A077221 -id:A077221 - cp's OEIS frontend

A195148 Concentric 20-gonal numbers.

0, 1, 20, 41, 80, 121, 180, 241, 320, 401, 500, 601, 720, 841, 980, 1121, 1280, 1441, 1620, 1801, 2000, 2201, 2420, 2641, 2880, 3121, 3380, 3641, 3920, 4201, 4500, 4801, 5120, 5441, 5780, 6121, 6480, 6841, 7220, 7601, 8000, 8401, 8820, 9241, 9680, 10121

Offset: 0

Omar E. Pol, Sep 17 2011

Concentric icosagonal numbers.

Sequence found by reading the line from 0, in the direction 0, 20, ..., and the same line from 1, in the direction 1, 41, ..., in the square spiral whose vertices are the generalized dodecagonal numbers A195162. Main axis, perpendicular to A124080 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).

A195322 and A195317 interleaved.
Cf. A032528, A077221, A195142, A195143, A195145, A195146, A195147, A195149.
Cf. A032527, A195048, A195049. Column 20 of A195040. - Omar E. Pol, Sep 29 2011

[5*n^2+2*(-1)^n-2: n in [0..50]]; // Vincenzo Librandi, Sep 27 2011

LinearRecurrence[{2,0,-2,1},{0,1,20,41},50] (* Harvey P. Dale, Apr 08 2016 *)

a(n)=5*n^2+2*(-1)^n-2 \\ Charles R Greathouse IV, Sep 28 2015

From Vincenzo Librandi, Sep 27 2011: (Start)
a(n) = 5*n^2 + 2*(-1)^n-2;
a(n) = -a(n-1) + 10*n^2 - 10*n + 1. (End)
G.f.: x*(1+18*x+x^2)/((1+x)*(1-x)^3). - Bruno Berselli, Sep 27 2011
Sum_{n>=1} 1/a(n) = Pi^2/120 + tan(Pi/sqrt(5))*Pi/(8*sqrt(5)). - Amiram Eldar, Jan 17 2023

A194274 Concentric square numbers (see Comments lines for definition).

Original entry on oeis.org

0, 1, 4, 8, 12, 17, 24, 32, 40, 49, 60, 72, 84, 97, 112, 128, 144, 161, 180, 200, 220, 241, 264, 288, 312, 337, 364, 392, 420, 449, 480, 512, 544, 577, 612, 648, 684, 721, 760, 800, 840, 881, 924, 968, 1012, 1057, 1104, 1152, 1200, 1249, 1300, 1352, 1404

Offset: 0

Omar E. Pol, Aug 20 2011

Cellular automaton on the first quadrant of the square grid. The sequence gives the number of cells "ON" in the structure after n-th stage. A098181 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 A194273, A194275 and A032528.
Also, union of A046092 and A077221, the bisections of this sequence.
Also row sums of an infinite square array T(n,k) in which column k lists 4*k-1 zeros followed by the numbers A008574 (see example).

			Using the numbers A008574 we can write:
0, 1, 4, 8, 12, 16, 20, 24, 28, 32, 36, ...
0, 0, 0, 0, 0,  1,   4,  8, 12, 16, 20, ...
0, 0, 0, 0, 0,  0,   0,  0,  0,  1,  4, ...
And so on.
===========================================
The sums of the columns give this sequence:
0, 1, 4, 8, 12, 17, 24, 32, 40, 49, 60, ...
...
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   o o   o o o   o o o o   o o o o o   o o o o o o
.
. 1    4      8        12         17           24

Arkadiusz Wesolowski, Table of n, a(n) for n = 0..10000
Index entries for sequences related to cellular automata
Index entries for linear recurrences with constant coefficients, signature (3,-4,4,-3,1).

Cf. A000290, A005563, A008574, A056594, A032528, A046092.

Cf. A077221, A085250, A098181, A194273, A194275.

