A122709 -id:A122709 - cp's OEIS frontend

A038764 a(n) = (9n^2 + 3n + 2)/2.

1, 7, 22, 46, 79, 121, 172, 232, 301, 379, 466, 562, 667, 781, 904, 1036, 1177, 1327, 1486, 1654, 1831, 2017, 2212, 2416, 2629, 2851, 3082, 3322, 3571, 3829, 4096, 4372, 4657, 4951, 5254, 5566, 5887, 6217, 6556, 6904, 7261, 7627, 8002, 8386, 8779, 9181

Offset: 0

N. J. A. Sloane, May 03 2000

Coefficients of x^2 of certain rook polynomials (for n>=1; see p. 18 of the Riordan paper). - Emeric Deutsch, Mar 08 2004

a(n) is also the least weight of self-conjugate partitions having n+1 different parts such that each part is congruent to 1 modulo 3. The first such self-conjugate partitions, corresponding to a(n) = 0, 1, 2, 3, are 1, 4+3, 7+4+4+4+3, 10+7+7+7+4+4+4+3. - Augustine O. Munagi, Dec 18 2008

J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23.

Colin Barker, Table of n, a(n) for n = 0..1000
S. J. Cyvin, B. N. Cyvin, and J. Brunvoll, Unbranched catacondensed polygonal systems containing hexagons and tetragons, Croatica Chem. Acta, 69 (1996), 757-774.
A. O. Munagi, Pairing conjugate partitions by residue classes, Discrete Math., 308 (2008), 2492-2501.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Reflection of A060544 in A081272.
Second column of A024462. Also = A064641(n+1, 2).
Shallow diagonal of triangular spiral in A051682.
Cf. A027468, A080855, A283394.
Partial sums of A122709.

LinearRecurrence[{3, -3, 1}, {1, 7, 22}, 50] (* Paolo Xausa, Jul 03 2025 *)

a(n)=n*(9*n+3)/2+1 \\ Charles R Greathouse IV, Jun 17 2017

Vec((1 + 2*x)^2 / (1 - x)^3 + O(x^60)) \\ Colin Barker, Jan 22 2018

a = lambda n: hypergeometric([-n, -2], [1], 3)
print([simplify(a(n)) for n in range(46)]) # Peter Luschny, Nov 19 2014

a(n) = binomial(n,0) + 6*binomial(n,1) + 9*binomial(n,2).
From Paul Barry, Mar 15 2003: (Start)
G.f.: (1 + 2*x)^2/(1 - x)^3.
Binomial transform of (1, 6, 9, 0, 0, 0, ...). (End)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2. -  Colin Barker, Jan 22 2018
a(n) = a(n-1) + 3*(3*n-1) for n>0, a(0)=1. - Vincenzo Librandi, Nov 17 2010
a(n) = hypergeometric([-n, -2], [1], 3). - Peter Luschny, Nov 19 2014
E.g.f.: exp(x)*(2 + 12*x + 9*x^2)/2. - Stefano Spezia, Mar 07 2023

More terms from James Sellers, May 03 2000

Entry revised by N. J. A. Sloane, Jan 23 2018

A179805 a(0) = 1, a(1) = 3, a(2) = 6 and a(n) = 2*a(n-1) - a(n-2) for n > 3.

Original entry on oeis.org

1, 3, 6, 15, 24, 33, 42, 51, 60, 69, 78, 87, 96, 105, 114, 123, 132, 141, 150, 159, 168, 177, 186, 195, 204, 213, 222, 231, 240, 249, 258, 267, 276, 285, 294, 303, 312, 321, 330, 339, 348, 357, 366, 375, 384, 393, 402

Offset: 0

Gary W. Adamson, Jul 27 2010

Apart from the second term, the same as A122709. - R. J. Mathar, Jul 30 2010

For n > 1, a(n) is the maximum value of the sum of the vertices in a normal magic triangle of order n (see formula 10 in Trotter). - Stefano Spezia, Mar 03 2021

			a(4) = 24 = 9 + a(3) = 9 + 15.
a(4) = 24 = 2*a(3) - a(2) = 2*15 - 6.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Terrel Trotter, Normal Magic Triangles of Order n, Journal of Recreational Mathematics Vol. 5, No. 1, 1972, pp. 28-32.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A122709.

LinearRecurrence[{2,-1},{1,3,6,15},50] (* Harvey P. Dale, Sep 25 2018 *)

(1 + 3*x + 6*x^2 + 15*x^3 + ...) = (1 + 3*x^2 + 3*x^3 + 3*x^4 + ...) * (1 + 3*x + 3*x^2 + 3*x^3 + 3*x^4 + ...).
a(0) = 1, a(1) = 3, a(2) = 6 and a(n) = 2*a(n-1) - a(n-2) for n > 3.
a(n) = a(n-1) + 9 for n > 2.
For n > 1, a(n) == 6 (mod 9).
From Colin Barker, Oct 28 2012: (Start)
a(n) = 9*n - 12 for n > 1.
G.f.: (2*x+1)*(3*x^2-x+1)/(x-1)^2. (End)
E.g.f.: 13 + 6*x + 3*exp(x)*(3x - 4). - Stefano Spezia, Mar 03 2021

A122016 Riordan array(1, x(1+2x)/(1-x)).

