A247792 -id:A247792 - cp's OEIS frontend

A016778 a(n) = (3*n+1)^2.

1, 16, 49, 100, 169, 256, 361, 484, 625, 784, 961, 1156, 1369, 1600, 1849, 2116, 2401, 2704, 3025, 3364, 3721, 4096, 4489, 4900, 5329, 5776, 6241, 6724, 7225, 7744, 8281, 8836, 9409, 10000, 10609, 11236, 11881, 12544, 13225, 13924, 14641, 15376, 16129, 16900, 17689

Offset: 0

N. J. A. Sloane

From Paul Curtz, Mar 28 2019: (Start)
Sequence is a spoke of the hexagonal spiral built from the terms of A016777:
.
        \
       100--97--94--91
          \           \
          49--46--43  88
          / \       \   \
        52  16--13  40  85
        /   / \   \   \   \
      55  19   1  10  37  82
      /   /   /   /   /   /
    58  22   4---7  34  79
      \   \         /   /
      61  25--28--31  76
        \             /
        64--67--70--73
(End)

Shawn A. Broyles, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000212, A000290, A016777, A016779, A016780, A016781, A214550, A247792, A262178.

(3*Range[0,50]+1)^2 (* or *) LinearRecurrence[{3,-3,1},{1,16,49},50] (* Harvey P. Dale, Mar 03 2013 *)

A016778(n):=(3*n+1)^2$
makelist(A016778(n),n,0,20); /* Martin Ettl, Nov 12 2012 */

a(n)=(3*n+1)^2 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = a(n-1) + 3*(6*n-1); a(0)=1. - Vincenzo Librandi, Nov 20 2010
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=1, a(1)=16, a(2)=49. - Harvey P. Dale, Mar 03 2013
a(n) = A247792(n) + 6*n. - Miquel Cerda, Oct 23 2016
G.f.: (1 + 13*x + 4*x^2)/(1 - x)^3. - Ilya Gutkovskiy, Oct 23 2016
a(n) = A000212(3*n) + A000212(1+3*n) + A000212(2+3*n). - Paul Curtz, Mar 28 2019
From Amiram Eldar, Nov 12 2020: (Start)
Sum_{n>=0} 1/a(n) = A214550.
Sum_{n>=0} (-1)^n/a(n) = A262178. (End)
From Elmo R. Oliveira, May 29 2025: (Start)
E.g.f.: exp(x)*(1 + 15*x + 9*x^2).
a(n) = A000290(A016777(n)) = A016777(n)^2. (End)

A069131 Centered 18-gonal numbers.

Original entry on oeis.org

1, 19, 55, 109, 181, 271, 379, 505, 649, 811, 991, 1189, 1405, 1639, 1891, 2161, 2449, 2755, 3079, 3421, 3781, 4159, 4555, 4969, 5401, 5851, 6319, 6805, 7309, 7831, 8371, 8929, 9505, 10099, 10711, 11341, 11989, 12655, 13339, 14041, 14761, 15499, 16255, 17029, 17821

Offset: 1

Terrel Trotter, Jr., Apr 07 2002

Equals binomial transform of [1, 18, 18, 0, 0, 0, ...]. Example: a(3) = 55 = (1, 2, 1) dot (1, 18, 18) = (1 + 36 + 18). - Gary W. Adamson, Aug 24 2010
Narayana transform (A001263) of [1, 18, 0, 0, 0, ...]. - Gary W. Adamson, Jul 28 2011
From Lamine Ngom, Aug 19 2021: (Start)
Sequence is a spoke of the hexagonal spiral built from the terms of A016777 (see illustration in links section).
a(n) is a bisection of A195042.
a(n) is a trisection of A028387.
a(n) + 1 is promic (A002378).
a(n) + 2 is a trisection of A002061.
a(n) + 9 is the arithmetic mean of its neighbors.
4*a(n) + 5 is a square: A016945(n)^2. (End)

			a(5) = 181 because 9*5^2 - 9*5 + 1 = 225 - 45 + 1 = 181.

Ivan Panchenko, Table of n, a(n) for n = 1..1000
John Elias, Illustration of Initial Terms: Triangular & Hexagonal Configurations.
Lamine Ngom, An origin of A069131 (illustration).
Leo Tavares, Illustration: Tri-Hexagons.
Eric Weisstein's World of Mathematics, Centered Polygonal Numbers.
R. Yin, J. Mu, and T. Komatsu, The p-Frobenius Number for the Triple of the Generalized Star Numbers, Preprints 2024, 2024072280. See p. 2.
Index entries for sequences related to centered polygonal numbers.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. centered polygonal numbers listed in A069190.
Cf. A000217, A028387, A195042, A016945, A002378, A082040, A304163, A003215, A247792, A016777,A016778, A016790, A010008, A008600, A002061.
Cf. A000290, A139278, A069129, A062786, A033996, A060544, A027468, A016754, A124080, A069099, A152740, A049598, A005891, A152741, A001844, A163756, A005448, A194715.

