A056107 -id:A056107 - cp's OEIS frontend

A113653 Isolated semiprimes in the hexagonal spiral.

6, 51, 69, 82, 91, 183, 194, 221, 249, 265, 287, 289, 309, 314, 319, 323, 355, 371, 403, 417, 437, 469, 478, 511, 517, 519, 533, 579, 589, 649, 681, 689, 731, 749, 758, 807, 838, 849, 926, 943, 951, 961, 965, 979, 1011, 1018, 1037, 1055, 1057, 1067, 1077, 1099, 1126, 1145, 1149, 1154, 1159

Offset: 1

Jonathan Vos Post, Jan 16 2006

nonn

Isolated semiprimes in the hexagonal spiral of A003215 and A001399, which is centered on 0. Of course such a spiral can be constructed beginning with any integer. Centering on 0 gives the interesting partition and multigraph equalities of A001399.
Integers in A001358 which are not adjacent in any of six directions to any other integer in A001358 when arranged in the hexagonal spiral.
An analog of  A113688 "Isolated semiprimes in the [square] spiral," and of the hexagonal prime spiral of [Abbott 2005; Weisstein, "Prime Spiral", MathWorld].
Unfortunately the original submission (which has been preserved as the "dead" sequence A335704) omitted the number 44 from the spiral, which has caused an enormous amount of trouble. - N. J. A. Sloane, Jun 27 2020

			The spiral begins:
                120-119-118-117-116-115-114
                 /                         \
              121  85--84--83-*82*-81--80 113
               /   /                     \   \
            122  86  56--55--54--53--52  79 112
             /   /   /                 \   \   \
          123  87  57  33--32--31--30 *51* 78 111
           /   /   /   /             \   \   \   \
        124  88  58  34  16--15--14  29  50  77 110
         /   /   /   /   /         \   \   \   \   \
      125  89  59  35  17   5---4  13  28  49  76 109
       /   /   /   /   /   /     \   \   \   \   \   \
    126  90  60  36  18  *6*  0   3  12  27  48  75 108
     /   /   /   /   /   /   /   /   /   /   /   /   /
  127 *91* 61  37  19   7   1---2  11  26  47  74 107 146
     \   \   \   \   \   \         /   /   /   /   /   /
    128  92  62  38  20   8---9--10  25  46  73 106 145
       \   \   \   \   \             /   /   /   /   /
      129  93  63  39  21--22--23--24  45  72 105 144
         \   \   \   \                 /   /   /   /
        130  94  64  40--41--42--43--44  71 104 143
           \   \   \                     /   /   /
          131  95  65--66--67--68-*69*-70 103 142
             \   \                         /   /
            132  96--97--98--99-100-101-102 141
               \                             /
              133-134-135-136-137-138-139-140

Abbott, P. (Ed.). "Mathematica One-Liners: Spiral on an Integer Lattice." Mathematica J. 1, 39, 1990.

P. Abbott, Re: Hexagonal Spiral, (alt link), May 11, 2005
H. Bottomley, Spokes of a Hexagonal Spiral.
R. J. Mathar, Maple program for A113653
Eric Weisstein's World of Mathematics, Prime Spiral.

Cf. A001358, A001399, A113688.
For the sequence of isolated primes see A335916.
Related sequences:
A113519 Semiprimes in 1st spoke of a hexagonal spiral starting at 1 (A056105).
A113524 Semiprimes in 2nd spoke of a hexagonal spiral (A056106).
A113525 Semiprimes in 3rd spoke of a hexagonal spiral (A056107).
A113527 Semiprimes in 4th spoke of a hexagonal spiral (A056108).
A113528 Semiprimes in 5th spoke of a hexagonal spiral (A056109).
A113530 Semiprimes in 6th spoke of a hexagonal spiral (A003215).

Corrected and edited by N. J. A. Sloane, Jun 27 2020. Thanks to Jeffrey K. Aronson for pointing out the error in the original submission.

Terms a(4) onwards corrected by R. J. Mathar, Jun 29 2020

A271675 Numbers m such that 3*m + 4 is a square.

