A270710 -id:A270710 - cp's OEIS frontend

A322570 Positive integers k such that A270710(k) (= (k+1)(3k-1)) have only 1 or 2 different digits in base 10.

1, 2, 3, 4, 5, 6, 12, 16, 17, 33, 34, 48, 54, 285, 333, 334, 365, 385, 430, 471, 516, 816, 1049, 3333, 3334, 33333, 33334, 333333, 333334, 483048, 3333333, 3333334, 33333333, 33333334, 333333333, 333333334, 3333333333, 3333333334, 33333333333, 33333333334

Offset: 1

Seiichi Manyama, Aug 29 2019

Giovanni Resta, Table of n, a(n) for n = 1..50

Cf. A002277, A016069, A093137, A213517 (in case of triangular numbers), A270710, A322571.

[k:k in [1..10000000]| #Set(Intseq((k+1)*(3*k-1))) le 2]; // Marius A. Burtea, Aug 29 2019

Select[Range@ 50000, Length@ Union@ IntegerDigits[3 #^2 + 2 # - 1] <= 2 &] (* Giovanni Resta, Sep 04 2019 *)

for(k=1, 1e8, if(#Set(digits(3*k^2+2*k-1))<=2, print1(k", ")))

For k > 0, A002277(k) is a term.

For k >= 0, A002277(k) + 1 (= A093137(k)) is a term.

a(35)-a(36) from Jinyuan Wang, Aug 30 2019

a(37)-a(40) from Giovanni Resta, Sep 04 2019

A322571 Positive integers A270710(k) (= (k+1)(3k-1)) which have only 1 or 2 different digits in base 10.

Original entry on oeis.org

4, 15, 32, 55, 84, 119, 455, 799, 900, 3332, 3535, 7007, 8855, 244244, 333332, 335335, 400404, 445444, 555559, 666464, 799799, 1999199, 3303300, 33333332, 33353335, 3333333332, 3333533335, 333333333332, 333335333335, 700007077007, 33333333333332, 33333353333335, 3333333333333332, 3333333533333335

Offset: 1

Seiichi Manyama, Aug 29 2019

Giovanni Resta, Table of n, a(n) for n = 1..50

Cf. A002277, A018885 (in case of squares), A213516 (in case of triangular numbers), A270710, A322570, A323639.

[a:k in [1..10000000]| #Set(Intseq(a)) le 2 where a is (k+1)*(3*k-1)]; // Marius A. Burtea, Aug 29 2019

Select[Table[(n+1)(3n-1),{n,3334*10^4}],Count[DigitCount[#],0]>7&] (* Harvey P. Dale, Jun 12 2022 *)

for(k=1, 1e8, if(#Set(digits(j=3*k^2+2*k-1))<=2, print1(j", ")))

a(n) = A270710(A322570(n)).

For k > 0, A002277(2*k) - 1 is a term.

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

A326728 A(n, k) = n(k - 1)k/2 - k, square array for n >= 0 and k >= 0 read by ascending antidiagonals.

Original entry on oeis.org

0, 0, -1, 0, -1, -2, 0, -1, -1, -3, 0, -1, 0, 0, -4, 0, -1, 1, 3, 2, -5, 0, -1, 2, 6, 8, 5, -6, 0, -1, 3, 9, 14, 15, 9, -7, 0, -1, 4, 12, 20, 25, 24, 14, -8, 0, -1, 5, 15, 26, 35, 39, 35, 20, -9, 0, -1, 6, 18, 32, 45, 54, 56, 48, 27, -10

Offset: 0

Peter Luschny, Aug 04 2019

A formal extension of the figurative numbers A139600 to negative n.

			[0] 0, -1, -2, -3, -4, -5, -6,  -7,  -8,  -9, -10, ... A001489
[1] 0, -1, -1,  0,  2,  5,  9,  14,  20,  27,  35, ... A080956
[2] 0, -1,  0,  3,  8, 15, 24,  35,  48,  63,  80, ... A067998
[3] 0, -1,  1,  6, 14, 25, 39,  56,  76,  99, 125, ... A095794
[4] 0, -1,  2,  9, 20, 35, 54,  77, 104, 135, 170, ... A014107
[5] 0, -1,  3, 12, 26, 45, 69,  98, 132, 171, 215, ... A326725
[6] 0, -1,  4, 15, 32, 55, 84, 119, 160, 207, 260, ... A270710
[7] 0, -1,  5, 18, 38, 65, 99, 140, 188, 243, 305, ...

Peter Luschny, Figurate number — a very short introduction. With plots from Stefan Friedrich Birkner.

