A014601 -id:A014601 - cp's OEIS frontend

A114383 Length of all-prime chain of prime(n) + successive even triangular numbers.

1, 2, 3, 2, 4, 2, 1, 2, 1, 9, 3, 2, 1, 2, 2, 1, 12, 2, 1, 8, 1, 2, 1, 3, 2, 5, 2, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 2, 2, 1, 1, 2, 2, 1, 1, 1, 4, 2, 1, 2, 1, 1, 2, 2, 2, 1, 3, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 6, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 2, 1

Offset: 2

Jonathan Vos Post, Feb 11 2006

a(1) is undefined, as prime(1) is the only even prime, for which the length-5 chain is of 2 + successive odd triangular numbers A014493: 2 prime, 2+1 = 3 prime, 2+3 = 5 prime, 2+15 = 17 prime, 2+21 = 23 prime, 2+45 = 47 prime and then 2+55 = 57 = 3*19 nonprime.

a(18) = 12. The next n for which a(n) >= 10 is a(22509) = 10. What is the next for which a(n) >= 12? Such n > 5761440. - Robert Israel, Jun 14 2016

			a(2) = 1 because prime(2) = 3 is prime, but prime(2) + EvenTriangular(1) = 3 + 6 = 9 = 3^2 is nonprime, giving a chain of just 1 successive prime.
a(3) = 2 because prime(3) + EvenTriangular(1) = 5 + 6 = 11 is prime, but prime(3) + EvenTriangular(2) = 5 + 10 = 15 = 3*5 is nonprime, giving a chain of 2 successive primes.
a(4) = 3 because 7 is prime, 7+6 = 13 is prime, 7+10 = 17 is prime, but 7+28 = 35 = 5*7 is nonprime, for a chain of 3 successive primes.
a(6) = 4 because 13 is prime, 13+6 = 19 is prime, 13+10 = 23 is prime, 13+28 = 41 is prime, but 13+36 = 49 = 7^2 is nonprime.
a(11) = 9 because 31 is prime, as is 31+6 = 37; 31+10 = 41; 31+28 = 59; 31+36 = 67; 31+66 = 97; 31+78 = 109; 31+120 = 151; 31+136 = 167; but 31+190 = 221 = 13*17 is nonprime.
a(18) = 11 because of the prime chain 61; 61+6 = 67; 61+10 = 71; 61+28 = 89; 61+36 = 97; 61+66 = 127; 61+78 = 139; 61+120 = 181; 61+136= 197; 61+190 = 251; 61+276 = 337; but 61+300 = 361 = 19^2 is nonprime.

Robert Israel, Table of n, a(n) for n = 2..10000

Cf. A000040, A000217, A014601.

f:= proc(n) local p,k,j,count;
p:= ithprime(n);
count:= -1;
for k from 0 do
  for j in [0,3] do
     count:= count+1;
     if not isprime (p + 1/2*(4*k+j)*(4*k+j+1)) then return count fi;
od od
end proc:
map(f, [$2..100]); # Robert Israel, Jun 14 2016

evt = Select[(# + 1) #/2 &@Range[200], EvenQ]; a[n_] := Block[{s = Prime@n, c = 1}, While[PrimeQ[s + evt[[c]]], c++]; c]; a /@ Range[2, 90] (* Giovanni Resta, Jun 14 2016 *)

a(n) = k = length of chain prime[n] + A014493(1) + ... + A014493(k) such that each term in the chain is prime. a(n) = k = length of chain A000040(n) + A000217(A014601(1)) + ... + A000217(A014601(k)) such that each term in the chain is in A000040.

Corrected and extended by Giovanni Resta, Jun 14 2016

A137595 Integers k such that the run lengths of identical digits in their binary expansion are palindromic after the rightmost bit is duplicated.