[9*n^2 - 9*n + 1 : n in [1..50]]; // Wesley Ivan Hurt, May 05 2021

FoldList[#1 + #2 &, 1, 18 Range@ 45] (* Robert G. Wilson v, Feb 02 2011 *)
LinearRecurrence[{3,-3,1},{1,19,55},50] (* Harvey P. Dale, Jan 20 2014 *)

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

a(n) = 9*n^2 - 9*n + 1.
a(n) = 18*n + a(n-1) - 18 (with a(1)=1). - Vincenzo Librandi, Aug 08 2010
G.f.: ( x*(1+16*x+x^2) ) / ( (1-x)^3 ). - R. J. Mathar, Feb 04 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(1)=1, a(2)=19, a(3)=55. - Harvey P. Dale, Jan 20 2014
From Amiram Eldar, Jun 21 2020: (Start)
Sum_{n>=1} 1/a(n) = Pi*tan(sqrt(5)*Pi/6)/(3*sqrt(5)).
Sum_{n>=1} a(n)/n! = 10*e - 1.
Sum_{n>=1} (-1)^n * a(n)/n! = 10/e - 1. (End)
From Lamine Ngom, Aug 19 2021: (Start)
a(n) = 18*A000217(n) + 1 = 9*A002378(n) + 1.
a(n) = 3*A003215(n) - 2.
a(n) = A247792(n) - 9*n.
a(n) = A082040(n) + A304163(n) - a(n-1) = A016778(n) + A016790(n) - a(n-1), n > 0.
a(n) + a(n+1) = 2*A247792(n) = A010008(n), n > 0.
a(n+1) - a(n) = 18*n = A008600(n). (End)
From Leo Tavares, Oct 31 2021: (Start)
a(n)= A000290(n) + A139278(n-1)
a(n) = A069129(n) + A002378(n-1)
a(n) = A062786(n) + 8*A000217(n-1)
a(n) = A062786(n) + A033996(n-1)
a(n) = A060544(n) + 9*A000217(n-1)
a(n) = A060544(n) + A027468(n-1)
a(n) = A016754(n-1) + 10*A000217(n-1)
a(n) = A016754(n-1) + A124080
a(n) = A069099(n) + 11*A000217(n-1)
a(n) = A069099(n) + A152740(n-1)
a(n) = A003215(n-1) + 12*A000217(n-1)
a(n) = A003215(n-1) + A049598(n-1)
a(n) = A005891(n-1) + 13*A000217(n-1)
a(n) = A005891(n-1) + A152741(n-1)
a(n) = A001844(n) + 14*A000217(n-1)
a(n) = A001844(n) + A163756(n-1)
a(n) = A005448(n) + 15*A000217(n-1)
a(n) = A005448(n) + A194715(n-1). (End)
E.g.f.: exp(x)*(1 + 9*x^2) - 1. - Nikolaos Pantelidis, Feb 06 2023

A158591 a(n) = 36*n^2 + 1.

Original entry on oeis.org

1, 37, 145, 325, 577, 901, 1297, 1765, 2305, 2917, 3601, 4357, 5185, 6085, 7057, 8101, 9217, 10405, 11665, 12997, 14401, 15877, 17425, 19045, 20737, 22501, 24337, 26245, 28225, 30277, 32401, 34597, 36865, 39205, 41617, 44101, 46657, 49285, 51985, 54757, 57601

Offset: 0

Vincenzo Librandi, Mar 22 2009

The identity (36*n^2 + 1)^2 - (324*n^2 + 18)*(2*n)^2 = 1 can be written as a(n)^2 - A158590(n)*A005843(n)^2 = 1.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Vincenzo Librandi, X^2-AY^2=1, Math Forum, 2007. [Wayback Machine link]
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005843, A158590, A247792.

[36*n^2+1: n in [0..40]]; // Vincenzo Librandi, Sep 11 2013

CoefficientList[Series[- (1 + 34 x + 37 x^2) / (x - 1)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Sep 11 2013 *)
36*Range[0,40]^2+1 (* or *) LinearRecurrence[{3,-3,1},{1,37,145},40] (* Harvey P. Dale, Jul 02 2019 *)

a(n)=36*n^2+1 \\ Charles R Greathouse IV, Oct 16 2015

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: -(1+34*x+37*x^2)/(x-1)^3.
From Amiram Eldar, Mar 14 2023: (Start)
Sum_{n>=0} 1/a(n) = (coth(Pi/6)*Pi/6 + 1)/2.
Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/6)*Pi/6 + 1)/2. (End)
From Elmo R. Oliveira, Jan 16 2025: (Start)
E.g.f.: exp(x)*(1 + 36*x + 36*x^2).
a(n) = A247792(2*n). (End)

Comment rewritten, formula replaced by R. J. Mathar, Oct 28 2009

A010008 a(0) = 1, a(n) = 18*n^2 + 2 for n>0.

