A139592 -id:A139592 - cp's OEIS frontend

A035608 Expansion of g.f. x(1 + 3x)/((1 + x)*(1 - x)^3).

0, 1, 5, 10, 18, 27, 39, 52, 68, 85, 105, 126, 150, 175, 203, 232, 264, 297, 333, 370, 410, 451, 495, 540, 588, 637, 689, 742, 798, 855, 915, 976, 1040, 1105, 1173, 1242, 1314, 1387, 1463, 1540, 1620, 1701, 1785, 1870, 1958, 2047, 2139, 2232, 2328, 2425, 2525, 2626

Offset: 0

N. J. A. Sloane

Maximum value of Voronoi's principal quadratic form of the first type when variables restricted to {-1,0,1}. - Michael Somos, Mar 10 2004
This is the main row of a version of the "square spiral" when read alternatively from left to right (see link). See also A001107, A007742, A033954, A033991. It is easy to see that the only prime in the sequence is 5. - Emilio Apricena (emilioapricena(AT)yahoo.it), Feb 08 2009
From Mitch Phillipson, Manda Riehl, Tristan Williams, Mar 06 2009: (Start)
a(n) gives the number of elements of S_2 \wr C_k that avoid the pattern 12, using the following ordering:
In S_j, a permutation p avoids a pattern q if it has no subsequence that is order-isomorphic to q. For example, p avoids the pattern 132 if it has no subsequence abc with a < c < b. We extend this notion to S_j \wr C_n as follows. Element psi =[ alpha_1^beta_1, ... alpha_j^beta_j ] avoids tau = [ a_1 ... a_m ] (tau in S_m) if psi' = [ alpha_1*beta_1 ... alpha_j*beta_j ] avoids tau in the usual sense. For n=2, there are 5 elements of S_2 \wr C_2 that avoid the pattern 12. They are: [ 2^1,1^1 ], [ 2^2,1^1 ], [ 2^2,1^2 ], [ 2^1,1^2 ], [ 1^2,2^1 ].
For example, if psi = [2^1,1^2], then psi'=[2,2] which avoids tau=[1,2] because no subsequence ab of psi' has a < b. (End)

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 115.

William A. Tedeschi, Table of n, a(n) for n = 0..10000
Emilio Apricena, A version of the Ulam spiral (This is called the square spiral. - _N. J. A. Sloane_, Jul 27 2018)
Emanuele Munarini, Topological indices for the antiregular graphs, Le Mathematiche, Vol. 76, No. 1 (2021), see p. 283.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Partial sums of A042948.
Cf. A002265, A004526, A006578, A011848, A133983, A139592, A173562, A253146.
Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.
Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.

[n^2 + n - 1 - Floor((n-1)/2): n in [0..25]]; // G. C. Greubel, Oct 29 2017

A035608:=n->floor((n + 1/4)^2): seq(A035608(n), n=0..100); # Wesley Ivan Hurt, Oct 29 2017

Table[n^2 + Floor[n/2], {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, Apr 12 2011 *)
CoefficientList[Series[x (1 + 3 x)/((1 + x) (1 - x)^3), {x, 0, 60}], x] (* or *) LinearRecurrence[{2, 0, -2, 1}, {0, 1, 5, 10}, 60] (* Harvey P. Dale, Feb 21 2013 *)

PARI
```
a(n)=n^2+n-1-(n-1)\2
```

a(n) = n^2 + n - 1 - floor((n-1)/2).
a(n) = A011848(2*n+1).
a(n) = A002378(n) - A004526(n+1). - Reinhard Zumkeller, Jan 27 2010
a(n) = 2*A006578(n) - A002378(n)/2 = A139592(n)/2. - Reinhard Zumkeller, Feb 07 2010
a(n) = A002265(n+2) + A173562(n). - Reinhard Zumkeller, Feb 21 2010
a(n) = floor((n + 1/4)^2). - Reinhard Zumkeller, Jan 27 2010
a(n) = (-1)^n*Sum_{i=0..n} (-1)^i*(2*i^2 + 3*i + 1). Omits the leading 0. - William A. Tedeschi, Aug 25 2010
a(n) = n^2 + floor(n/2), from Mathematica section. - Vladimir Joseph Stephan Orlovsky, Apr 12 2011
a(0)=0, a(1)=1, a(2)=5, a(3)=10; for n > 3, a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4). - Harvey P. Dale, Feb 21 2013
For n > 1: a(n) = a(n-2) + 4*n - 3; see also row sums of triangle A253146. - Reinhard Zumkeller, Dec 27 2014
a(n) = 3*A002620(n) + A002620(n+1). - R. J. Mathar, Jul 18 2015
From Amiram Eldar, Mar 20 2022: (Start)
Sum_{n>=1} 1/a(n) = 4 - 2*log(2) - Pi/3.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*Pi/3 - 4*(1-log(2)). (End)
E.g.f.: (x*(2*x + 3)*cosh(x) + (2*x^2 + 3*x - 1)*sinh(x))/2. - Stefano Spezia, Apr 24 2024

A139593 A139276(n) followed by A139272(n+1).

