A081268 -id:A081268 - cp's OEIS frontend

A081267 Diagonal of triangular spiral in A051682.

1, 9, 26, 52, 87, 131, 184, 246, 317, 397, 486, 584, 691, 807, 932, 1066, 1209, 1361, 1522, 1692, 1871, 2059, 2256, 2462, 2677, 2901, 3134, 3376, 3627, 3887, 4156, 4434, 4721, 5017, 5322, 5636, 5959, 6291, 6632, 6982, 7341, 7709, 8086, 8472, 8867, 9271

Offset: 0

Paul Barry, Mar 15 2003

Binomial transform of (1, 8, 9, 0, 0, 0, ...).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Milan Janjić, Two Enumerative Functions
Richard J. Mathar, Bivariate generating functions enumerating non-bonding dominoes on rectangular boards, arXiv:2404.18806 (2024) Table 2.
Richard J. Mathar, Bivariate Generating Functions Enumerating Non-Bonding Dominoes on Rectangular Boards, arXiv:2404.18806 [math.CO], 2024. See p. 7.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000326, A050509, A051682, A081268.

Cf. A220083 for a list of numbers of the form n*P(s,n)-(n-1)*P(s,n-1), where P(s,n) is the n-th polygonal number with s sides.

[(9*n^2 + 7*n + 2)/2: n in [0..50]]; // Vincenzo Librandi, Aug 14 2014

LinearRecurrence[{3,-3,1},{1,9,26},50] (* Harvey P. Dale, Aug 13 2014 *)
CoefficientList[Series[(1 + 6 x + 2 x^2)/(1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Aug 14 2014 *)

a(n)=(9*n^2+7*n+2)/2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = C(n, 0) + 8*C(n, 1) + 9*C(n, 2).
a(n) = (9*n^2 + 7*n + 2)/2.
G.f.: (1 + 6*x + 2*x^2)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), for n > 2. a(n) = right term in M^n * [1 1 1], where M = the 3 X 3 matrix [1 0 0 / 3 1 0 / 5 3 1]. M^n * [1 1 1] = [1 3n+1 a(n)]. - Gary W. Adamson, Dec 22 2004
a(n) = 9*n + a(n-1) - 1 with n > 0, a(0)=1. - Vincenzo Librandi, Aug 08 2010
a(n) = (n+1)*A000326(n+1) - (n)*A000326(n). - Bruno Berselli, Dec 10 2012
a(n) = A050509(n) - A050509(n-1). - Bill McEachen, Nov 01 2020
E.g.f.: exp(x)*(2 + 16*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

A081270 Diagonal of triangular spiral in A051682.

Original entry on oeis.org

3, 16, 38, 69, 109, 158, 216, 283, 359, 444, 538, 641, 753, 874, 1004, 1143, 1291, 1448, 1614, 1789, 1973, 2166, 2368, 2579, 2799, 3028, 3266, 3513, 3769, 4034, 4308, 4591, 4883, 5184, 5494, 5813, 6141, 6478, 6824, 7179, 7543, 7916, 8298, 8689, 9089, 9498, 9916

Offset: 0

Paul Barry, Mar 15 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Hacène Belbachir, Toufik Djellal, Jean-Gabriel Luque, On the self-convolution of generalized Fibonacci numbers, arXiv:1703.00323 [math.CO], 2017.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A064226, A081268.

[(9*n^2+17*n+6)/2: n in [0..50]]; // Vincenzo Librandi, Jul 08 2012

CoefficientList[Series[(3+7x-x^2)/(1-x)^3,{x,0,50}],x] (* Vincenzo Librandi, Jul 08 2012 *)

a(n)=(9*n^2+17*n+6)/2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = A064226(n) + 2*n.
a(n) = 3*binomial(n,0) + 13*binomial(n,1) + 9*binomial(n,2); binomial transform of (3, 13, 9, 0, 0, 0, ...).
a(n) = (9*n^2 + 17*n + 6)/2.
G.f.: (3 + 7*x - x^2)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Jul 08 2012
E.g.f.: exp(x)*(6 + 26*x + 9*x^2)/2. - Elmo R. Oliveira, Nov 13 2024

A198392 a(n) = (6n(3n+7)+(2n+13)*(-1)^n+3)/16 + 1.

Original entry on oeis.org

2, 4, 12, 18, 31, 41, 59, 73, 96, 114, 142, 164, 197, 223, 261, 291, 334, 368, 416, 454, 507, 549, 607, 653, 716, 766, 834, 888, 961, 1019, 1097, 1159, 1242, 1308, 1396, 1466, 1559, 1633, 1731, 1809, 1912, 1994, 2102, 2188, 2301, 2391, 2509, 2603, 2726, 2824, 2952

Offset: 0

Bruno Berselli, Oct 25 2011

For an origin of this sequence, see the triangular spiral illustrated in the Links section.

First bisection gives A117625 (without the initial term).

Bruno Berselli, Table of n, a(n) for n = 0..1000
Bruno Berselli, Illustration of initial terms.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A152832 (by Superseeker).

Cf. sequences related to the triangular spiral: A022266, A022267, A027468, A038764, A045946, A051682, A062708, A062725, A062728, A062741, A064225, A064226, A081266-A081268, A081270-A081272, A081275 [incomplete list].

[(6*n*(3*n+7)+(2*n+13)*(-1)^n+3)/16+1: n in [0..50]];

LinearRecurrence[{1,2,-2,-1,1},{2,4,12,18,31},60] (* Harvey P. Dale, Jun 15 2022 *)

for(n=0, 50, print1((6*n*(3*n+7)+(2*n+13)*(-1)^n+3)/16+1", "));

G.f.: (2+2*x+4*x^2+2*x^3-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)-a(-n-1) = A168329(n+1).
a(n)+a(n-1) = A102214(n).
a(2n)-a(2n-1) = A016885(n).
a(2n+1)-a(2n) = A016825(n).

cp's OEIS Frontend

A081267 Diagonal of triangular spiral in A051682.

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

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A081270 Diagonal of triangular spiral in A051682.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A198392 a(n) = (6*n*(3*n+7)+(2*n+13)*(-1)^n+3)/16 + 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

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

A198392 a(n) = (6n(3n+7)+(2n+13)*(-1)^n+3)/16 + 1.