A152750 -id:A152750 - cp's OEIS frontend

A195320 7 times hexagonal numbers: a(n) = 7n(2*n-1).

0, 7, 42, 105, 196, 315, 462, 637, 840, 1071, 1330, 1617, 1932, 2275, 2646, 3045, 3472, 3927, 4410, 4921, 5460, 6027, 6622, 7245, 7896, 8575, 9282, 10017, 10780, 11571, 12390, 13237, 14112, 15015, 15946, 16905, 17892, 18907, 19950, 21021, 22120, 23247, 24402, 25585

Offset: 0

Omar E. Pol, Sep 18 2011

Sequence found by reading the line from 0, in the direction 0, 7, ..., in the square spiral whose vertices are the generalized enneagonal numbers A118277.

Also sequence found by reading the same line (mentioned above) in the Pythagorean spiral whose edges have length A195019 and whose vertices are the numbers A195020. This is the one of the semi-diagonals of the square spiral, which is related to the primitive Pythagorean triple [3, 4, 5]. - Omar E. Pol, Oct 13 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Bisection of A024966.

Cf. A000384, A118277, A152746, A152750, A185019, A193053, A195019, A195020, A198017, A316466.

[7*n*(2*n-1): n in [0..50]]; // Vincenzo Librandi, Sep 28 2011

Table[7*n*(2*n - 1), {n, 0, 50}] (* Paolo Xausa, Jan 29 2025 *)
7*PolygonalNumber[6, Range[0, 50]] (* Paolo Xausa, Jan 29 2025 *)

a(n)=7*n*(2*n-1) \\ Charles R Greathouse IV, Sep 28 2015

a(n) = 14*n^2 - 7*n = 7*A000384(n).
G.f.: -7*x*(1+3*x)/(x-1)^3. - R. J. Mathar, Sep 27 2011
From Elmo R. Oliveira, Dec 27 2024: (Start)
E.g.f.: 7*exp(x)*x*(2*x + 1).
a(n) = A316466(n) - n = A024966(2*n+1).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A187586 T(n,k)=Number of n-step E, S, NW and NE-moving king's tours on a kXk board summed over all starting positions.

Original entry on oeis.org

1, 4, 0, 9, 6, 0, 16, 20, 8, 0, 25, 42, 48, 5, 0, 36, 72, 120, 84, 0, 0, 49, 110, 224, 286, 106, 0, 0, 64, 156, 360, 604, 578, 104, 0, 0, 81, 210, 528, 1038, 1484, 1069, 78, 0, 0, 100, 272, 728, 1588, 2794, 3514, 1708, 34, 0, 0, 121, 342, 960, 2254, 4508, 7480, 7666, 2309, 13

Offset: 1

R. H. Hardin Mar 11 2011

Table starts
.1.4...9...16....25.....36.....49......64......81.....100.....121.....144
.0.6..20...42....72....110....156.....210.....272.....342.....420.....506
.0.8..48..120...224....360....528.....728.....960....1224....1520....1848
.0.5..84..286...604...1038...1588....2254....3036....3934....4948....6078
.0.0.106..578..1484...2794...4508....6626....9148...12074...15404...19138
.0.0.104.1069..3514...7480..12874...19696...27946...37624...48730...61264
.0.0..78.1708..7666..19104..35832...57592...84384..116208..153064..194952
.0.0..34.2309.15056..45718..95776..164135..250132..353767..475040..613951
.0.0..13.2792.27252.103108.246792..458018..732810.1069534.1468190.1928778
.0.0...0.3108.45960.219432.609070.1243461.2111652.3201436.4508924

			Some k=4 solutions for 4X4
..0..0..0..0....3..0..0..0....0..0..0..0....0..4..0..0....0..0..0..3
..0..0..0..0....4..2..0..0....4..2..0..0....3..0..0..0....0..0..2..4
..0..1..2..0....1..0..0..0....0..3..1..0....0..2..0..0....0..0..0..1
..0..0..3..4....0..0..0..0....0..0..0..0....1..0..0..0....0..0..0..0

R. H. Hardin, Table of n, a(n) for n = 1..200

Row 2 is A002943(n-1)

Row 3 is A152750(n-1)

Empirical: T(1,k) = k^2
Empirical: T(2,k) = 4*k^2 - 6*k + 2
Empirical: T(3,k) = 16*k^2 - 40*k + 24
Empirical: T(4,k) = 58*k^2 - 204*k + 174 for k>2
Empirical: T(5,k) = 202*k^2 - 912*k + 994 for k>3
Empirical: T(6,k) = 714*k^2 - 3888*k + 5104 for k>4
Empirical: T(7,k) = 2516*k^2 - 15980*k + 24408 for k>5
Empirical: T(8,k) = 8819*k^2 - 63926*k + 111127 for k>6
Empirical: T(9,k) = 30966*k^2 - 251630*k + 489234 for k>7
Empirical: T(10,k) = 108852*k^2 - 978404*k + 2100276 for k>8

A067239 a(0)=1, a(n) = 8n(2*n-1).

