 Coordination sequence for C_5 lattice.

Original entry on oeis.org

1, 50, 450, 1970, 5890, 14002, 28610, 52530, 89090, 142130, 216002, 315570, 446210, 613810, 824770, 1086002, 1404930, 1789490, 2248130, 2789810, 3424002, 4160690, 5010370, 5984050, 7093250, 8350002

Offset: 0

mbaake(AT)sunelc3.tphys.physik.uni-tuebingen.de (Michael Baake)

Seiichi Manyama, Table of n, a(n) for n = 0..10000
M. Baake and U. Grimm, Coordination sequences for root lattices and related graphs, arXiv:cond-mat/9706122, Zeit. f. Kristallographie, 212 (1997), 253-256.
R. Bacher, P. de la Harpe and B. Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A103884 (row 5). For coordination sequences of other C_n lattices see A022144 (C_2), A010006 (C3), A019560 - A019564 (C_4 through C_8), A035746 - A035787 (C_9 through C_50).

Cf. A008413, A137513.

LinearRecurrence[{5,-10,10,-5,1},{1,50,450,1970,5890,14002},30] (* Harvey P. Dale, Nov 21 2021 *)

G.f.: (1+45*x+210*x^2+210*x^3+45*x^4+x^5)/(1-x)^5 = 1+2*x*(5+10*x+x^2)^2/(1-x)^5.
G.f. for sequence with interpolated zeros: cosh(10*arctanh(x)) = 1/2*( ((1 + x)/(1 - x))^5 + ((1 - x)/(1 + x))^5 ) = 1 + 50*x^2 + 450*x^4 + 1970*x^6 + .... - Peter Bala, Apr 09 2017
a(n) = A008413(2*n). - Seiichi Manyama, Jun 08 2018

A103883 Square array A(n,k) read by antidiagonals: coordination sequence for lattice B_n.

Original entry on oeis.org

1, 1, 8, 1, 18, 16, 1, 32, 74, 24, 1, 50, 224, 170, 32, 1, 72, 530, 768, 306, 40, 1, 98, 1072, 2562, 1856, 482, 48, 1, 128, 1946, 6968, 8130, 3680, 698, 56, 1, 162, 3264, 16394, 28320, 20082, 6432, 954, 64, 1, 200, 5154, 34624, 83442, 85992, 42130, 10304, 1250, 72

Offset: 2

Ralf Stephan, Feb 20 2005

			Array, A(n, k), begins:
  1,   8,    16,     24,      32,       40,        48, ... A022144;
  1,  18,    74,    170,     306,      482,       698, ... A022145;
  1,  32,   224,    768,    1856,     3680,      6432, ... A022146;
  1,  50,   530,   2562,    8130,    20082,     42130, ... A022147;
  1,  72,  1072,   6968,   28320,    85992,    214864, ... A022148;
  1,  98,  1946,  16394,   83442,   307314,    907018, ... A022149;
  1, 128,  3264,  34624,  216448,   954880,   3301952, ... A022150;
  1, 162,  5154,  67266,  507906,  2653346,  10666146, ... A022151;
  1, 200,  7760, 122264, 1099040,  6728168,  31208560, ... A022152;
  1, 242, 11242, 210474, 2224178, 15804866,  83999962, ... A022153;
  1, 288, 15776, 346304, 4254912, 34792672, 210482016, ... A022154;
  ...
Antidiagonals, T(n, k), begin as:
  1;
  1,   8;
  1,  18,   16;
  1,  32,   74,    24;
  1,  50,  224,   170,    32;
  1,  72,  530,   768,   306,    40;
  1,  98, 1072,  2562,  1856,   482,   48;
  1, 128, 1946,  6968,  8130,  3680,  698,  56;
  1, 162, 3264, 16394, 28320, 20082, 6432, 954, 64;

G. C. Greubel, Antidiagonals n = 2..50, flattened
M. Baake and U. Grimm, Coordination sequences for root lattices and related graphs, arXiv:cond-mat/9706122 [cond-mat.stat-mech], 1997.
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
Joan Serra-Sagrista, Enumeration of lattice points in l_1 norm, Inf. Proc. Lett. 76 (1-2) (2000) 39-44.

Rows include A022144, A022145, A022146, A022147, A022148, A022149, A022150, A022151, A022152, A022153, A022154.
Columns include A001105.
Cf. A103881, A103884.