Original entry on oeis.org

0, 4, 7, 15, 20, 32, 39, 55, 64, 84, 95, 119, 132, 160, 175, 207, 224, 260, 279, 319, 340, 384, 407, 455, 480, 532, 559, 615, 644, 704, 735, 799, 832, 900, 935, 1007, 1044, 1120, 1159, 1239, 1280, 1364, 1407, 1495, 1540, 1632, 1679, 1775, 1824, 1924, 1975, 2079, 2132, 2240, 2295, 2407

Offset: 1

Juri-Stepan Gerasimov, Apr 12 2016

7 is the unique prime in this sequence. If m is in this sequence, then 3*m + 4 = k^2 for k is nonzero integer, that is, m = (k^2 - 4)/3 = (k-2)*(k+2)/3. So m can be only prime if one of divisors is prime and another one is 1. Otherwise there should be more than 1 prime divisors, that is n must be composite. - Altug Alkan, Apr 12 2016
From Ray Chandler, Apr 12 2016: (Start)
Square roots of resulting squares gives A001651 (with a different starting point).
Sequence is the union of (positive terms) in A140676 and A270710. (End)
The sequence terms are the exponents in the expansion of Sum_{n >= 0} q^n*(1 - q)*(1 - q^3)*...*(1 - q^(2*n+1)) = 1 - q^4 - q^7 + q^15 + q^20 - q^32 - q^50 + + - - .... - Peter Bala, Dec 19 2024

			a(4) = 32 because 3*32 + 4 = 100 = 10*10.

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A001651, A140676, A270710.

Cf. numbers n such that 3*n + k is a square: A120328 (k=-6), A271713 (k=-5), A056107 (k=-3), A257083 (k=-2), A033428 (k=0), A001082 (k=1), A080663 (k=3), this sequence (k=4), A100536 (k=6).

Magma
```
[n: n in [0..4000] | IsSquare(3*n+4)];
```

Select[Range[0,2500], IntegerQ@ Sqrt[3 # + 4] &] (* Michael De Vlieger, Apr 12 2016 *)
LinearRecurrence[{1,2,-2,-1,1},{0,4,7,15,20},60] (* Harvey P. Dale, Dec 09 2016 *)

from gmpy2 import is_square
for n in range(0,10**5):
    if(is_square(3*n+4)):print(n)
# Soumil Mandal, Apr 12 2016

O.g.f.: x^2*(4 + 3*x - x^3)/((1 + x)^2*(1 - x)^3).
E.g.f.: 1 + (1 - 2*x)*exp(-x)/8 - 3*(3 - 4*x - 2*x^2)*exp(x)/8.
a(n) = A001082(n+1) - 1 = (6*n*(n+1) + (2*n + 1)*(-1)^n - 1)/8 - 1. Therefore: a(2*k+1) = k*(3*k+4), a(2*k) = (k+1)*(3*k-1).
Sum_{n>=2} 1/a(n) = 19/16 - Pi/(4*sqrt(3)). - Amiram Eldar, Jul 26 2024

Edited and extended by Bruno Berselli, Apr 12 2016

A359364 Triangle read by rows. The Motzkin triangle, the coefficients of the Motzkin polynomials. M(n, k) = binomial(n, k) * CatalanNumber(k/2) if k is even, otherwise 0.

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 1, 0, 3, 0, 1, 0, 6, 0, 2, 1, 0, 10, 0, 10, 0, 1, 0, 15, 0, 30, 0, 5, 1, 0, 21, 0, 70, 0, 35, 0, 1, 0, 28, 0, 140, 0, 140, 0, 14, 1, 0, 36, 0, 252, 0, 420, 0, 126, 0, 1, 0, 45, 0, 420, 0, 1050, 0, 630, 0, 42, 1, 0, 55, 0, 660, 0, 2310, 0, 2310, 0, 462, 0

Offset: 0

Peter Luschny, Jan 09 2023

The generalized Motzkin numbers M(n, k) are a refinement of the Motzkin numbers M(n) (A001006) in the sense that they are coefficients of polynomials M(n, x) = Sum_{n..k} M(n, k) * x^k that take the value M(n) at x = 1. The coefficients of x^n are the aerated Catalan numbers A126120.
Variants are the irregular triangle A055151 with zeros deleted, A097610 with reversed rows, A107131 and A080159.
In the literature the name 'Motzkin triangle' is also used for the triangle A026300, which is generated from the powers of the generating function of the Motzkin numbers.