Original entry on oeis.org

1, 8, 48, 120, 224, 360, 528, 728, 960, 1224, 1520, 1848, 2208, 2600, 3024, 3480, 3968, 4488, 5040, 5624, 6240, 6888, 7568, 8280, 9024, 9800, 10608, 11448, 12320, 13224, 14160, 15128, 16128, 17160, 18224, 19320, 20448, 21608, 22800, 24024

Offset: 0

Benoit Cloitre, Mar 03 2002

Engel expansion of cosh(1/2).

Also, 8 times hexagonal numbers (8*A000384(n) = A152750(n)), for n > 0. - Omar E. Pol, Dec 14 2008

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A006784 for Engel expansion definition.

Cf. A000384, A152750.

s=0;lst={s};Do[s+=n++ +8;AppendTo[lst, s], {n, 0, 8!, 32}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 16 2008 *)

Vec((1+5*x+27*x^2-x^3)/(1-x)^3+O(x^99)) \\ Charles R Greathouse IV, Apr 13 2012

a(n) = a(n-1) + 32*n - 24 (with a(1)=8). - Vincenzo Librandi, Dec 15 2010
From Colin Barker, Apr 13 2012: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3.
G.f.: (1 + 5*x + 27*x^2 - x^3)/(1-x)^3. (End)
E.g.f.: 1 + 8*exp(x)*x*(1 + 2*x). - Elmo R. Oliveira, Dec 15 2024
From Amiram Eldar, May 05 2025: (Start)
Sum_{n>=0} 1/a(n) = log(2)/4 + 1.
Sum_{n>=0} (-1)^n/a(n) = 1 - Pi/16 + log(2)/8. (End)

A267522 a(n) = 4(n + 1)(n + 2)(4n + 3)/3.

Original entry on oeis.org

8, 56, 176, 400, 760, 1288, 2016, 2976, 4200, 5720, 7568, 9776, 12376, 15400, 18880, 22848, 27336, 32376, 38000, 44240, 51128, 58696, 66976, 76000, 85800, 96408, 107856, 120176, 133400, 147560, 162688, 178816, 195976, 214200, 233520, 253968, 275576, 298376, 322400

Offset: 0

Ilya Gutkovskiy, Apr 09 2016

Partial sums of A152750.

			a(0) = (0 + 2)*(1 + 3) = 8;
a(1) = (0 + 2)*(1 + 3) + (2 + 4)*(3 + 5) = 56;
a(2) = (0 + 2)*(1 + 3) + (2 + 4)*(3 + 5) + (4 + 6)*(5 + 7) = 176;
a(3) = (0 + 2)*(1 + 3) + (2 + 4)*(3 + 5) + (4 + 6)*(5 + 7) + (6 + 8)*(7 + 9) = 400, etc

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A002412, A152750, A268684.

Table[(4 (n + 1)) (n + 2) ((4 n + 3)/3), {n, 0, 38}]
LinearRecurrence[{4, -6, 4, -1}, {8, 56, 176, 400}, 39]

a(n) = 4*(n + 1)*(n + 2)*(4*n + 3)/3; \\ Michel Marcus, Apr 10 2016

x='x+O('x^99); Vec(8*(1+3*x)/(1-x)^4) \\ Altug Alkan, Apr 10 2016

G.f.: 8*(1 + 3*x)/(1 - x)^4.
E.g.f.: (4/3)*exp(x)*(6 + 36*x + 27*x^2 + 4*x^3).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) = 4*A268684(n + 1).
Sum_{n>=0} 1/a(n) = -3*(2*Pi - 12*log(2) + 1)/20 = 0.15518712893...
a(n) = 8*A002412(n+1). - Yasser Arath Chavez Reyes, Feb 23 2024

cp's OEIS Frontend

A195320 7 times hexagonal numbers: a(n) = 7*n*(2*n-1).

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A187586 T(n,k)=Number of n-step E, S, NW and NE-moving king's tours on a kXk board summed over all starting positions.

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A067239 a(0)=1, a(n) = 8*n*(2*n-1).

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Author

Keywords

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Mathematica

PARI

Formula

A267522 a(n) = 4*(n + 1)*(n + 2)*(4*n + 3)/3.

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Examples

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Crossrefs

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Mathematica

PARI

PARI

Formula

A195320 7 times hexagonal numbers: a(n) = 7n(2*n-1).

A067239 a(0)=1, a(n) = 8n(2*n-1).

A267522 a(n) = 4(n + 1)(n + 2)(4n + 3)/3.