A038764 -id:A038764 - cp's OEIS frontend

A081272 Downward vertical of triangular spiral in A051682.

1, 25, 85, 181, 313, 481, 685, 925, 1201, 1513, 1861, 2245, 2665, 3121, 3613, 4141, 4705, 5305, 5941, 6613, 7321, 8065, 8845, 9661, 10513, 11401, 12325, 13285, 14281, 15313, 16381, 17485, 18625, 19801, 21013, 22261, 23545, 24865, 26221, 27613, 29041, 30505

Offset: 0

Paul Barry, Mar 15 2003

Reflection of A081271 in the horizontal A051682.
Binomial transform of (1, 24, 36, 0, 0, 0, .....).
One of the six verticals of a triangular spiral which starts with 1 (see the link). Other verticals are A060544, A081589, A080855, A157889, A038764. - Yuriy Sibirmovsky, Sep 18 2016.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Kaie Kubjas, Luca Sodomaco, and Elias Tsigaridas, Exact solutions in low-rank approximation with zeros, arXiv:2010.15636 [math.AG], 2020.
Yuriy Sibirmovsky, Six verticals of the triangular spiral.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A051682.

Cf. A062741, A060544, A081589, A080855, A157889, A038764.

Table[n^2 + (n + 1)^2, {n, 0, 300, 3}] (* or *) LinearRecurrence[{3, -3, 1}, {1, 25, 85}, 80] (* Vladimir Joseph Stephan Orlovsky, Feb 17 2012 *)
Table[n^2 + (n + 1)^2, {n, 0, 150, 3}] (* Vincenzo Librandi, Aug 07 2013 *)

x='x+O('x^99); Vec((1+22*x+13*x^2)/(1-x)^3) \\ Altug Alkan, Sep 18 2016

a(n) = C(n, 0) + 24*C(n, 1) + 36*C(n, 2).
a(n) = 18*n^2 + 6*n + 1.
G.f.: (1 + 22*x + 13*x^2)/(1 - x)^3.
E.g.f.: exp(x)*(1 + 24*x + 18*x^2). - Stefano Spezia, Mar 07 2023

A081583 Third row of Pascal-(1,2,1) array A081577.

Original entry on oeis.org

1, 10, 46, 136, 307, 586, 1000, 1576, 2341, 3322, 4546, 6040, 7831, 9946, 12412, 15256, 18505, 22186, 26326, 30952, 36091, 41770, 48016, 54856, 62317, 70426, 79210, 88696, 98911, 109882, 121636, 134200, 147601, 161866, 177022, 193096

Offset: 0

Paul Barry, Mar 23 2003

Equals binomial transform of [1, 9, 27, 27, 0, 0, 0, ...] where (1, 9, 27, 27) = row 3 of triangle A013610. - Gary W. Adamson, Jul 19 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A013610, A038764, A081584, A081577.

[(2+9*n+9*n^3)/2: n in [0..40]]; // Vincenzo Librandi, Aug 09 2013

seq((2+9*n+9*n^3)/2, n=0..40); # G. C. Greubel, May 25 2021

CoefficientList[Series[(1+2x)^3/(1-x)^4, {x,0,50}], x] (* Vincenzo Librandi, Aug 09 2013 *)
LinearRecurrence[{4,-6,4,-1},{1,10,46,136},60] (* Harvey P. Dale, Oct 01 2021 *)

a = lambda n: hypergeometric([-n, -3], [1], 3)
[simplify(a(n)) for n in range(36)] # Peter Luschny, Nov 19 2014

a(n) = (2 + 9*n + 9*n^3)/2.
G.f.: (1+2*x)^3/(1-x)^4.
a(n) = hypergeommetric2F1([-n, -3], [1], 3). - Peter Luschny, Nov 19 2014
E.g.f.: (1/2)*(2 + 18*x + 27*x^2 + 9*x^3)*exp(x). - G. C. Greubel, May 25 2021

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).