Cf. A001489 (n=0), A080956 (n=1), A067998 (n=2), A095794 (n=3), A014107 (n=4), A326725 (n=5), A270710 (n=6).
Columns include A008585, A016933, A017329.
Cf. A139600.

A := (n, k) -> n*(k - 1)*k/2 - k:
seq(seq(A(n - k, k), k=0..n), n=0..11);

def A326728Row(n):
    x, y = 1, 1
    yield 0
    while True:
        yield -x
        x, y = x + y - n, y - n
for n in range(8):
    R = A326728Row(n)
print([next(R) for _ in range(11)])

A273894 Table T(n,k), n >= 0, k = 1..2^n, read by rows, giving coefficients of iterations of polynomial x^2-x: see Comments for precise definition.

Original entry on oeis.org

1, -1, 1, 1, 0, -2, 1, -1, 1, 2, -5, 2, 4, -4, 1, 1, 0, -4, 2, 12, -14, -20, 48, -14, -50, 60, -10, -28, 24, -8, 1, -1, 1, 4, -10, -8, 54, -24, -180, 270, 270, -960, 150, 2064, -2040, -2352, 5871, -1566, -7236, 8880, 120, -9120, 7980, 120, -5340, 4212, -756

Offset: 0

Robert Israel, Jun 02 2016

Let p(0) = t, p(n) = p(n-1)^2 - p(n-1) for i >= 1.
T(n,k) is coefficient of t^k in p(n).
Rows sum to 0, except for row 0. - David A. Corneth, Jun 02 2016

			Table starts:
   1;
  -1, 1;
   1, 0, -2,  1;
  -1, 1,  2, -5,  2,   4,  -4,  1;
   1, 0, -4,  2, 12, -14, -20, 48, -14, -50, 60, -10, -28, 24, -8, 1;
  ...

Robert Israel, Table of n, a(n) for n = 0..11212

Cf. A000035, A033999, A137501, A161680, A270710.

P[0]:= t:
for n from 1 to 8 do
  P[n]:= expand(P[n-1]^2 - P[n-1])
od:
seq(seq(coeff(P[n],t,j),j=1..2^n),n=0..8);

CoefficientList[NestList[Expand[#^2-#]&, x, 5]/x, x] // Flatten (* Jean-François Alcover, Apr 29 2019 *)

T(n,k) = -T(n-1,k) + Sum_{j=1..k-1} T(n-1,j) T(n-1,k-j).
Column k is of the form
T(n,k) = b_k(n) + (-1)^n*c_k(n)
where b_k and c_k seem to be polynomials of degree floor(k/2) - 1 and floor((k-1)/2) respectively (except b_1 = 0).
Leading coefficient of b_k(n) + (-1)^n*c_k(n) seems to be
  -(-2)^(k/2-2) - binomial(-3/2,k/2-1)*2^(k/2-2)*(-1)^n if k is even,
  2^((k-1)/2)*binomial(-1/2,(k-1)/2)*(-1)^n if k is odd.
T(n,1) = (-1)^n = A033999(n).
T(n,2) = 1/2 + (-1)^n/2 = A000035(n)
T(n,3) = -1/2 + (-n + 1/2)*(-1)^n = -A137501(n).
T(n,4) = -n + 5/4 + (3*n/2 - 5/4)*(-1)^n
  = n/2 if n is even, -5*(n-1)/2 if n is odd.
T(n,5) = 2*n - 11/4 + (3*n^2/2 - 5*n + 11/4)*(-1)^n
  = 12*A161680(n/2) if n is even, -2*A270710((n-3)/2) if n >= 3 is odd.
T(n, 2^n) = 1 = A000012(n). - David A. Corneth, Jun 02 2016

cp's OEIS Frontend

A322570 Positive integers k such that A270710(k) (= (k+1)*(3*k-1)) have only 1 or 2 different digits in base 10.

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A322571 Positive integers A270710(k) (= (k+1)*(3*k-1)) which have only 1 or 2 different digits in base 10.

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Magma

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

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Magma

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Python

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A326728 A(n, k) = n*(k - 1)*k/2 - k, square array for n >= 0 and k >= 0 read by ascending antidiagonals.

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Maple

Python

A273894 Table T(n,k), n >= 0, k = 1..2^n, read by rows, giving coefficients of iterations of polynomial x^2-x: see Comments for precise definition.

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Keywords

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Maple

Mathematica

Formula

A322570 Positive integers k such that A270710(k) (= (k+1)(3k-1)) have only 1 or 2 different digits in base 10.

A322571 Positive integers A270710(k) (= (k+1)(3k-1)) which have only 1 or 2 different digits in base 10.

A326728 A(n, k) = n(k - 1)k/2 - k, square array for n >= 0 and k >= 0 read by ascending antidiagonals.