			Triangle M(n, k) starts:
[0] 1;
[1] 1, 0;
[2] 1, 0,  1;
[3] 1, 0,  3, 0;
[4] 1, 0,  6, 0,   2;
[5] 1, 0, 10, 0,  10, 0;
[6] 1, 0, 15, 0,  30, 0,   5;
[7] 1, 0, 21, 0,  70, 0,  35, 0;
[8] 1, 0, 28, 0, 140, 0, 140, 0,  14;
[9] 1, 0, 36, 0, 252, 0, 420, 0, 126, 0;

M. Artioli, G. Dattoli, S. Licciardi, and S. Pagnutti, Motzkin Numbers: an Operational Point of View, arXiv:1703.07262 [math.CO], 2017, (Table 1 on p. 3).

Variants: A055151, A080159, A097610, A107131.
Columns M(n, 2*k): A000012, A000217, A034827, A000910, A088625, A088626, A213380.
Cf. A001006 (Motzkin numbers), A026300 (Motzkin gf. triangle), A126120 (aerated Catalan), A000108 (Catalan).
Cf. A138364, A107587, A002457, A002522, A025179, A025235, A056107, A014531, A023426, A091147, A189912, A343386, A343773, A359647, A359649.

CatalanNumber := n -> binomial(2*n, n)/(n + 1):
M := (n, k) -> ifelse(irem(k, 2) = 1, 0, CatalanNumber(k/2)*binomial(n, k)):
for n from 0 to 9 do seq(M(n, k), k = 0..n) od;
# Alternative, as coefficients of polynomials:
p := n -> hypergeom([(1 - n)/2, -n/2], [2], (2*x)^2):
seq(print(seq(coeff(simplify(p(n)), x, k), k = 0..n)), n = 0..9);
# Using the exponential generating function:
egf := exp(x)*BesselI(1, 2*x*t)/(x*t): ser:= series(egf, x, 11):
seq(print(seq(coeff(simplify(n!*coeff(ser, x, n)), t, k), k = 0..n)), n = 0..9);

from functools import cache
@cache
def M(n: int, k: int) -> int:
    if k %  2: return 0
    if n <  3: return 1
    if n == k: return (2 * (n - 1) * M(n - 2, n - 2)) // (n // 2 + 1)
    return (M(n - 1, k) * n) // (n - k)
for n in range(10): print([M(n, k) for k in range(n + 1)])

Let p(n, x) = hypergeom([(1 - n)/2, -n/2], [2], (2*x)^2).
p(n, 1) = A001006(n); p(n, sqrt(2)) = A025235(n); p(n, 2) = A091147(n).
p(2, n) = A002522(n); p(3, n) = A056107(n).
p(n, n) = A359649(n); 2^n*p(n, 1/2) = A000108(n+1).
M(n, k) = [x^k] p(n, x).
M(n, k) = [t^k] (n! * [x^n] exp(x) * BesselI(1, 2*t*x) / (t*x)).
M(n, k) = [t^k][x^n] ((1 - x - sqrt((x-1)^2 - (2*t*x)^2)) / (2*(t*x)^2)).
M(n, n) = A126120(n).
M(n, n-1) = A138364(n), the number of Motzkin n-paths with exactly one flat step.
M(2*n, 2*n) = A000108(n), the number of peakless Motzkin paths having a total of n up and level steps.
M(4*n, 2*n) = A359647(n), the central terms without zeros.
M(2*n+2, 2*n) = A002457(n) = (-4)^n * binomial(-3/2, n).
Sum_{k=0..n} M(n - k, k) = A023426(n).
Sum_{k=0..n} k * M(n, k) = 2*A014531(n-1) = 2*GegenbauerC(n - 2, -n, -1/2).
Sum_{k=0..n} i^k*M(n, k) = A343773(n), (i the imaginary unit), is the excess of the number of even Motzkin n-paths (A107587) over the odd ones (A343386).
Sum_{k=0..n} Sum_{j=0..k} M(n, j) = A189912(n).
Sum_{k=0..n} Sum_{j=0..k} M(n, n-j) = modified A025179(n).
For a recursion see the Python program.

