A235332 -id:A235332 - cp's OEIS frontend

A022267 a(n) = n(9n + 1)/2.

0, 5, 19, 42, 74, 115, 165, 224, 292, 369, 455, 550, 654, 767, 889, 1020, 1160, 1309, 1467, 1634, 1810, 1995, 2189, 2392, 2604, 2825, 3055, 3294, 3542, 3799, 4065, 4340, 4624, 4917, 5219, 5530, 5850, 6179

Offset: 0

N. J. A. Sloane

From Floor van Lamoen, Jul 21 2001: (Start)
Write 0, 1, 2, 3, 4, ... in a triangular spiral; then a(n) is the sequence found by reading the line from 0 in the direction 0, 5, ... . The spiral begins:
.
            15
            / \
          16  14
          /     \
        17   3  13
        /   / \   \
      18   4   2  12
      /   /     \   \
    19   5   0---1  11
    /   /             \
  20   6---7---8---9--10
.
(End)
a(n) is the sum of n consecutive integers starting from 4*n+1: (5), (9+10), (13+14+15), ... - Klaus Purath, Jul 07 2020
a(n) with n>0 are the numbers with the periodic length 3 in the Bulgarian and Mancala solitaire. - Paul Weisenhorn, Jan 29 2022

Lei Zhou, Table of n, a(n) for n = 0..10000
Milan Janjic, Two Enumerative Functions
Leo Tavares, Illustration: X Triangles
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A051682, A110449, A235332.
Cf. numbers of the form n*(d*n+10-d)/2: A008587, A056000, A028347, A140090, A014106, A028895, A045944, A186029, A007742, A033429, A022268, A049452, A186030, A135703, A152734, A139273.
Cf. similar sequences listed in A254963.
Cf. similar sequences listed in A022289.
Cf. A060544, A016813.

seq(binomial(9*n+1,2)/9, n=0..37); # Zerinvary Lajos, Jan 21 2007

Table[ n (9 n + 1)/2, {n, 0, 40}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 5, 19}, 40] (* Harvey P. Dale, Jul 01 2013 *)

vector(100,n,(n-1)*(9*n-8)/2) \\ Derek Orr, Feb 06 2015

a(n) = A110449(n, 4) for n>3.
From Bruno Berselli, Feb 11 2011: (Start)
G.f.: x*(5 + 4*x)/(1 - x)^3.
a(n) = 4*A000217(n) + A000566(n). (End)
a(n) = 9*n + a(n-1) - 4 with n>0, a(0)=0. - Vincenzo Librandi, Aug 04 2010
a(n) = A218470(9*n+4). - Philippe Deléham, Mar 27 2013
a(n) = A000217(5*n) - A000217(4*n). - Bruno Berselli, Oct 13 2016
E.g.f.: (1/2)*(9*x^2 + 10*x)*exp(x). - G. C. Greubel, Jul 17 2017
a(n) = A060544(n+1) - A016813(n). - Leo Tavares, Mar 20 2022

A022266 a(n) = n(9n - 1)/2.

Original entry on oeis.org

0, 4, 17, 39, 70, 110, 159, 217, 284, 360, 445, 539, 642, 754, 875, 1005, 1144, 1292, 1449, 1615, 1790, 1974, 2167, 2369, 2580, 2800, 3029, 3267, 3514, 3770, 4035, 4309, 4592, 4884, 5185, 5495, 5814, 6142, 6479, 6825, 7180, 7544, 7917, 8299, 8690, 9090, 9499

Offset: 0

N. J. A. Sloane

From Floor van Lamoen, Jul 21 2001: (Start)
Write 0,1,2,3,4,... in a triangular spiral, then a(n) is the sequence found by reading the line from 0 in the direction 0,4,...
The spiral begins:
            15
            / \
          16  14
          /     \
        17   3  13
        /   / \   \
      18   4   2  12
      /   /     \   \
    19   5   0---1  11
    /   /             \
  20   6---7---8---9--10
(End)
a(n) with n>0 are the numbers with period length 3 in Bulgarian and Mancala solitaire. - Paul Weisenhorn Jan 29 2022

G. C. Greubel, Table of n, a(n) for n = 0..5000
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).
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A000326, A022267, A049452, A051682, A218470, A235332.

Cf. similar sequences listed in A022288.