Original entry on oeis.org

0, 3, 11, 22, 38, 57, 81, 108, 140, 175, 215, 258, 306, 357, 413, 472, 536, 603, 675, 750, 830, 913, 1001, 1092, 1188, 1287, 1391, 1498, 1610, 1725, 1845, 1968, 2096, 2227, 2363, 2502, 2646, 2793, 2945, 3100, 3260, 3423, 3591, 3762

Offset: 0

Omar E. Pol, May 03 2008

Sequence found by reading the line from 0, in the direction 0, 3, ... and the same line from 0, in the direction 0, 11, ..., in the square spiral whose vertices are the triangular numbers A000217.

A139593 appears (both numerically and via back of an envelope algebra, but not a publishable proof) to be the cumulative sum of A047470. - Markus J. Q. Roberts, Jul 12 2009

			Array begins:
   0,   3;
  11,  22;
  38,  57;
  81, 108;

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A000217, A046092, A139272, A139276, A077221, A139591, A139592, A139594, A139595, A139596, A139597, A139598, A047470.

LinearRecurrence[{2,0,-2,1},{0,3,11,22},50] (* Harvey P. Dale, Feb 09 2019 *)

Array read by rows: row n gives 8*n^2 + 3n, 8*(n+1)^2 - 5(n+1).
From Colin Barker, Sep 15 2013: (Start)
a(n) = (-1 + (-1)^n + 6*n + 8*n^2)/4.
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
G.f.: -x*(5*x+3) / ((x-1)^3*(x+1)). (End)

Edited by Omar E. Pol, Jul 13 2009

A139596 A033587(n) followed by even hexagonal number A014635(n+1).

Original entry on oeis.org

0, 6, 14, 28, 44, 66, 90, 120, 152, 190, 230, 276, 324, 378, 434, 496, 560, 630, 702, 780, 860, 946, 1034, 1128, 1224, 1326, 1430, 1540, 1652, 1770, 1890, 2016, 2144, 2278, 2414, 2556, 2700, 2850, 3002, 3160, 3320, 3486, 3654, 3828

Offset: 0

Omar E. Pol, May 03 2008

Sequence found by reading the line from 0, in the direction 0, 6,... and the same line from 0, in the direction 0, 14,..., in the square spiral whose vertices are the triangular numbers A000217.

			Array begins:
0, 6
14, 28
44, 66
90, 120

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A000217, A014635, A033587, A046092, A077221, A139591, A139592, A139593, A139595, A139597, A139598.

LinearRecurrence[{2,0,-2,1},{0,6,14,28},50] (* Harvey P. Dale, Jan 20 2024 *)

Array read by rows: row n gives 8*n^2 + 6*n, 8*(n+1)^2 - 2(n+1).
O.g.f.: -2*x*(x+3)/((x-1)^3*(1+x)). - R. J. Mathar, May 06 2008
a(n) = 2*A156859(n). - R. J. Mathar, Feb 28 2018

A139598 A035008(n) followed by A139098(n+1).

Original entry on oeis.org

0, 8, 16, 32, 48, 72, 96, 128, 160, 200, 240, 288, 336, 392, 448, 512, 576, 648, 720, 800, 880, 968, 1056, 1152, 1248, 1352, 1456, 1568, 1680, 1800, 1920, 2048, 2176, 2312, 2448, 2592, 2736, 2888, 3040, 3200, 3360, 3528, 3696, 3872

Offset: 0

Omar E. Pol, May 03 2008

Sequence found by reading the line from 0, in the direction 0, 8, ... and the line from 16, in the direction 16, 48, ..., in the square spiral whose vertices are the triangular numbers A000217.

Also represents the minimum number of segments in the smooth Jordan curve which crosses every edge of an n X n square lattice exactly once. For example, the curve for a 3 X 3 lattice would have at least 32 segments. - Nikolas Novakovic, Aug 28 2022

			Array begins:
   0,   8;
  16,  32;
  48,  72;
  96, 128;

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A000217, A035008, A046092, A139098, A077221, A139591, A139592, A139593, A139595, A139596, A139597.