A103883:= func< n,k | (&+[Binomial(n-j-1,n-k-1)*(Binomial(2*n-2*k+1,2*j) - 2*j*Binomial(n-k,j)) : j in [0..k]]) >;
[A103883(n,k): k in [0..n-2], n in [2..14]]; // G. C. Greubel, May 24 2023

offset = 2;
T[n_, k_] := SeriesCoefficient[Sum[(Binomial[2n + 1, 2i] - 2i Binomial[n, i]) x^i, {i, 0, n}]/(1 - x)^n, {x, 0, k}];
Table[T[n - k, k], {n, offset, 11}, {k, 0, n - offset}] // Flatten (* Jean-François Alcover, Feb 13 2019 *)

def A103883(n,k): return sum(binomial(n-j-1,n-k-1)*(binomial(2*n-2*k+1,2*j) - 2*j*binomial(n-k,j)) for j in range(k+1))
flatten([[A103883(n,k) for k in range(n-1)] for n in range(2,15)]) # G. C. Greubel, May 24 2023

G.f. of n-th row: (Sum_{i=0..n} (C(2n+1, 2*i) - 2*i*C(n, i))*x^i)/(1-x)^n.
From G. C. Greubel, May 24 2023: (Start)
A(n, k) = Sum_{j=0..k} binomial(n+k-j-1, n-1)*(binomial(2*n+1, 2*j) - 2*j*binomial(n, j)) (array).
T(n, k) = Sum_{j=0..k} binomial(n-j-1, n-k-1)*(binomial(2*n-2*k+1, 2*j) - 2*j*binomial(n-k, j)) (antidiagonals). (End)

A181390 Absolute difference between (sum of previous terms) and (n-th-odd square) with a(1) = 1.

Original entry on oeis.org

1, 0, 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112, 120, 128, 136, 144, 152, 160, 168, 176, 184, 192, 200, 208, 216, 224, 232, 240, 248, 256, 264, 272, 280, 288, 296, 304, 312, 320, 328, 336, 344, 352, 360, 368, 376, 384, 392, 400, 408, 416, 424

Offset: 1

Giovanni Teofilatto, Oct 17 2010

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

Cf. A008590, A022144.

Module[{lst={1}},Do[AppendTo[lst,Abs[Total[lst]-n^2]],{n,1,111,2}];lst] (* or *) Join[{1},LinearRecurrence[{2,-1},{0,8},60]] (* Harvey P. Dale, Aug 23 2012 *)
ad[{t_,n_,a_}]:=Module[{c=Abs[t-(2n-1)^2]},{t+c,n+1,c}]; NestList[ad,{1,1,1},60][[All,3]] (* or  *) Join[{1}, NestList[8 + # &, 0, 60]] (* Harvey P. Dale, May 14 2019 *)

a(n)=if(n>1,8*n-8,1) \\ Charles R Greathouse IV, Jul 31 2013

a(n) = 8*(n-2) = A008590(n-2), n>1. - R. J. Mathar, Oct 18 2010
G.f.: x*(1 - 2*x + 9*x^2)/(-1 + x)^2. -Alexander R. Povolotsky, Oct 18 2010
a(1)=1, a(2)=0, a(3)=8, a(n)=2*a(n-1)-a(n-2). -Harvey P. Dale, Aug 23 2012
E.g.f.: 16 + 9*x + 8*exp(x)*(x - 2). - Stefano Spezia, Apr 03 2023

Adapted g.f. to the offset from Bruno Berselli, Aug 01 2013

A281813 a(0) = 3, a(n) = 8*n + 4 for n > 0.

Original entry on oeis.org

3, 12, 20, 28, 36, 44, 52, 60, 68, 76, 84, 92, 100, 108, 116, 124, 132, 140, 148, 156, 164, 172, 180, 188, 196, 204, 212, 220, 228, 236, 244, 252, 260, 268, 276, 284, 292, 300, 308, 316, 324, 332, 340, 348, 356, 364, 372, 380, 388, 396, 404

Offset: 0

Rayan Ivaturi, Jan 30 2017

Consider a 1 X S rectangle on an infinite grid and surround it successively with the minimum number of 1 X 1 tiles: the initial S on step 0, 2S + 6 on step 1, 2S + 14 on step 2, and so on. This sequence is case S = 3. See Ivaturi link for a connection to sieving for primes.

Rayan Ivaturi, Ripple Numbers
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A017113.

Other 'ripple sequences': A022144 (s=1), A017089 (s=2).

a(n)=if(n>0, 8*n+4, 3) \\ Charles R Greathouse IV, Feb 07 2017

G.f.: (3 + 6*x - x^2)/(1 - x)^2.
a(n) = A017113(n) for n>0, a(0) = 3.
a(n) = A086570(n+1) for n>=1. - R. J. Mathar, Jun 21 2025

Entry revised by Editors of OEIS, Feb 09 2017

cp's OEIS Frontend

A108998 Square array, read by antidiagonals, where row n equals the coordination sequence of B_n lattice, for n >= 0.

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A201279 a(n) = 6n^2 + 10n + 5.

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A217873 a(n) = 4*n*(n^2 + 2)/3.

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A109001 Triangle, read by rows, where g.f. of row n equals the product of (1-x)^n and the g.f. of the coordination sequence for root lattice B_n, for n >= 0.

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GAP

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A019561 Coordination sequence for C_5 lattice.

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A103883 Square array A(n,k) read by antidiagonals: coordination sequence for lattice B_n.

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Magma

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A181390 Absolute difference between (sum of previous terms) and (n-th-odd square) with a(1) = 1.

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A217873 a(n) = 4n(n^2 + 2)/3.