[n*(9*n-1)/2 : n in [0..50]]; // Wesley Ivan Hurt, Dec 04 2016

[seq(binomial(9*n,2)/9, n=0..37)]; # Zerinvary Lajos, Jan 02 2007
seq(n*(6*n-1)-n*(3*n-1)/2, n=0..37); # Zerinvary Lajos, Jun 12 2007

Table[n (9 n - 1)/2, {n, 0, 40}] (* Bruno Berselli, Oct 17 2016 *)
LinearRecurrence[{3,-3,1},{0,4,17},50] (* Harvey P. Dale, Aug 06 2023 *)

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

a(n) = binomial(9*n,2)/9 for n >= 0. - Zerinvary Lajos, Jan 02 2007
a(n) = A049452(n) - A000326(n). - Zerinvary Lajos, Jun 12 2007
a(n) = 9*n + a(n-1) - 5 for n > 0, a(0)=0. - Vincenzo Librandi, Aug 04 2010
G.f.: x*(4 + 5*x)/(1 - x)^3. - Colin Barker, Feb 14 2012
a(n) = A218470(9*n+3). - Philippe Deléham, Mar 27 2013
a(n) = A000217(5*n-1) - A000217(4*n-1). - Bruno Berselli, Oct 17 2016
From Wesley Ivan Hurt, Dec 04 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.
a(n) = (1/7) * Sum_{i=n..(8*n-1)} i. (End)
E.g.f.: (x/2)*(9*x + 8)*exp(x). - G. C. Greubel, Aug 24 2017
a(n) = A000326(3*n) / 3. - Joerg Arndt, May 04 2021

A062123 a(n) = (9n^2 + 9n + 4)/2.

Original entry on oeis.org

2, 11, 29, 56, 92, 137, 191, 254, 326, 407, 497, 596, 704, 821, 947, 1082, 1226, 1379, 1541, 1712, 1892, 2081, 2279, 2486, 2702, 2927, 3161, 3404, 3656, 3917, 4187, 4466, 4754, 5051, 5357, 5672, 5996, 6329, 6671, 7022, 7382, 7751, 8129, 8516, 8912, 9317

Offset: 0

Vladeta Jovovic, Jun 04 2001

Third column of A046741.

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983,(2.3.14).

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

Cf. dumbbells: A002940, A002941, A002889, A046741, A055608, A062124-A062127.

Cf. A235332.

List([0..50], n -> 2 +9*n*(1+n)/2); # G. C. Greubel, Jan 31 2019

[2 +9*n*(1+n)/2: n in [0..50]]; // G. C. Greubel, Jan 31 2019

Table[2 +9*n*(1+n)/2, {n,0,50}] (* G. C. Greubel, Jan 31 2019 *)
LinearRecurrence[{3,-3,1},{2,11,29},50] (* Harvey P. Dale, Jan 12 2020 *)

for (n=0, 1000, write("b062123.txt", n, " ", 2 + (n + n^2)*9/2) ) \\ Harry J. Smith, Aug 02 2009

[2 +9*n*(1+n)/2 for n in range(50)] # G. C. Greubel, Jan 31 2019

G.f.: (1+2*x)*(2+x)/(1-x)^3. Generally, g.f. for k-th column of A046741 is coefficient of y^k in expansion of (1-y)/((1-y-y^2)*(1-y)-(1+y)*x).
a(n) = 9*n + a(n-1), with n>0, a(0)=2. - Vincenzo Librandi, Aug 07 2010
E.g.f.: (4 +18*x +9*x^2)*exp(x)/2. - G. C. Greubel, Jan 31 2019

More terms from Larry Reeves (larryr(AT)acm.org), Jun 06 2001

A064225 a(n) = (9n^2 + 5n + 2)/2.

Original entry on oeis.org

1, 8, 24, 49, 83, 126, 178, 239, 309, 388, 476, 573, 679, 794, 918, 1051, 1193, 1344, 1504, 1673, 1851, 2038, 2234, 2439, 2653, 2876, 3108, 3349, 3599, 3858, 4126, 4403, 4689, 4984, 5288, 5601, 5923, 6254, 6594, 6943, 7301, 7668, 8044, 8429, 8823, 9226

Offset: 0

N. J. A. Sloane, Sep 22 2001

Diagonal of triangular spiral in A051682. - Michael Somos, Jul 22 2006
Ehrhart polynomial of closed quadrilateral with vertices (0,2),(2,3),(3,1),(2,0). - Michael Somos, Jul 22 2006
In the natural number array A000027 this sequence is the first knight moves diagonal (A081267 is the second, A001844 is the main diagonal). It can be used to define this diagonal for any array: A007318(A064225-1)=A005809 (Subtraction by 1 because A007318 is defined with offset 0.) - Tilman Piesk, Mar 24 2012
Or positions of pentagonal numbers, such that p(a(n)) = p(a(n)-1) + p(3*n+1), where p=A000326. - Vladimir Shevelev, Jan 21 2014