A359760 Triangle read by rows. The Kummer triangle, the coefficients of the Kummer polynomials. K(n, k) = binomial(n, k) * oddfactorial(k/2) if k is even, otherwise 0, where oddfactorial(z) := (2z)!/(2^zz!).

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 1, 0, 3, 0, 1, 0, 6, 0, 3, 1, 0, 10, 0, 15, 0, 1, 0, 15, 0, 45, 0, 15, 1, 0, 21, 0, 105, 0, 105, 0, 1, 0, 28, 0, 210, 0, 420, 0, 105, 1, 0, 36, 0, 378, 0, 1260, 0, 945, 0, 1, 0, 45, 0, 630, 0, 3150, 0, 4725, 0, 945, 1, 0, 55, 0, 990, 0, 6930, 0, 17325, 0, 10395, 0

Offset: 0

Peter Luschny, Jan 13 2023

The Kummer numbers K(n, k) are a refinement of the oddfactorial numbers (A001147) in the sense that they are the coefficients of polynomials K(n, x) = Sum_{n..k} K(n, k) * x^k that take the value oddfactorial(n) at x = 1. The coefficients of x^n are the aerated oddfactorial numbers A123023.

These numbers appear in many different versions (see the crossrefs). They are the coefficients of the Chebyshev-Hermite polynomials in signed form when ordered in decreasing powers. Our exposition is based on the seminal paper by Kummer, which preceded the work of Chebyshev and Hermite for more than 20 years. They are also referred to as Bessel numbers of the second kind (Mansour et al.) when the odd powers are omitted.

			Triangle K(n, k) starts:
 [0] 1;
 [1] 1, 0;
 [2] 1, 0,  1;
 [3] 1, 0,  3, 0;
 [4] 1, 0,  6, 0,   3;
 [5] 1, 0, 10, 0,  15, 0;
 [6] 1, 0, 15, 0,  45, 0,   15;
 [7] 1, 0, 21, 0, 105, 0,  105, 0;
 [8] 1, 0, 28, 0, 210, 0,  420, 0, 105;
 [9] 1, 0, 36, 0, 378, 0, 1260, 0, 945, 0;

John Riordan, Introduction to Combinatorial Analysis, Dover (2002), pp. 85-86.

Pierre Humbert, Monographie des polynômes de Kummer, Nouvelles annales de mathématiques, journal des candidats aux écoles polytechnique et normale, Serie 5, Volume 1 (1922), pp. 81-92.
E. E. Kummer, Über die hypergeometrische Reihe, Journal für die reine und angewandte Mathematik 15 (1836): 39-83.
T. Mansour, M. Schork and M. Shattuck, The Generalized Stirling and Bell Numbers Revisited, Journal of Integer Sequences, Vol. 15 (2012), #12.8.3.
Ladislav Truksa, Hypergeometric orthogonal systems of polynomials III, Aktuárské vědy, Vol. 2 (1931), No. 4, 177-203, (see p.200).

Variants: Signed version: A073278. Other variants are the irregular triangle A100861 with zeros deleted, A066325 and A099174 with reversed rows, A111924, A144299, A104556.
Cf. A000085, A001147, A123023, A047974, A115329, A002522, A056107, A359739,
A001879, A001464, A005425, A344501, A123022, A359761.

oddfactorial := proc(z) (2*z)! / (2^z*z!) end:
K := (n, k) -> ifelse(irem(k, 2) = 1, 0, binomial(n, k) * oddfactorial(k/2)):
seq(seq(K(n, k), k = 0..n), n = 0..11);
# Alternative, as coefficients of polynomials:
p := (n, x) -> 2^(n/2)*(-1/x^2)^(-n/2)*KummerU(-n/2, 1/2, -1/(2*x^2)):
seq(print(seq(coeff(simplify(p(n, x)), x, k), k = 0..n)), n = 0 ..9);
# Using the exponential generating function:
egf := exp(x + (t*x)^2 / 2): ser := series(egf, x, 12):
seq(print(seq(coeff(n! * coeff(ser, x, n), t, k), k = 0..n)), n = 0..9);