[n le 2 select n-1 else (n-1)^2 - Self(n-2): n in [1..61]]; // G. C. Greubel, Jan 31 2024

Table[Floor[3*n/4] + Floor[(n*(n + 2) + 1)/2] - Floor[(3*n + 1)/4], {n, 0, 52}] (* Arkadiusz Wesolowski, Nov 08 2011 *)
RecurrenceTable[{a[0]==0,a[1]==1,a[n]==n^2-a[n-2]},a,{n,60}] (* or *) LinearRecurrence[{3,-4,4,-3,1},{0,1,4,8,12},60] (* Harvey P. Dale, Sep 11 2013 *)

prpr = 0
prev = 1
for n in range(2,777):
    print(str(prpr), end=", ")
    curr = n*n - prpr
    prpr = prev
    prev = curr
# Alex Ratushnyak, Aug 03 2012

def A194274(n): return (3*n>>2)+(n*(n+2)+1>>1)-(3*n+1>>2) # Chai Wah Wu, Jul 15 2023

def A194274(n): return n if n<2 else n^2 - A194274(n-2)
[A194274(n) for n in range(41)] # G. C. Greubel, Jan 31 2024

a(n) = n^2 - a(n-2), with a(0)=0, a(1)=1. - Alex Ratushnyak, Aug 03 2012
From R. J. Mathar, Aug 22 2011: (Start)
G.f.: x*(1 + x)/((1 + x^2)*(1 - x)^3).
a(n) = (A005563(n) - A056594(n-1))/2. (End)
a(n) = a(-n-2) = (2*n*(n+2) + (1-(-1)^n)*i^(n+1))/4, where i=sqrt(-1). - Bruno Berselli, Sep 22 2011
a(n) = floor(3*n/4) + floor((n*(n+2)+1)/2) - floor((3*n+1)/4). - Arkadiusz Wesolowski, Nov 08 2011
a(n) = 3*a(n-1) - 4*a(n-2) + 4*a(n-3) - 3*a(n-4) + a(n-5), with a(0)=0, a(1)=1, a(2)=4, a(3)=8, a(4)=12. - Harvey P. Dale, Sep 11 2013
E.g.f.: (exp(x)*x*(3 + x) - sin(x))/2. - Stefano Spezia, Feb 26 2023

A195042 Concentric 9-gonal numbers.

Original entry on oeis.org

0, 1, 9, 19, 36, 55, 81, 109, 144, 181, 225, 271, 324, 379, 441, 505, 576, 649, 729, 811, 900, 991, 1089, 1189, 1296, 1405, 1521, 1639, 1764, 1891, 2025, 2161, 2304, 2449, 2601, 2755, 2916, 3079, 3249, 3421, 3600, 3781, 3969, 4159, 4356, 4555, 4761, 4969, 5184, 5401, 5625

Offset: 0

Omar E. Pol, Sep 27 2011

Also concentric enneagonal numbers or concentric nonagonal numbers.
A016766 and A069131 interleaved.
Partial sums of A056020. - Reinhard Zumkeller, Jan 07 2012

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 9 of A195040.

Cf. A016766, A069131, A077221, A195041, A195142, A195043.

a195042 n = a195042_list !! n
a195042_list = scanl (+) 0 a056020_list
-- Reinhard Zumkeller, Jan 07 2012

[(9*n^2+5/2*((-1)^n-1))/4: n in [0..50]]; // Vincenzo Librandi, Sep 29 2011

LinearRecurrence[{2,0,-2,1},{0,1,9,19},60] (* Harvey P. Dale, Nov 24 2019 *)

a(n)=(9*n^2+5/2*((-1)^n-1))/4 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = (9*n^2 + 5/2*((-1)^n - 1))/4.
From R. J. Mathar, Sep 28 2011: (Start)
G.f.: -x*(1+7*x+x^2) / ( (1+x)*(x-1)^3 ).
a(n) + a(n+1) = A060544(n+1). (End)
Sum_{n>=1} 1/a(n) = Pi^2/54 + tan(sqrt(5)*Pi/6)*Pi/(3*sqrt(5)). - Amiram Eldar, Jan 16 2023

A139592 A033585(n) followed by A139271(n+1).