A361682 Array read by descending antidiagonals. A(n, k) is the number of multiset combinations of {0, 1} whose type is defined in the comments. Also A(n, k) = hypergeom([-k, -2], [1], n).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 6, 5, 1, 1, 10, 13, 7, 1, 1, 15, 25, 22, 9, 1, 1, 21, 41, 46, 33, 11, 1, 1, 28, 61, 79, 73, 46, 13, 1, 1, 36, 85, 121, 129, 106, 61, 15, 1, 1, 45, 113, 172, 201, 191, 145, 78, 17, 1, 1, 55, 145, 232, 289, 301, 265, 190, 97, 19, 1

Offset: 0

Peter Luschny, Mar 21 2023

A combination of a multiset M is an unordered selection of k objects of M, where every object can appear at most as many times as it appears in M.
A(n, k) = Cardinality(Union_{j=0..k} Combination(MultiSet(1^[j*n], 0^[(k-j)*n]))), where MultiSet(r^[s], u^[v]) denotes a set that contains the element r with multiplicity s and the element u with multiplicity v; thus the multisets under consideration have n*k elements. Since the base set is {1, 0} the elements can be represented as binary strings. Applying the combination operator to the multisets results in a set of binary strings where '0' resp. '1' can appear at most j*n resp. (k-j)*n times. 'At most' means that they do not have to appear; in other words, the resulting set always includes the empty string ''.
In contrast to the procedure in A361045 we consider here the cardinality of the set union and not the sum of the individual cardinalities. If you want to exclude the empty string, you will find the sequences listed in A361521. The same construction with multiset permutations instead of multiset combinations results in A361043.
A different view can be taken if one considers the hypergeometric representation, hypergeom([-k, -m], [1], n). This is a family of arrays that includes the 'rascal' triangle: the all 1's array A000012 (m = 0), the rascal array A077028 (m = 1), this array (m = 2), and A361731 (m = 3).

			Array A(n, k) starts:
   [0] 1,  1,   1,    1,   1,   1,   1,    1, ...  A000012
   [1] 1,  3,   6,   10,  15,  21,  28,   36, ...  A000217
   [2] 1,  5,  13,   25,  41,  61,  85,  113, ...  A001844
   [3] 1,  7,  22,   46,  79, 121, 172,  232, ...  A038764
   [4] 1,  9,  33,   73, 129, 201, 289,  393, ...  A081585
   [5] 1, 11,  46,  106, 191, 301, 436,  596, ...  A081587
   [6] 1, 13,  61,  145, 265, 421, 613,  841, ...  A081589
   [7] 1, 15,  78,  190, 351, 561, 820, 1128, ...  A081591
   000012  | A028872 | A239325 |
       A005408    A100536   A069133
.
Triangle T(n, k) starts:
   [0] 1;
   [1] 1,  1;
   [2] 1,  3,   1;
   [3] 1,  6,   5,   1;
   [4] 1, 10,  13,   7,   1;
   [5] 1, 15,  25,  22,   9,   1;
   [6] 1, 21,  41,  46,  33,  11,   1;
   [7] 1, 28,  61,  79,  73,  46,  13,  1;
   [8] 1, 36,  85, 121, 129, 106,  61, 15,  1;
   [9] 1, 45, 113, 172, 201, 191, 145, 78, 17, 1.
.
Row 4 of the triangle:
A(0, 4) =  1 = card('').
A(1, 3) = 10 = card('', 0, 00, 000, 1, 10, 100, 11, 110, 111).
A(2, 2) = 13 = card('', 0, 00, 000, 0000, 1, 10, 100, 11, 110, 1100, 111, 1111).
A(3, 1) =  7 = card('', 0, 00, 000, 1, 11, 111).
A(4, 0) =  1 = card('').

Rows: A000012, A000217, A001844, A038764, A081585, A081587, A081589, A081591.
Columns: A000012, A005408, A028872, A100536, A239325, A069133.
Cf. A239592 (main diagonal), A239331 (transposed array).
Cf. A361045, A361043, A361521, A361731, A077028.