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 3, 6, 1, 0, 3, 15, 9, 1, 0, 3, 24, 36, 12, 1, 0, 3, 33, 90, 66, 15, 1, 0, 3, 42, 171, 228, 105, 18, 1, 0, 3, 51, 279, 579, 465, 153, 21, 1, 0, 3, 60, 414, 1200, 1500, 828, 210, 24, 1, 0, 3, 69, 576, 2172, 3858, 3258, 1344, 276, 27, 1

Offset: 0

Philippe Deléham, Sep 24 2006

Triangle T(n,k), 0 <= k <= n, read by rows given by [0,3,-2,0,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. Rising and falling diagonals are A078010 and A122552.

			Triangle begins:
  1;
  0, 1;
  0, 3,  1;
  0, 3,  6,   1;
  0, 3, 15,   9,    1;
  0, 3, 24,  36,   12,    1;
  0, 3, 33,  90,   66,   15,   1;
  0, 3, 42, 171,  228,  105,  18,   1;
  0, 3, 51, 279,  579,  465, 153,  21,  1;
  0, 3, 60, 414, 1200, 1500, 828, 210, 24, 1;

Huyile Liang, Jinyang Zhang, and Yu Wang, Some properties of the matrix related to q-coloured coordination number, Filomat (2024) Vol. 38, No. 4, 1465-1477. See p. 1466.

Cf. Diagonals A000012, A008585, A062741, columns A000007, A122553, A122709, row sums A026150.

Cf. A078010, A084938, A102900, A117575, A122117, A122552, A147518.

T[n_,k_]:=SeriesCoefficient[(1-x)/(1-(y+1)*x-2*y*x^2),{x,0,n},{y,0,k}]; Table[T[n,k],{n,0,10},{k,0,n}]//Flatten (* Stefano Spezia, Dec 27 2023 *)

Sum_{k=0..n} T(n,k)*x^(n-k) = A026150(n), A102900(n) for x = 1, 2.
T(n,k) = T(n-1,k) + T(n-1,k-1) + 2*T(n-2,k-1). - Philippe Deléham, Sep 25 2006
G.f.: (1-x)/(1-(y+1)*x-2*y*x^2). - Philippe Deléham, Jan 31 2012
Sum_{k=0..n} T(n,k)*x^k = A117575(n+1), A000007(n), A026150(n), A122117(n), A147518(n) for x = -1, 0, 1, 2, 3 respectively. - Philippe Deléham, Jan 31 2012

More terms from Stefano Spezia, Dec 27 2023

A179000 Array T(n,k) read by antidiagonals: coefficient [x^k] of (1 + n*Sum_{i>=1} x^i)^2, k >= 0.

Original entry on oeis.org

1, 1, 2, 1, 4, 3, 1, 6, 8, 4, 1, 8, 15, 12, 5, 1, 10, 24, 24, 16, 6, 1, 12, 35, 40, 33, 20, 7, 1, 14, 48, 60, 56, 42, 24, 8, 1, 16, 63, 84, 85, 72, 51, 28, 9, 1, 18, 80, 112, 120, 110, 88, 60, 32, 10

Offset: 1

Gary W. Adamson, Jan 03 2011

Antidiagonal sums are in A136396.

			First few rows of the array:
  1   2   3   4   5   6   7   8   9  10  11  A000027
  1   4   8  12  16  20  24  28  32  36  40  A008574
  1   6  15  24  33  42  51  60  69  78  87  A122709
  1   8  24  40  56  72  88 104 120 136 152  A051062
  1  10  35  60  85 110 135 160 185 210 235
  1  12  48  84 120 156 192 228 264 300 336
  1  14  63 112 161 210 259 308 357 406 455
  1  16  80 144 208 272 336 400 464 528 592
  1  18  99 180 261 342 423 504 585 666 747
Row n=3 is generated by (1 + 3x + 3x^2 + 3x^3 + 3x^4 + ...)^2 = 1 + 6x + 15x^2 + 24x^3 + ..., for example.

Cf. A136396, A179901.

A179000 := proc(n,k) if k = 0 then 1; else 2*n+n^2*(k-1) ; end if; end proc: # R. J. Mathar, Jan 05 2011

T(n,0) = 1; T(n,k) = n*(2+n*(k-1)), k > 0. - R. J. Mathar, Jan 05 2011

cp's OEIS Frontend

A038764 a(n) = (9*n^2 + 3*n + 2)/2.

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A179805 a(0) = 1, a(1) = 3, a(2) = 6 and a(n) = 2*a(n-1) - a(n-2) for n > 3.

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A122016 Riordan array(1, x*(1+2*x)/(1-x)).

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A179000 Array T(n,k) read by antidiagonals: coefficient [x^k] of (1 + n*Sum_{i>=1} x^i)^2, k >= 0.

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Formula

A038764 a(n) = (9n^2 + 3n + 2)/2.

A122016 Riordan array(1, x(1+2x)/(1-x)).