Original entry on oeis.org

1, 3, 6, 7, 13, 15, 25, 26, 28, 31, 49, 53, 59, 63, 97, 102, 106, 109, 115, 116, 120, 127, 193, 201, 213, 221, 227, 235, 247, 255, 385, 398, 406, 409, 421, 426, 434, 445, 451, 460, 468, 475, 487, 488, 496, 511, 769, 785, 809, 825, 837, 853, 877, 893, 899, 915

Offset: 1

Gary W. Adamson, Jan 29 2008

Decimal-binary representations of palindromic continued fractions.
Using the conversion rules, the first 14 fractions in the Stern-Brocot infinite Farey tree, (rational fractions k, 0 < k < 1) with palindromic continued fraction representations are: 1/2, 1/3, 2/5, 1/4, 3/8, 1/5, 5/12, 5/13, 3/10, 1/6, 7/16, 8/21, 4/15, 1/7.
In other words, this sequence encodes the positive fractions less than 1 with palindromic continued fractions by the order that they appear in the Stern-Brocot infinite Farey tree. The terms of the continued fraction [0; c1, c2, ..., cN] are encoded in the run lengths of the binary digits of k. However, the last term in a continued fraction is always greater than 1, so instead of encoding cN, we encode cN-1. - Dominic McCarty, Mar 04 2025

			26 in binary is 11010. Appending a duplicate of the rightmost digit, 0, to the right gives 110100. The run lengths of consecutive identical binary digits is 2,1,1,2, which is a palindrome, so 26 is in the sequence.
The fraction corresponding to the encoded continued fraction [0;2,1,1,2] is 5/13.

Dominic McCarty, Table of n, a(n) for n = 1..10000

Cf. A014601.

from itertools import groupby
def ok(n):
    if n == 0: return False
    d = [len(list(g[1])) for g in groupby(bin(n)[2:])]
    d[-1] += 1
    return all(d[i]==d[-i-1] for i in range(len(d)//2))
print((str([n for n in range(100) if ok(n)]))) # Dominic McCarty, Mar 04 2025

Edited by Franklin T. Adams-Watters, Mar 29 2014

Name edited by Dominic McCarty, Mar 04 2025

A139691 Discriminants of the normalized general quintic polynomials with nonnegative coefficients.

Original entry on oeis.org

0, 12, 40, 48, 49, 69, 84, 92, 93, 117, 124, 125, 128, 132, 144, 161, 176, 184, 189, 217, 229, 240, 245, 256, 257, 272, 312, 320, 324, 332, 333, 340, 348, 392, 400, 432, 448, 456, 472, 512, 549, 588, 592, 605, 609, 688, 697, 708, 725, 761, 804, 832, 836, 837

Offset: 1

Artur Jasinski, Apr 29 2008

nonn

Possible discriminants of the general normalized quintic polynomial x^5+b*x^4+c*x^3+d*x^2+e*x+f with b,c,d,e,f>=0

Mohammad K. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics, Vol. 36, No. 2, 2007, pp. 251-257. Mathematical Reviews, MR2312537. Zentralblatt MATH, Zbl 1133.11012.

Cf. A014601, A042948.