			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 o
.                   o o     o o o o o
.                      o       o o o o
.                                 o o
.                                    o
.
.   1     8        24           49
- _Aaron David Fairbanks_, Feb 23 2025

Harry J. Smith, Table of n, a(n) for n = 0..1000
National Security Agency, Intrigued? (advertisement), Notices of the Amer. Math. Soc., vol. 49 (2002), p. 216.
J. A. Siehler, Selections without adjacency on a rectangular grid, arXiv:1409.3869, Table 3, k=2 (different offset)
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000027, A000326, A001844, A005809, A007318, A051682, A064226, A081267, A235332.

Column w=2 of A371967.

Table[(9n^2+5n+2)/2,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{1,8,24},51] (* Harvey P. Dale, Sep 13 2011 *)

{a(n) = 1 + n * (9*n + 5) / 2}; /* Michael Somos, Jul 22 2006 */

(define (A064225 n) (/ (+ (* 9 n n) (* 5 n) 2) 2))

a(n) = 9*n+a(n-1)-2, with n>0, a(0) = 1. - Vincenzo Librandi, Aug 07 2010
a(0)=1, a(1)=8, a(2)=24, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Sep 13 2011
G.f.: (1+5*x+3*x^2)/(1-x)^3. - Colin Barker, Feb 23 2012
A064226(n) = a(-1-n). - Michael Somos, Jul 22 2006 (While the sequence itself is only one-way infinite, this identity works, as the defining formula (in the Name-field) produces integers also for the negative values of n, -1, -2, -3, etc.) - Antti Karttunen, Mar 24 2012
E.g.f.: exp(x)*(2 + 14*x + 9*x^2)/2. - Stefano Spezia, Dec 25 2022

A064226 a(n) = (9n^2 + 13n + 6)/2.

Original entry on oeis.org

3, 14, 34, 63, 101, 148, 204, 269, 343, 426, 518, 619, 729, 848, 976, 1113, 1259, 1414, 1578, 1751, 1933, 2124, 2324, 2533, 2751, 2978, 3214, 3459, 3713, 3976, 4248, 4529, 4819, 5118, 5426, 5743, 6069, 6404, 6748, 7101, 7463, 7834, 8214, 8603, 9001, 9408, 9824

Offset: 0

N. J. A. Sloane, Sep 22 2001

Diagonal of triangular spiral in A051682. - Paul Barry, Mar 15 2003

Ehrhart polynomial of open quadrilateral with vertices (0,2),(2,3),(3,1),(2,0). - Michael Somos, Jul 22 2006

Harry J. Smith, Table of n, a(n) for n = 0..1000
Milan Janjic, Two Enumerative Functions.
National Security Agency, Intrigued? (advertisement), Notices of the American Mathematical Society, Vol. 49 (2002), p. 216.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A051682, A064225, A081267, A081268, A235332.

I:=[3,14,34]; [n le 3 select I[n] else 3*Self(n-1) - 3*Self(n-2) + Self(n-3): n in [1..50]]; // Vincenzo Librandi, Jul 19 2015

A064226:=n-> (9*n^2 + 13*n + 6) / 2; seq(A064226(n), n=0..50); # Wesley Ivan Hurt, May 08 2014

Table[(9 n^2 + 13 n + 6)/2, {n, 0, 50}] (* Wesley Ivan Hurt, May 08 2014 *)
LinearRecurrence[{3, -3, 1}, {3, 14, 34}, 50] (* Vincenzo Librandi, Jul 19 2015 *)

{a(n) = 3 + n * (9*n + 13) / 2}; /* Michael Somos, Jul 22 2006 */

From Paul Barry, Mar 15 2003: (Start)
a(n) = 3*C(n,0) + 11*C(n,1) + 9*C(n,2); binomial transform of (3, 11, 9, 0, 0, 0, ...).
G.f.: (3 + 5*x + x^2)/(1-x)^3.
a(n) = A081268(n) + 2. (End)
A064225(n) = a(-1-n). -  Michael Somos, Jul 22 2006
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Wesley Ivan Hurt, Apr 16 2023
E.g.f.: (3 + 11*x + 9*x^2/2)*exp(x). - Elmo R. Oliveira, Oct 21 2024

A178977 a(n) = (3n+2)(3*n+5)/2.