A := (n, k) -> 1 + n*k*(4 + n*(k - 1))/2:
for n from 0 to 7 do seq(A(n, k), k = 0..7) od;
# Alternative:
ogf := n -> (1 + (n - 1)*x)^2 / (1 - x)^3:
ser := n -> series(ogf(n), x, 12):
row := n -> seq(coeff(ser(n), x, k), k = 0..9):
seq(print(row(n)), n = 0..7);

def A(m: int, steps: int) -> int:
    if m == 0: return 1
    size = m * steps
    cset = set()
    for a in range(0, size + 1, m):
        S = [str(int(i < a)) for i in range(size)]
        C = Combinations(S)
        cset.update("".join(i for i in c) for c in C)
    return len(cset)
def ARow(n: int, size: int) -> list[int]:
    return [A(n, k) for k in range(size + 1)]
for n in range(8): print(ARow(n, 7))

A(n, k) = 1 + n*k*(4 + n*(k - 1))/2.
T(n, k) = 1 + k*(n - k)*(4 + k*(n - k - 1))/2.
A(n, k) = [x^k] (1 + (n - 1)*x)^2 / (1 - x)^3.
A(n, k) = hypergeom([-k, -2], [1], n).
A(n, k) = A361521(n, k) + 1.

A239331 Square array, read by antidiagonals: column k has g.f. (1+(k-1)*x)^2/(1-x)^3.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 6, 1, 1, 7, 13, 10, 1, 1, 9, 22, 25, 15, 1, 1, 11, 33, 46, 41, 21, 1, 1, 13, 46, 73, 79, 61, 28, 1, 1, 15, 61, 106, 129, 121, 85, 36, 1, 1, 17, 78, 145, 191, 201, 172, 113, 45, 1, 1, 19, 97, 190, 265, 301, 289, 232, 145, 55, 1, 1, 21

Offset: 0

Philippe Deléham, Mar 16 2014

			Square array begins:
n\k : 0......1......2......3......4......5......6......7......8......9
======================================================================
.0||  1......1......1......1......1......1......1......1......1......1
.1||  1......3......5......7......9.....11.....13.....15.....17.....19
.2||  1......6.....13.....22.....33.....46.....61.....78.....97....118
.3||  1.....10.....25.....46.....73....106....145....190....241....298
.4||  1.....15.....41.....79....129....191....265....351....449....559
.5||  1.....21.....61....121....201....301....421....561....721....901
.6||  1.....28.....85....172....289....436....613....820...1057...1324
.7||  1.....36....113....232....393....596....841...1128...1457...1828
.8||  1.....45....145....301....513....781...1105...1485...1921...2413
.9||  1.....55....181....379....649....991...1405...1891...2449...3079
10||  1.....66....221....466....801...1226...1741...2346...3041...3826
11||  1.....78....265....562....969...1486...2113...2850...3697...4654

Cf. Columns: A000012, A000217, A001844, A038764, A081585, A081587, A081589, A081591, A081593.

Cf. Rows: A000012, A005408, A028872, A100536, A239325, A069133.

T(n,k) = 3*T(n-1,k) - 3*T(n-2,k) + T(n-3,k).
T(n,k) = 3*T(n,k-1) - 3*T(n,k-2) + T(n,k-3).
T(n,k) = (T(n,k-1) + T(n,k+1))/2 - A161680(n).
T(n,k) = (T(n-1,k) + T(n+1,k) - A000290(n))/2.

cp's OEIS Frontend

A081272 Downward vertical of triangular spiral in A051682.

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A081583 Third row of Pascal-(1,2,1) array A081577.

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

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Magma

Mathematica

PARI

Formula

A361682 Array read by descending antidiagonals. A(n, k) is the number of multiset combinations of {0, 1} whose type is defined in the comments. Also A(n, k) = hypergeom([-k, -2], [1], n).

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Maple

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A239331 Square array, read by antidiagonals: column k has g.f. (1+(k-1)*x)^2/(1-x)^3.

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Formula

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