aa = {}; a = 1; Do[Print[f]; Do[Do[Do[Do[k = b^2 c^2 d^2 e^2 - 4 a c^3 d^2 e^2 - 4 b^3 d^3 e^2 + 18 a b c d^3 e^2 - 27 a^2 d^4 e^2 - 4 b^2 c^3 e^3 + 16 a c^4 e^3 + 18 b^3 c d e^3 - 80 a b c^2 d e^3 - 6 a b^2 d^2 e^3 + 144 a^2 c d^2 e^3 - 27 b^4 e^4 + 144 a b^2 c e^4 - 128 a^2 c^2 e^4 - 192 a^2 b d e^4 + 256 a^3 e^5 - 4 b^2 c^2 d^3 f + 16 a c^3 d^3 f + 16 b^3 d^4 f - 72 a b c d^4 f + 108 a^2 d^5 f + 18 b^2 c^3 d e f - 72 a c^4 d e f - 80 b^3 c d^2 e f + 356 a b c^2 d^2 e f + 24 a b^2 d^3 e f - 630 a^2 c d^3 e f - 6 b^3 c^2 e^2 f + 24 a b c^3 e^2 f + 144 b^4 d e^2 f - 746 a b^2 c d e^2 f + 560 a^2 c^2 d e^2 f + 1020 a^2 b d^2 e^2 f - 36 a b^3 e^3 f + 160 a^2 b c e^3 f - 1600 a^3 d e^3 f - 27 b^2 c^4 f^2 + 108 a c^5 f^2 + 144 b^3 c^2 d f^2 - 630 a b c^3 d f^2 - 128 b^4 d^2 f^2 + 560 a b^2 c d^2 f^2 + 825 a^2 c^2 d^2 f^2 - 900 a^2 b d^3 f^2 - 192 b^4 c e f^2 + 1020 a b^2 c^2 e f^2 - 900 a^2 c^3 e f^2 + 160 a b^3 d e f^2 - 2050 a^2 b c d e f^2 + 2250 a^3 d^2 e f^2 - 50 a^2 b^2 e^2 f^2 + 2000 a^3 c e^2 f^2 + 256 b^5 f^3 - 1600 a b^3 c f^3 + 2250 a^2 b c^2 f^3 + 2000 a^2 b^2 d f^3 - 3750 a^3 c d f^3 - 2500 a^3 b e f^3 + 3 125 a^4 f^4; If[k > 0 && k < 1000, AppendTo[aa, k]], {b, 0, 30}], {c, 0, 30}], {d, 0, 30}], {e, 0, 30}], {f, 0, 30}]; Union[aa] (*Artur Jasinski*)

A154708 Numbers a such that b and c exist with b <= a < c and a*(a+1) + b^2 = c^2.

Original entry on oeis.org

3, 4, 7, 8, 11, 12, 15, 16, 19, 20, 23, 24, 27, 28, 31, 32, 35, 36, 39, 40, 43, 44, 47, 48, 51, 52, 55, 56, 59, 60, 63, 64, 67, 68, 71, 72, 75, 76, 79, 80, 83, 84, 87, 88, 91, 92, 95, 96, 99, 100, 103, 104, 107, 108, 111, 112, 115, 116, 119, 120, 123, 124, 127, 128, 131, 132, 135, 136, 139, 140

Offset: 1

Pierre CAMI, Jan 14 2009, Jan 15 2009

Numbers a such that a*(a+1) = c2 - b2 with b <= a < c let a(1)=3 then a(2*n) = a(2*n-1) + 1 and a(2*n+1) = a(2*n) + 3. [Pierre CAMI, Jan 15 2009]

			4*5 + 4*4 = 6*6; 7*8 + 5*5 = 9*9; 8*9 + 7*7 = 11*11;
3*4 = 4*4 - 2*2; a(1)=3; 4*5 = 6*6 - 4*4; a(2) = 4; 7*6 = 9*9 - 5*5; a(3)=7. [_Pierre CAMI_, Jan 15 2009]

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

Essentially the same as A014601. [R. J. Mathar, Nov 01 2009]

LinearRecurrence[{1,1,-1},{3,4,7},70] (* Harvey P. Dale, Jan 07 2016 *)

a(1)=4; thereafter a(2*n) = a(2*n-1) + 3, a(2*n+1) = a(2*n) + 1.
a(n) = 4*n - a(n-1) - 1 (with a(1)=3). [Vincenzo Librandi, Nov 26 2010]
From Colin Barker, Mar 06 2013: (Start)
a(n) = (1 - (-1)^n + 4*n)/2.
a(n) = a(n-1) + a(n-2) - a(n-3).
G.f.: x*(x+3) / ((x-1)^2*(x+1)). (End)

More terms from Vincenzo Librandi, Nov 26 2010

A225478 Triangle read by rows, 4^k*s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.

