A139596 -id:A139596 - cp's OEIS frontend

A014635 a(n) = 2n(4*n - 1).

0, 6, 28, 66, 120, 190, 276, 378, 496, 630, 780, 946, 1128, 1326, 1540, 1770, 2016, 2278, 2556, 2850, 3160, 3486, 3828, 4186, 4560, 4950, 5356, 5778, 6216, 6670, 7140, 7626, 8128, 8646, 9180, 9730, 10296, 10878, 11476, 12090, 12720, 13366, 14028, 14706

Offset: 0

Mohammad K. Azarian

Even hexagonal numbers.
Number of edges in the join of two complete graphs of order 3n and n, K_3n * K_n - Roberto E. Martinez II, Jan 07 2002
Bisection of A000384. Also, this sequence arises from reading the line from 0, in the direction 0, 6, ..., in the square spiral whose vertices are the triangular numbers A000217. Perfect numbers are members of this sequence because a(A134708(n)) = A000396(n). Also, positive members are a bisection of A139596. - Omar E. Pol, May 07 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..880
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.
Amelia Carolina Sparavigna, The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences, Politecnico di Torino, Italy (2019), [math.NT].
Amelia Carolina Sparavigna, The groupoid of the Triangular Numbers and the generation of related integer sequences, Politecnico di Torino, Italy (2019).
Leo Tavares, Illustration: Diamond Cut Hexagons
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A000384, A000396, A134708, A139596.

Cf. A154105, A016754.

[2*n*(4*n-1): n in [0..50]]; // Vincenzo Librandi, Apr 25 2011

[seq(binomial(4*n,2),n=0..43)]; # Zerinvary Lajos, Jan 02 2007

s=0;lst={s};Do[s+=n++ +6;AppendTo[lst, s], {n, 0, 7!, 16}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 16 2008 *)
Table[2*n*(4*n - 1), {n,0,50}] (* G. C. Greubel, Jul 18 2017 *)
PolygonalNumber[6,Range[0,90,2]] (* or *) LinearRecurrence[{3,-3,1},{0,6,28},50] (* Harvey P. Dale, Jan 21 2023 *)

a(n)=2*n*(4*n-1) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = C(4*n,2), n>=0. - Zerinvary Lajos, Jan 02 2007
O.g.f.: 2*x*(3+5*x)/(1-x)^3. - R. J. Mathar, May 06 2008
a(n) = 8*n^2 - 2*n. - Omar E. Pol, May 07 2008
a(n) = a(n-1) + 16*n - 10 (with a(0)=0). - Vincenzo Librandi, Nov 20 2010
E.g.f.: (8*x^2 + 6*x)*exp(x). - G. C. Greubel, Jul 18 2017
From Vaclav Kotesovec, Aug 18 2018: (Start)
Sum_{n>=1} 1/a(n) = 3*log(2)/2 - Pi/4.
Sum_{n>=1} (-1)^n / a(n) = log(2)/2 + log(1+sqrt(2))/sqrt(2) - Pi / 2^(3/2). (End)
a(n) = A154105(n-1) - A016754(n-1). - Leo Tavares, May 02 2023

More terms from Erich Friedman

A200192 T(n,k)=Number of -k..k arrays x(0..n-1) of n elements with zero sum, adjacent elements differing by more than one, and elements alternately increasing and decreasing.

Original entry on oeis.org

1, 1, 2, 1, 4, 0, 1, 6, 6, 2, 1, 8, 14, 14, 0, 1, 10, 28, 48, 24, 2, 1, 12, 44, 120, 144, 54, 0, 1, 14, 66, 242, 506, 482, 104, 2, 1, 16, 90, 426, 1298, 2240, 1534, 230, 0, 1, 18, 120, 688, 2794, 7266, 9856, 5148, 464, 2, 1, 20, 152, 1040, 5300, 18838, 40632, 44562, 16826

Offset: 1

R. H. Hardin Nov 14 2011

Table starts
.1....1.....1......1.......1........1.........1.........1..........1..........1
.2....4.....6......8......10.......12........14........16.........18.........20
.0....6....14.....28......44.......66........90.......120........152........190
.2...14....48....120.....242......426.......688......1040.......1494.......2066
.0...24...144....506....1298.....2794......5300......9220......14974......23094
.2...54...482...2240....7266....18838.....41938.....83600.....153278.....263198
.0..104..1534...9856...40632...127800....334278....765598....1585416....3034396
.2..230..5148..44562..231916...881008...2702398...7100700...16594066...35377058
.0..464.16826.199932.1320876..6081086..21910764..66127278..174522934..414666246
.2.1028.56918.914676.7630236.42452472.179391752.621239172.1850379990.4897547876

			Some solutions for n=6 k=5
.-4....0....5....2....4...-4....5....4....1....4....1...-3...-3...-1....4....0
..3....3...-4....4...-3....2...-5...-2...-3...-3...-1....5....0....3...-3...-2
.-4...-4....1...-5....5...-2....4....1....3....3....3...-5...-3...-1....0....3
..2....2...-4....1...-5....4...-5...-2...-1...-4...-4....5....5....1...-2...-2
..0...-5....4...-5....4...-3....4....3....1....4....5...-2...-3...-2....2....4
..3....4...-2....3...-5....3...-3...-4...-1...-4...-4....0....4....0...-1...-3

R. H. Hardin, Table of n, a(n) for n = 1..871

Row 3 is A139596(n-1)

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

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

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]

cp's OEIS Frontend

A014635 a(n) = 2*n*(4*n - 1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions

A200192 T(n,k)=Number of -k..k arrays x(0..n-1) of n elements with zero sum, adjacent elements differing by more than one, and elements alternately increasing and decreasing.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

A014635 a(n) = 2n(4*n - 1).