Original entry on oeis.org

1, 20, 74, 164, 290, 452, 650, 884, 1154, 1460, 1802, 2180, 2594, 3044, 3530, 4052, 4610, 5204, 5834, 6500, 7202, 7940, 8714, 9524, 10370, 11252, 12170, 13124, 14114, 15140, 16202, 17300, 18434, 19604, 20810, 22052, 23330, 24644, 25994, 27380, 28802, 30260

Offset: 0

N. J. A. Sloane

The identity (18*n^2+2)^2-(9*n^2+2)*(6*n)^2=4 can be written as a(n+1)^2-A010002(n+1)*A008588(n+1)^2=4. - Vincenzo Librandi, Feb 07 2012

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

After 20, all terms are in A000408.

Cf. A206399.

[1] cat [18*n^2+2: n in [1..50]]; // Vincenzo Librandi, Aug 03 2015

Join[{1}, 18 Range[41]^2 + 2] (* Bruno Berselli, Feb 06 2012 *)
Join[{1}, LinearRecurrence[{3, -3, 1}, {20, 74, 164}, 50]] (* Vincenzo Librandi, Aug 03 2015 *)

G.f.: (1+x)*(1+16*x+x^2)/(1-x)^3. - Bruno Berselli, Feb 06 2012
a(n) = (3*n-1)^2+(3*n+1)^2 = (n-1)^2+(n+1)^2+(4*n)^2 for n>0. - Bruno Berselli, Feb 06 2012
E.g.f.: (x*(x+1)*18+2)*e^x-1. - Gopinath A. R., Feb 14 2012
Sum_{n>=0} 1/a(n) = 3/4+ (1/12)*Pi*coth(Pi/3) = 1.0853330948... - R. J. Mathar, May 07 2024
a(n) = 2*A247792(n), n>0. - R. J. Mathar, May 07 2024
a(n) = A069131(n)+A069131(n+1). - R. J. Mathar, May 07 2024

More terms from Bruno Berselli, Feb 06 2012

A157888 a(n) = 81*n^2 + 9.

Original entry on oeis.org

90, 333, 738, 1305, 2034, 2925, 3978, 5193, 6570, 8109, 9810, 11673, 13698, 15885, 18234, 20745, 23418, 26253, 29250, 32409, 35730, 39213, 42858, 46665, 50634, 54765, 59058, 63513, 68130, 72909, 77850, 82953, 88218, 93645, 99234, 104985, 110898, 116973, 123210

Offset: 1

Vincenzo Librandi, Mar 08 2009

The identity (18*n^2 + 1)^2 - (81*n^2 + 9)*(2*n)^2 = 1 can be written as A157889(n)^2 - a(n)*A005843(n+1)^2 = 1. - Vincenzo Librandi, Feb 05 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Vincenzo Librandi, X^2-AY^2=1, Math Forum, 2007. [Wayback Machine link]
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005843, A157889, A247792.

I:=[90, 333, 738]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]]; // Vincenzo Librandi, Feb 05 2012

LinearRecurrence[{3, -3, 1}, {90, 333, 738}, 40] (* Vincenzo Librandi, Feb 05 2012 *)
81*Range[40]^2+9 (* Harvey P. Dale, Aug 05 2015 *)

for(n=1, 40, print1(81*n^2 + 9", ")); \\ Vincenzo Librandi, Feb 05 2012

From Vincenzo Librandi, Feb 05 2012: (Start)
G.f: x*(90 + 63*x + 9*x^2)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
From Amiram Eldar, Mar 07 2023: (Start)
Sum_{n>=1} 1/a(n) = (coth(Pi/3)*Pi/3 - 1)/18.
Sum_{n>=1} (-1)^(n+1)/a(n) = (1 - cosech(Pi/3)*Pi/3)/18. (End)
From Elmo R. Oliveira, Jan 16 2025: (Start)
E.g.f.: 9*(exp(x)*(9*x^2 + + 9*x + 1) - 1).
a(n) = 9*A247792(n). (End)

cp's OEIS Frontend

A016778 a(n) = (3*n+1)^2.

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A069131 Centered 18-gonal numbers.

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A158591 a(n) = 36*n^2 + 1.

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A010008 a(0) = 1, a(n) = 18*n^2 + 2 for n>0.

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A157888 a(n) = 81*n^2 + 9.

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