K[n_, k_] := K[n, k] = Which[OddQ[k], 0, k == 0, 1, n == k, K[n - 1, n - 2], True, K[n - 1, k] n/(n - k)];
Table[K[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 25 2023 *)

from functools import cache
@cache
def K(n: int, k: int) -> int:
    if k %  2: return 0
    if n <  3: return 1
    if n == k: return K(n - 1, n - 2)
    return (K(n - 1, k) * n) // (n - k)
for n in range(10): print([K(n, k) for k in range(n + 1)])

Let p(n, x) = 2^(n/2)*(-1/x^2)^(-n/2)*KummerU(-n/2, 1/2, -1/(2*x^2)).
p(n, 1) = A000085(n); p(n, sqrt(2)) = A047974(n); p(n, 2) = A115329(n);
p(2, n) = A002522(n) (n >= 1); p(3, n) = A056107(n) (n >= 1);
p(n, n) = A359739(n) (n >= 1); 2^n*p(n, 1/2) = A005425(n).
K(n, k) = [x^k] p(n, x).
K(n, k) = [t^k] (n! * [x^n] exp(x + (t*x)^2 / 2)).
K(n, n) = A123023(n).
K(n, n-1) = A123023(n + 1).
K(2*n, 2*n) = A001147(n).
K(4*n, 2*n) = A359761, the central terms without zeros.
K(2*n+2, 2*n) = A001879.
Sum_{k=0..n} (-1)^n * i^k * K(n, k) = A001464(n), ((the number of even involutions) - (the number of odd involutions) in the symmetric group S_n (Robert Israel)).
Sum_{k=0..n} Sum_{j=0..k} K(n, j) = A000085(n + 1).
For a recursion see the Python program.

A102728 Array read by antidiagonals: T(n, k) = ((n+1)^k-(n-1)^k)/2.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 0, 1, 2, 1, 0, 1, 4, 4, 0, 0, 1, 6, 13, 8, 1, 0, 1, 8, 28, 40, 16, 0, 0, 1, 10, 49, 120, 121, 32, 1, 0, 1, 12, 76, 272, 496, 364, 64, 0, 0, 1, 14, 109, 520, 1441, 2016, 1093, 128, 1, 0, 1, 16, 148, 888, 3376, 7448, 8128, 3280, 256, 0, 0, 1, 18, 193, 1400, 6841

Offset: 0

Lambert Klasen (lambert.klasen(AT)gmx.net) and Gary W. Adamson, Feb 07 2005

Consider a 2 X 2 matrix M = [N, 1] / [1, N]. The n-th row of the array contains the values of the non-diagonal elements of M^k, k=0,1,.... (Corresponding diagonal entry = non-diagonal entry + (N-1)^k.) Table:
N: row sequence g.f. cross references.
(1^n-(-1)^n)/2 x/((1+1x)(1-1x)) A000035
(2^n-0^n)/2 x/(1-2x) A000079
(3^n-1^n)/2 x/((1-1x)(1-3x)) A003462
(4^n-2^n)/2 x/((1-2x)(1-4x)) A006516
(7^n-3^n)/2 x/((1-3x)(1-5x)) A005059
(6^n-4^n)/2 x/((1-4x)(1-6x)) A016149
(7^n-5^n)/2 x/((1-5x)(1-7x)) A016161 A081200
(8^n-6^n)/2 x/((1-6x)(1-8x)) A016170 A081201
(9^n-7^n)/2 x/((1-7x)(1-9x)) A016178 A081202
(10^n-8^n)/2 x/((1-8x)(1-10x)) A016186 A081203
(11^n-9^n)/2 x/((1-9x)(1-11x)) A016190
(12^n-10^n)/2 x/((1-10x)(1-12x)) A016196
...
Characteristic polynomial x^2-2nx+(n^2-1) has roots n+-1, so if r(n) denotes a row sequence, r(n+1)/r(n) converges to n+1.
Columns follow polynomials with certain binomial coefficients:
column: polynomial
0
1
2n
3n^2+ 1 (see A056107)
4n^3+ 4n (= 8*A006003(n))
5n^4+ 10n^2+ 1
6n^5+ 20n^3+ 6n
7n^6+ 35n^4+ 21n^2+ 1
8; 8n^7+ 56n^5+ 56n^3+ 8n
9n^8+ 84n^6+126n^4+ 36n^2+ 1
10n^9+ 120n^7+252n^5+120n^3+ 10n
11n^10+165n^8+462n^6+330n^4+ 55n^2+ 1

			Array begins:
0,1,0,1,0,1...
0,1,2,4,8,16...
0,1,4,13,40,121...
0,1,6,28,120,496...
0,1,8,49,272,1441...
...