LinearRecurrence[{2,0,-2,1},{0,8,16,32},50] (* Harvey P. Dale, Sep 27 2019 *)

Array read by rows: row n gives 8*n^2 + 8*n, 8*(n+1)^2.
From Colin Barker, Jul 22 2012: (Start)
a(n) = (1 - (-1)^n + 4*n + 2*n^2).
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
G.f.: 8*x/((1-x)^3*(1+x)). (End)
a(n) = 8*A002620(n+1). - R. J. Mathar, May 04 2014

A139591 A139275(n) followed by 18-gonal number A051870(n+1).

Original entry on oeis.org

0, 1, 9, 18, 34, 51, 75, 100, 132, 165, 205, 246, 294, 343, 399, 456, 520, 585, 657, 730, 810, 891, 979, 1068, 1164, 1261, 1365, 1470, 1582, 1695, 1815, 1936, 2064, 2193, 2329, 2466, 2610, 2755, 2907, 3060, 3220, 3381, 3549, 3718, 3894, 4071, 4255, 4440, 4632

Offset: 0

Omar E. Pol, May 03 2008

Sequence found by reading the line from 0, in the direction 0, 9, ... and the line from 1, in the direction 1, 18, ..., in the square spiral whose vertices are the triangular numbers A000217.

			Array begins:
   0,   1;
   9,  18;
  34,  51;
  75, 100;
  ...

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000217, A046092, A051870, A139275, A077221, A139592, A139593, A139595, A139596, A139597, A139598.

Partial sums of A047393.

Array read by rows: row n gives 8*n^2 + n, 8*(n+1)^2 - 7*(n+1).
G.f.: -x*(7*x+1)/((x-1)^3*(x+1)). - Colin Barker, Oct 16 2012
a(n) = 2*n^2 + (7/2)*n + (3/4)*((-1)^n-1). - Sean A. Irvine, Jul 14 2022

A195605 a(n) = (4n(n+2)+(-1)^n+1)/2 + 1.

Original entry on oeis.org

2, 7, 18, 31, 50, 71, 98, 127, 162, 199, 242, 287, 338, 391, 450, 511, 578, 647, 722, 799, 882, 967, 1058, 1151, 1250, 1351, 1458, 1567, 1682, 1799, 1922, 2047, 2178, 2311, 2450, 2591, 2738, 2887, 3042, 3199, 3362, 3527, 3698, 3871, 4050, 4231, 4418, 4607, 4802

Offset: 0

Bruno Berselli, Sep 21 2011 - based on remarks and sequences by Omar E. Pol

Sequence found by reading the numbers in increasing order on the vertical line containing 2 of the square spiral whose vertices are the triangular numbers (A000217) - see Pol's comments in other sequences visible in this numerical spiral.

Also A077591 (without first term) and A157914 interleaved.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Bruno Berselli, Illustration of initial terms: An origin of A195605.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A000217, A077591, A157914.
Cf. A047621 (contains first differences), A016754 (contains the sum of any two consecutive terms).
Cf. A033585, A069129, A077221, A102083, A139098, A139271-A139277, A139592, A139593, A188135, A194268, A194431, A195241 [incomplete list].

[(4*n*(n+2)+(-1)^n+3)/2: n in [0..48]];

CoefficientList[Series[(2 + 3 x + 4 x^2 - x^3) / ((1 + x) (1 - x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 19 2013 *)
LinearRecurrence[{2,0,-2,1},{2,7,18,31},50] (* Harvey P. Dale, Jan 21 2017 *)

for(n=0, 48, print1((4*n*(n+2)+(-1)^n+3)/2", "));

G.f.: (2+3*x+4*x^2-x^3)/((1+x)*(1-x)^3).
a(n) = a(-n-2) = 2*a(n-1)-2*a(n-3)+a(n-4).
a(n) = A047524(A000982(n+1)).
Sum_{n>=0} 1/a(n) = 1/2 + Pi^2/16 - cot(Pi/(2*sqrt(2)))*Pi/(4*sqrt(2)). - Amiram Eldar, Mar 06 2023

A139595 A139277(n) followed by A139273(n+1).

Original entry on oeis.org

0, 5, 13, 26, 42, 63, 87, 116, 148, 185, 225, 270, 318, 371, 427, 488, 552, 621, 693, 770, 850, 935, 1023, 1116, 1212, 1313, 1417, 1526, 1638, 1755, 1875, 2000, 2128, 2261, 2397, 2538, 2682, 2831, 2983, 3140, 3300, 3465, 3633, 3806

Offset: 0

Omar E. Pol, May 03 2008

Sequence found by reading the line from 0, in the direction 0, 5,... and the same line from 0, in the direction 0, 13,..., in the square spiral whose vertices are the triangular numbers A000217.