Original entry on oeis.org

1, 3, 4, 21, 40, 16, 231, 524, 336, 64, 3465, 8784, 7136, 2304, 256, 65835, 180756, 170720, 72320, 14080, 1024, 1514205, 4420728, 4649584, 2346240, 613120, 79872, 4096, 40883535, 125416476, 143221680, 81946816, 25939200, 4609024, 430080, 16384, 1267389585, 4051444896, 4941537984, 3113238016, 1131902464, 246636544, 31768576, 2228224, 65536

Offset: 0

Peter Luschny, May 17 2013

Triangle T(n,k), read by rows, given by (3, 4, 7, 8, 11, 12, 15, 16, ... (A014601)) DELTA (4, 0, 4, 0, 4, 0, 4, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, May 14 2015.

			[n\k][    0,       1,       2,       3,      4,     5,    6 ]
[0]       1,
[1]       3,       4,
[2]      21,      40,      16,
[3]     231,     524,     336,      64,
[4]    3465,    8784,    7136,    2304,    256,
[5]   65835,  180756,  170720,   72320,  14080,  1024,
[6] 1514205, 4420728, 4649584, 2346240, 613120, 79872, 4096.

Peter Luschny, Generalized Eulerian polynomials.
Peter Luschny, The Stirling-Frobenius numbers.

T(n, 0) ~ A008545; T(n, n) ~ A000302; T(n, n-1) ~ A002700.
row sums ~ A034176; alternating row sums ~ A008545.
Cf. A225471, A132393 (m=1), A028338 (m=2), A225477 (m=3).

s[][0, 0] = 1; s[m][n_, k_] /; (k > n || k < 0) = 0; s[m_][n_, k_] := s[m][n, k] = s[m][n - 1, k - 1] + (m*n - 1)*s[m][n - 1, k];
T[n_, k_] := 4^k*s[4][n, k];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 02 2019 *)

@CachedFunction
def SF_CS(n, k, m):
    if k > n or k < 0 : return 0
    if n == 0 and k == 0: return 1
    return m*SF_CS(n-1, k-1, m) + (m*n-1)*SF_CS(n-1, k, m)
for n in (0..8): [SF_CS(n, k, 4) for k in (0..n)]

For a recurrence see the Sage program.

T(n,k) = 4^k * A225471(n,k). - Philippe Deléham, May 14 2015.

A305494 Let s(D) = Sum_{(a,b,c)} j((-b+sqrt(D))/(2a)) where (a,b,c) is taken over all the primitive reduced binary quadratic forms ax^2+bxy+cy^2 with b^2-4*ac = D. This sequence is s(D) as D runs through the numbers -3, -4, -7, -8, -11, -12, ... .

Original entry on oeis.org

0, 1728, -3375, 8000, -32768, 54000, -191025, 287496, -884736, 1264000, -3491750, 4834944, -12288000, 16581375, -39491307, 52250000, -117964800, 153542016, -331531596, 425692800, -884736000, 1122662608, -2257834125, 2835810000, -5541101568, 6896880000, -13136684625

Offset: 1

Seiichi Manyama, Jun 02 2018

sign

			In the case D = -15,
j((1+sqrt(-15))/2) + j((1+sqrt(-15))/4) = (-191025-85995*sqrt(5))/2 + (-191025+85995*sqrt(5))/2 = -191025.
  ----+-------------------------------------------+---------
    D | Coefficients of Hilbert class polynomial  |   a(n)
  ----+-------------------------------------------+---------
   -3 |              0,            1;             |        0
   -4 |          -1728,            1;             |     1728
   -7 |           3375,            1;             |    -3375
   -8 |          -8000,            1;             |     8000
  -11 |          32768,            1;             |   -32768
  -12 |         -54000,            1;             |    54000
  -15 |     -121287375,       191025,        1;   |  -191025
  -16 |        -287496,            1;             |   287496
  -19 |         884736,            1;             |  -884736
  -20 |     -681472000,     -1264000,        1;   |  1264000
  -23 | 12771880859375,  -5151296875,  3491750, 1;| -3491750
  -24 |    14670139392,     -4834944,        1;   |  4834944

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A014601, A032354, A305474.