MM(n,N)=local(M);M=matrix(n,n);for(i=1,n, for(j=1,n,if(i==j,M[i,j]=N,M[i,j]=1)));M for(k=0,12, for(i=0,k,print1((MM(2,k-i)^i)[1,2],","))) T(n, k) = ((n+1)^k-(n-1)^k)/2 for(k=0,10, for(i=0,10,print1(T(k,i),","));print()) for(k=0,10, for(i=0,10,print1(((k+1)^i-(k-1)^i)/2,","));print()) for(k=0,10, for(i=0,10,print1(polcoeff(x/((1-(k-1)*x)*(1-(k+1)*x)),i),","));print())

A143804 Triangle read by rows, thrice the Connell numbers (A001614) - 2.

Original entry on oeis.org

1, 4, 10, 13, 19, 25, 28, 34, 40, 46, 49, 55, 61, 67, 73, 76, 82, 88, 94, 100, 106, 109, 115, 121, 127, 133, 139, 145, 148, 154, 160, 166, 172, 178, 184, 190, 193, 199, 205, 211, 217, 223, 229, 235, 241, 244, 250, 256, 262, 268, 274, 280, 286, 292, 298

Offset: 1

Gary W. Adamson, Sep 01 2008

Right border of the triangle = A100536: (1, 10, 25, 46, 73,...).
Left border = A056107: (1, 4, 13, 28, 49,...).
Row sums = A005915: (1, 14, 57, 148, 305,...).
n-th row = (right border then going to the left): (n-th term of A100536 followed by (n-1) operations of (-6), (-6), (-6),... As a Connell-like triangle, odd row terms are in the subset 6n-5; even row terms are in the set 6n-2.
Row 3 = (13, 19, 25) beginning with A100536(3) = 25 at the right then following the trajectory (-6), (-6).
Using the modular rules, the triangle begins (1; 4, 10; 13, 19, 25;...) since 1 == 6n-5, while 4 is the next higher term in the set 6n-2, then 10 also in the set 6n-2, being an even row.

			First few rows of the triangle:
  1;
  4, 10;
  13, 19, 25;
  28, 34, 40, 46;
  49, 55, 61, 67, 73;
  76, 82, 88, 94, 100, 106;
  ...

Cf. A001614, A100536, A056107, A005915.

from math import isqrt
def A143804(n): return 3*((m:=n<<1)-(k:=isqrt(m))-int(m>=k*(k+1)+1))-2 # Chai Wah Wu, Aug 01 2022

a(n) = 3*A001614(n) - 2.

A271723 Numbers k such that 3*k - 8 is a square.

Original entry on oeis.org

3, 4, 8, 11, 19, 24, 36, 43, 59, 68, 88, 99, 123, 136, 164, 179, 211, 228, 264, 283, 323, 344, 388, 411, 459, 484, 536, 563, 619, 648, 708, 739, 803, 836, 904, 939, 1011, 1048, 1124, 1163, 1243, 1284, 1368, 1411, 1499, 1544, 1636, 1683, 1779, 1828, 1928, 1979, 2083, 2136, 2244, 2299

Offset: 1

Juri-Stepan Gerasimov, Apr 13 2016

Square roots of resulting squares gives A001651. - Ray Chandler, Apr 14 2016

			a(1) = 3 because 3*3 - 8 = 1^2.

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A001651.

Cf. numbers n such that 3*n + k is a square: this sequence (k=-8), A120328 (k=-6), A271713 (k=-5), A056107 (k=-3), A257083 (k=-2), A033428 (k=0), A001082 (k=1), A080663 (k=3), A271675 (k=4), A100536 (k=6), A271741 (k=7), A067725 (k=9).