Original entry on oeis.org

0, 2, 10, 20, 36, 54, 78, 104, 136, 170, 210, 252, 300, 350, 406, 464, 528, 594, 666, 740, 820, 902, 990, 1080, 1176, 1274, 1378, 1484, 1596, 1710, 1830, 1952, 2080, 2210, 2346, 2484, 2628, 2774, 2926, 3080, 3240, 3402, 3570, 3740

Offset: 0

Omar E. Pol, May 03 2008

Sequence found by reading the line from 0, in the direction 0, 2,... and the same line from 0, in the direction 0, 10,..., in the square spiral whose vertices are the triangular numbers A000217.

a(n) = 2*A006578(n) - A002378(n)/2 = 2*A035608(n). [From Reinhard Zumkeller, Feb 07 2010]

			Array begins:
0, 2
10, 20
36, 54
78, 104

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, A033585, A046092, A139271, A077221, A139591, A139593, A139595, A139596, A139597, A139598.

Array read by rows: row n gives 8*n^2 + 2n, 8*(n+1)^2 - 6(n+1).
a(n) = 2*floor((n + 1/4)^2). [From Reinhard Zumkeller, Feb 07 2010]
G.f.: 2*x*(1+3*x)/((1-x)^3*(1+x)). [Colin Barker, Apr 26 2012]

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

A195041 Concentric heptagonal numbers.

Original entry on oeis.org

0, 1, 7, 15, 28, 43, 63, 85, 112, 141, 175, 211, 252, 295, 343, 393, 448, 505, 567, 631, 700, 771, 847, 925, 1008, 1093, 1183, 1275, 1372, 1471, 1575, 1681, 1792, 1905, 2023, 2143, 2268, 2395, 2527, 2661, 2800, 2941, 3087, 3235, 3388, 3543

Offset: 0

Omar E. Pol, Sep 27 2011

A033582 and A069127 interleaved.

Partial sums of A047336. - Reinhard Zumkeller, Jan 07 2012

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 7 of A195040.

Cf. A032527, A032528, A033582, A047336, A069099, A069127, A077221, A195042.

a195041 n = a195041_list !! n
a195041_list = scanl (+) 0 a047336_list
-- Reinhard Zumkeller, Jan 07 2012

[7*n^2/4+3*((-1)^n-1)/8: n in [0..50]]; // Vincenzo Librandi, Sep 29 2011

CoefficientList[Series[-((x (1+5 x+x^2))/((-1+x)^3 (1+x))),{x,0,80}],x] (* or *) LinearRecurrence[{2,0,-2,1},{0,1,7,15},80] (* Harvey P. Dale, Jan 18 2021 *)

a(n)=7*n^2\4 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 7*n^2/4 + 3*((-1)^n - 1)/8.
From R. J. Mathar, Sep 28 2011: (Start)
G.f.: -x*(1+5*x+x^2) / ( (1+x)*(x-1)^3 ).
a(n) + a(n+1) = A069099(n+1). (End)
a(n) = n^2 + floor(3*n^2/4). - Bruno Berselli, Aug 08 2013
Sum_{n>=1} 1/a(n) = Pi^2/42 + tan(sqrt(3/7)*Pi/2)*Pi/sqrt(21). - Amiram Eldar, Jan 16 2023

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

cp's OEIS Frontend

A195148 Concentric 20-gonal numbers.

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A194274 Concentric square numbers (see Comments lines for definition).

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A195042 Concentric 9-gonal numbers.

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A139592 A033585(n) followed by A139271(n+1).

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

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A195041 Concentric heptagonal numbers.

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

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