A354522 Square array A(n, k), n, k >= 0, read by antidiagonals; A(n, k) = g(f(n) + f(k)) where f denotes A001057 and g denotes its inverse.

Original entry on oeis.org

0, 1, 1, 2, 3, 2, 3, 0, 0, 3, 4, 5, 4, 5, 4, 5, 2, 1, 1, 2, 5, 6, 7, 6, 7, 6, 7, 6, 7, 4, 3, 0, 0, 3, 4, 7, 8, 9, 8, 9, 8, 9, 8, 9, 8, 9, 6, 5, 2, 1, 1, 2, 5, 6, 9, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 8, 7, 4, 3, 0, 0, 3, 4, 7, 8, 11, 12, 13, 12, 13, 12, 13, 12, 13, 12, 13, 12, 13, 12

Offset: 0

Rémy Sigrist, Sep 14 2022

This sequence is directly related to A355278.
The function f is a bijection from the nonnegative integers to the integers (Z).
The nonnegative integers, together with (x,y) -> A(x,y), form an abelian group isomorph to the additive group Z (f and g act as isomorphisms).
As a consequence, each row and each column is a permutation of the nonnegative integers.

			Array A(n, k) begins:
  n\k |  0   1   2   3   4   5   6   7   8   9  10  11  12
  ----+---------------------------------------------------
    0 |  0   1   2   3   4   5   6   7   8   9  10  11  12
    1 |  1   3   0   5   2   7   4   9   6  11   8  13  10
    2 |  2   0   4   1   6   3   8   5  10   7  12   9  14
    3 |  3   5   1   7   0   9   2  11   4  13   6  15   8
    4 |  4   2   6   0   8   1  10   3  12   5  14   7  16
    5 |  5   7   3   9   1  11   0  13   2  15   4  17   6
    6 |  6   4   8   2  10   0  12   1  14   3  16   5  18
    7 |  7   9   5  11   3  13   1  15   0  17   2  19   4
    8 |  8   6  10   4  12   2  14   0  16   1  18   3  20
    9 |  9  11   7  13   5  15   3  17   1  19   0  21   2
   10 | 10   8  12   6  14   4  16   2  18   0  20   1  22
   11 | 11  13   9  15   7  17   5  19   3  21   1  23   0
   12 | 12  10  14   8  16   6  18   4  20   2  22   0  24

Cf. A001057, A014601, A014681, A047264, A047521, A355278, A357144.

f(n) = - (-1)^n * ((n+1)\2)
g(n) = if (n<=0, -2*n, 2*n-1)
A(n, k) = g(f(n) + f(k))

A355278(n+1, k+1) = prime(1 + A(n, k)) (where prime(m) denotes the m-th prime number).
A(n, k) = A(k, n).
A(n, 0) = n.
A(n, A014681(n)) = 0.
A(m, A(n, k)) = A(A(m, n), k).
A(n, n) = A014601(n).
A(n, A(n, n)) = A047264(n+1).
A(A(n, n), A(n, n)) = A047521(n+1).

A355663 Square array A(n, k), n, k >= 0, read by antidiagonals; for any number n with runs in binary expansion (r_w, ..., r_0), let p(n) be the polynomial of a single indeterminate x where the coefficient of x^e is r_e for e = 0..w and otherwise 0, and let q be the inverse of p; A(n, k) = q(p(n) + p(k)).

Original entry on oeis.org

0, 1, 1, 2, 3, 2, 3, 4, 4, 3, 4, 7, 12, 7, 4, 5, 8, 8, 8, 8, 5, 6, 11, 24, 15, 24, 11, 6, 7, 12, 19, 16, 16, 19, 12, 7, 8, 15, 28, 23, 48, 23, 28, 15, 8, 9, 16, 16, 24, 39, 39, 24, 16, 16, 9, 10, 19, 48, 31, 56, 51, 56, 31, 48, 19, 10, 11, 20, 35, 32, 32, 35, 35, 32, 32, 35, 20, 11

Offset: 0

Rémy Sigrist, Jul 13 2022

In other words, A(n, k) encodes the sum of the polynomials encoded by n and k.