Magma
```
[n: n in [1..2400] | IsSquare(3*n-8)];
```

seq(seq(((3*m+k)^2+8)/3, k=1..2),m=0..50); # Robert Israel, Dec 05 2016

Select[Range@ 2400, IntegerQ@ Sqrt[3 # - 8] &] (* Bruno Berselli, Apr 14 2016 *)
LinearRecurrence[{1,2,-2,-1,1},{3,4,8,11,19},60] (* Harvey P. Dale, Oct 02 2020 *)

from gmpy2 import is_square
[n for n in range(3000) if is_square(3*n-8)] # Bruno Berselli, Dec 05 2016

[(6*(n-1)*n-(2*n-1)*(-1)**n+23)/8 for n in range(1, 60)] # Bruno Berselli, Dec 05 2016

From Ilya Gutkovskiy, Apr 13 2016: (Start)
G.f.: x*(3 + x - 2*x^2 + x^3 + 3*x^4)/((1 - x)^3*(1 + x)^2).
a(n) = (6*(n - 1)*n - (2*n - 1)*(-1)^n + 23)/8. (End)

A105373 Square array by antidiagonals of number of straight lines with n points in a k-dimensional hypercube with n points on each edge.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 28, 8, 1, 1, 120, 49, 10, 1, 1, 496, 272, 76, 12, 1, 1, 2016, 1441, 520, 109, 14, 1, 1, 8128, 7448, 3376, 888, 148, 16, 1, 1, 32640, 37969, 21280, 6841, 1400, 193, 18, 1, 1, 130816, 192032, 131776, 51012, 12496, 2080, 244, 20, 1, 1, 523776

Offset: 1

Henry Bottomley, Apr 02 2005

			Rows start:
  1,  1,   1,   1,    1,     1, ...;
  1,  6,  28, 120,  496,  2016, ...;
  1,  8,  49, 272, 1441,  7448, ...;
  1, 10,  76, 520, 3376, 21280, ...;
  1, 12, 109, 888, 6841, 51012, ...;
  etc.
T(5,3)=109 because in a 5 X 5 X 5 cube there are 25 columns, 25 linear rows in one direction, 25 linear rows in another direction, 5 short diagonals in each of 6 directions and 4 long diagonals; and 3*25 + 6*5 + 4 = 109.

See A102728. Rows essentially include A000012, A006516, A005059, A016149 or A081199, A016161 or A081200, A016170 or A081201, A016178 or A081202 etc. Columns essentially include A000012, A005843, A056107, A105373.

T(1, k)=1. For n>1: T(n, k) = ((n+2)^k-n^k)/2 = (n+2)*T(n, k-1)+n^(k-1) = A102728(k, n+1).

A109340 Expansion of x^2(1+x+4x^2)/((1+x+x^2)*(1-x)^3).

Original entry on oeis.org

0, 0, 1, 3, 9, 16, 24, 36, 49, 63, 81, 100, 120, 144, 169, 195, 225, 256, 288, 324, 361, 399, 441, 484, 528, 576, 625, 675, 729, 784, 840, 900, 961, 1023, 1089, 1156, 1224, 1296, 1369, 1443, 1521, 1600, 1680, 1764, 1849, 1935, 2025, 2116, 2208, 2304, 2401

Offset: 0

Creighton Dement, Aug 20 2005

From Gerhard Kirchner, Jan 20 2017: (Start)
According to the game "Mecanix":
In a triangular arrangement of wheel axles (n rows with 1, 2, ..., n axles), a connected set of unblocked gear wheels is installed such that the number of wheel quadruples forming half-hexagons is maximal.
a(n-1) is the maximum number.
Example:
Gear wheels (*) and free axles (·):
              ·
             * *
    *       * · *
   · *     · * * ·
  * * ·   * * · * *
   n=3       n=5
n=3: 1 half-hexagon, a(2)=1.
n=5: 3 half-hexagons and 1 full hexagon containing 6 half-hexagons -> a(4)=3+6*1=9.
See "Connected gear wheels" link.
Annotation: In such a configuration also the number of wheels is maximal. It is A007980(n). For n < 3, however, there is no half-hexagon. (End)
Floretion Algebra Multiplication Program, FAMP Code: 4tessumrokseq[A*B] with A = + .5'i + .5'j + .5'k + .5e and B = + .5i' + .5j' + .5k' + .5e; roktype: Y[15] = Y[15] + p; sumtype: Y[8] = (int)Y[6] - (int)Y[7] + Y[8] + sum (internal program code)

Gerhard Kirchner, Connected gear wheels
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Cf. A105770, A047240, A056107.