			Array begins:
0, 5
13, 26
42, 63
87, 116

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1)

Cf. A000217, A046092, A139273, A139277, A077221, A139591, A139592, A139593, A139596, A139597, A139598.

LinearRecurrence[{2,0,-2,1},{0,5,13,26},50] (* Harvey P. Dale, Jul 31 2021 *)

Array read by rows: row n gives 8*n^2 + 5n, 8*(n+1)^2 - 3(n+1).

G.f.: -x*(5+3*x) / ( (1+x)*(x-1)^3 ). - R. J. Mathar, Feb 13 2011

A139597 A139278(n) followed by A139274(n+1).

Original entry on oeis.org

0, 7, 15, 30, 46, 69, 93, 124, 156, 195, 235, 282, 330, 385, 441, 504, 568, 639, 711, 790, 870, 957, 1045, 1140, 1236, 1339, 1443, 1554, 1666, 1785, 1905, 2032, 2160, 2295, 2431, 2574, 2718, 2869, 3021, 3180, 3340, 3507, 3675, 3850

Offset: 0

Omar E. Pol, May 03 2008

Sequence found by reading the line from 0, in the direction 0, 7,... and the line from 15, in the direction 15, 46,..., in the square spiral whose vertices are the triangular numbers A000217.

			Array begins:
0, 7
15, 30
46, 69
93, 124

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A000217, A046092, A139274, A139278, A077221, A139591, A139592, A139593, A139595, A139596, A139598.

Array read by rows: row n gives 8*n^2 + 7n, 8*(n+1)^2 - (n+1).

a(n) = (3-3*(-1)^n+14*n+8*n^2)/4. a(n) = 2*a(n-1)-2*a(n-3)+a(n-4). G.f.: x*(7+x)/((1-x)^3*(1+x)). [Colin Barker, Jul 22 2012]

A195241 Expansion of (1-x+19x^3-3x^4)/(1-x)^3.

Original entry on oeis.org

1, 2, 3, 23, 59, 111, 179, 263, 363, 479, 611, 759, 923, 1103, 1299, 1511, 1739, 1983, 2243, 2519, 2811, 3119, 3443, 3783, 4139, 4511, 4899, 5303, 5723, 6159, 6611, 7079, 7563, 8063, 8579, 9111, 9659, 10223, 10803, 11399, 12011, 12639, 13283, 13943

Offset: 0

Bruno Berselli, Sep 13 2011 - based on remarks and sequences by Omar E. Pol

Sequence found by reading the line 1, 2, 3, 23,.. in the square spiral whose vertices are the triangular numbers (A000217) - see Pol's comments in other sequences visible in this numerical spiral.

This is a subsequence of A110326 (without signs) and A047838 (apart from the second term, 2).

Bruno Berselli, Table of n, a(n) for n = 0..1000
Bruno Berselli, Illustration of initial terms: An origin of A195241.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217.

Cf. A033585, A069129, A077221, A102083, A139098, A139271-A139277, A139592, A139593, A188135, A194268, A194431, A195605 [incomplete list].

m:=44; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x+19*x^3-3*x^4)/(1-x)^3));

CoefficientList[Series[(1 - x + 19 x^3 - 3 x^4)/(1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Mar 26 2013 *)
LinearRecurrence[{3,-3,1},{1,2,3,23,59},50] (* Harvey P. Dale, Dec 04 2022 *)

makelist(coeff(taylor((1-x+19*x^3-3*x^4)/(1-x)^3, x, 0, n), x, n), n, 0, 43);

Vec((1-x+19*x^3-3*x^4)/(1-x)^3+O(x^44))

G.f.: (1-x+19*x^3-3*x^4)/(1-x)^3.

a(n) = 8*n^2-20*n+11 for n>1; a(0)=1, a(1)=2.

cp's OEIS Frontend

A035608 Expansion of g.f. x*(1 + 3*x)/((1 + x)*(1 - x)^3).

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A139593 A139276(n) followed by A139272(n+1).

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A139596 A033587(n) followed by even hexagonal number A014635(n+1).

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A139598 A035008(n) followed by A139098(n+1).

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A139591 A139275(n) followed by 18-gonal number A051870(n+1).

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A195605 a(n) = (4*n*(n+2)+(-1)^n+1)/2 + 1.

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Magma

Mathematica

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A139595 A139277(n) followed by A139273(n+1).

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Mathematica

A035608 Expansion of g.f. x(1 + 3x)/((1 + x)*(1 - x)^3).

A195605 a(n) = (4n(n+2)+(-1)^n+1)/2 + 1.

A195241 Expansion of (1-x+19x^3-3x^4)/(1-x)^3.