Original entry on oeis.org

5, 20, 44, 77, 119, 170, 230, 299, 377, 464, 560, 665, 779, 902, 1034, 1175, 1325, 1484, 1652, 1829, 2015, 2210, 2414, 2627, 2849, 3080, 3320, 3569, 3827, 4094, 4370, 4655, 4949, 5252, 5564, 5885, 6215, 6554, 6902, 7259, 7625, 8000, 8384, 8777, 9179, 9590, 10010

Offset: 0

Paul Curtz, Jan 02 2011

Companion to A145910.

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

Cf. A008585, A145910, A168233, A168300, A016789, A178971, A235332.

[n*(n+3)/2: n in [2..135 by 3]]; // Bruno Berselli, Apr 14 2011

A178977:=n->(3*n+2)*(3*n+5)/2: seq(A178977(n), n=0..50); # Wesley Ivan Hurt, Oct 23 2014

f[n_] := (3 n + 2) (3 n + 5)/2; Array[f, 45, 0]
LinearRecurrence[{3,-3,1},{5,20,44},50] (* Harvey P. Dale, Apr 19 2013 *)

a(n)=(3*n+2)*(3*n+5)/2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = a(n-1) + 6 + 9*n.
a(n) = A178971(3*n+2).
a(n) = A145910(n) + 3 + 3*n = A145910(n) + A008585(n+1).
a(n) = A168233(n+1)*A168300(n+1).
G.f.: (-5-5*x+x^2)/(x-1)^3. [Adapted to the offset by Bruno Berselli, Apr 14 2011]
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Apr 19 2013
From Amiram Eldar, Mar 10 2022: (Start)
Sum_{n>=0} 1/a(n) = 1/3.
Sum_{n>=0} (-1)^n/a(n) = 4*Pi/(9*sqrt(3)) - 1/3 - 4*log(2)/9. (End)
From Elmo R. Oliveira, Oct 30 2024: (Start)
E.g.f.: exp(x)*exp(x)*(5 + 15*x + 9*x^2/2).
a(n) = A016789(n)*A016789(n+1)/2. (End)

A235537 Expansion of (6 + 13x - 8x^2 - 8x^3 + 6x^4)/((1 + x)^2*(1 - x)^3).

Original entry on oeis.org

6, 19, 23, 41, 49, 72, 84, 112, 128, 161, 181, 219, 243, 286, 314, 362, 394, 447, 483, 541, 581, 644, 688, 756, 804, 877, 929, 1007, 1063, 1146, 1206, 1294, 1358, 1451, 1519, 1617, 1689, 1792, 1868, 1976, 2056, 2169, 2253, 2371, 2459, 2582, 2674, 2802, 2898

Offset: 0

Bruno Berselli, Jan 23 2014

Bruno Berselli, Table of n, a(n) for n = 0..1000
Bruno Berselli, Triangular spiral which contains the terms of A235537 (see A051682).
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A235332.

[(6*n*(3*n+17)-(2*n+43)*(-1)^n+11)/16+8: n in [0..50]];

Table[(6 n (3 n + 17) - (2 n + 43) (-1)^n + 11)/16 + 8, {n, 0, 50}]
LinearRecurrence[{1,2,-2,-1,1},{6,19,23,41,49},80] (* Harvey P. Dale, Aug 22 2015 *)

G.f.: (6 + 13*x - 8*x^2 - 8*x^3 + 6*x^4)/((1 + x)^2*(1 - x)^3).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
a(n) = (6*n*(3*n + 17) - (2*n + 43)*(-1)^n + 11)/16 + 8. The terms a(2k) are in A235332; the closed form of the terms a(2k+1) is n*(9*n+35)/2+19.

cp's OEIS Frontend

A022267 a(n) = n*(9*n + 1)/2.

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A022266 a(n) = n*(9*n - 1)/2.

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A062123 a(n) = (9n^2 + 9n + 4)/2.

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A064225 a(n) = (9*n^2 + 5*n + 2)/2.

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A064226 a(n) = (9*n^2 + 13*n + 6)/2.

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A178977 a(n) = (3*n+2)*(3*n+5)/2.

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A022267 a(n) = n(9n + 1)/2.

A022266 a(n) = n(9n - 1)/2.

A064225 a(n) = (9n^2 + 5n + 2)/2.

A064226 a(n) = (9n^2 + 13n + 6)/2.

A178977 a(n) = (3n+2)(3*n+5)/2.

A235537 Expansion of (6 + 13x - 8x^2 - 8x^3 + 6x^4)/((1 + x)^2*(1 - x)^3).