CoefficientList[Series[x^2(1+x+4x^2)/((1+x+x^2)(1-x)^3),{x,0,50}],x] (* or *) LinearRecurrence[{2,-1,1,-2,1},{0,0,1,3,9},60] (* Harvey P. Dale, Jun 24 2013 *)

a(n+1) - a(n) = A047240(n);
a(n) + a(n+1) + a(n+2) = A056107(n);
a(n+2) - a(n+1) + a(n) = A105770(n).
a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5); a(0)=0, a(1)=0, a(2)=1, a(3)=3, a(4)=9. - Harvey P. Dale, Jun 24 2013
a(n) = (n-1)^2 - ((n+1) mod 3) mod 2, n >= 1. - Gerhard Kirchner, Jan 20 2017
E.g.f.: (exp(x)*(2 + 3*(x - 1)*x) - 2*exp(-x/2)*cos(sqrt(3)*x/2))/3. - Stefano Spezia, Dec 23 2022

A243813 Table read by antidiagonals: T(n,k) is the curvature (truncated to integer) of a circle in a variation of nested Pappus chains (see Comments for details).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 2, 9, 1, 1, 1, 1, 3, 13, 1, 1, 1, 1, 2, 5, 19, 1, 1, 1, 1, 1, 3, 7, 25, 1, 1, 1, 1, 1, 2, 4, 9, 33, 1, 1, 1, 1, 1, 1, 2, 5, 11, 41, 1, 1, 1, 1, 1, 1, 2, 3, 6, 14, 51, 1, 1, 1, 1, 1, 1, 1, 2, 4, 7, 17, 61, 1, 1, 1, 1, 1, 1, 1, 2, 3, 5, 9, 21

Offset: 0

Kival Ngaokrajang, Jun 11 2014

Refer to the construction rule used in A243618. For this case, the curvature is defined by (-1/k, 1/(k-1), 1), the circle radius will diverge to infinity (zero curvature). The integral curvatures appearing as periodic, i.e., 2, 6, 6, 10, 30, 42, 28, 12, ..., = A083482(k-1). The integral curvatures seem to align as some sequence, e.g., 3, 7, 13, 21, 31, 43, ..., = A002061(k) and 9, 25, 49, ..., = A016754(k-1). See illustration.

			Table begins:
  n/k  2   3   4   5   6   7  ...
   0   1   1   1   1   1   1  ...
   1   1   1   1   1   1   1  ...
   2   3   1   1   1   1   1  ...
   3   5   2   1   1   1   1  ...
   4   9   3   2   1   1   1  ...
   5  13   5   3   2   1   1  ...
   6  19   7   4   2   2   1  ...
   7  25   9   5   3   2   2  ...
   8  33  11   6   4   3   2  ...
   9  41  14   7   5   3   2  ...
  10  51  17   9   6   4   3  ...
  11  61  21  11   7   5   3  ...
  12  73  25  13   8   5   4  ...
  ...

Kival Ngaokrajang, Illustration for k = 2..7

Cf. A243618, A083482, A002061, A016754.
Cf. Column 1 = A080827(n), column 2 = A056827(n) + 1.
Cf. Integral curvature in column 1..6: [A058331, A227776, A056107, A212656, A158558, A158604].

cp's OEIS Frontend

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A271723 Numbers k such that 3*k - 8 is a square.

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A359760 Triangle read by rows. The Kummer triangle, the coefficients of the Kummer polynomials. K(n, k) = binomial(n, k) * oddfactorial(k/2) if k is even, otherwise 0, where oddfactorial(z) := (2z)!/(2^zz!).

A109340 Expansion of x^2(1+x+4x^2)/((1+x+x^2)*(1-x)^3).