			Array A(n, k) begins:
  n\k|   0   1   2   3    4    5    6   7    8    9   10   11   12
  ---+------------------------------------------------------------
    0|   0   1   2   3    4    5    6   7    8    9   10   11   12
    1|   1   3   4   7    8   11   12  15   16   19   20   23   24
    2|   2   4  12   8   24   19   28  16   48   35   44   39   56
    3|   3   7   8  15   16   23   24  31   32   39   40   47   48
    4|   4   8  24  16   48   39   56  32   96   71   88   79  112
    5|   5  11  19  23   39   51   35  47   79   99   76  103   71
    6|   6  12  28  24   56   35   60  48  112   67   92   71  120
    7|   7  15  16  31   32   47   48  63   64   79   80   95   96
    8|   8  16  48  32   96   79  112  64  192  143  176  159  224
    9|   9  19  35  39   71   99   67  79  143  195  156  199  135
   10|  10  20  44  40   88   76   92  80  176  156  204  152  184
   11|  11  23  39  47   79  103   71  95  159  199  152  207  143
   12|  12  24  56  48  112   71  120  96  224  135  184  143  240

Index entries for sequences related to binary expansion of n

Cf. A001196, A014601, A101211, A355664.

toruns(n) = { my (r=[]); while (n, my (v=valuation(n+n%2, 2)); n\=2^v; r=concat(v, r)); r }
fromruns(r) = { my (v=0); for (k=1, #r, v=(v+k%2)*2^r[k]-k%2); v }
A(n,k) = { fromruns(Vec(Pol(toruns(n)) + Pol(toruns(k)))) }

A(n, k) = A(k, n).
A(n, 0) = n.
A(n, 1) = A014601(n) for any n > 0.
A(n, n) = A001196(n).

A364168 Numbers that can be written in more than one way in the form (j+2k)^2-(j+k)^2-j^2 with j,k>0.

Original entry on oeis.org

15, 27, 32, 35, 36, 39, 51, 55, 60, 63, 64, 75, 84, 87, 91, 95, 96, 99, 100, 108, 111, 115, 119, 123, 128, 132, 135, 140, 143, 144, 147, 155, 156, 159, 160, 171, 175, 180, 183, 187, 192, 195, 196, 203, 204, 207, 215, 219, 220, 224, 228, 231, 235, 240, 243, 247, 252, 255

Offset: 1

Darío Clavijo, Jul 12 2023

nonn

From Darío Clavijo, Mar 05 2025: (Start)
Also, numbers h that can be written as a difference of squares such as h=4*y^2-x^2 where x=2*y-p and y=(p+q)/4 and p<3*q with p and q divisors of h.
a(n) == 0 or 3 (mod 4). (End).
From Darío Clavijo, Apr 22 2025: (Start)
Numbers that can be written in more than one way in the form (j+k) * (3k-j).
Every term is congruent to {0, 3, 4, 7, 11, 12, 15} (mod 16). (End).

			27 is a term since (6+2*3)^2 - (6+3)^2 - 6^2 = (20+2*7)^2 - (20+7)^2 - 20^2 = 27.

Project Euler, Problem 135: Same Differences.

Cf. A000005, A000290, A027750, A014601.

from math import isqrt
def isok(h):
    if (h & 15) not in [0, 3, 4, 7, 11, 12, 15]: return False
    c = 0
    for p in range(1, isqrt(h)+1):
        q, r = divmod(h,p)
        if r == 0 and (pq := p + q) & 3 == 0:
            t = pq >> 2;
            c += (t < p) + (p != q and t < q)
            if c > 1: return True
print([h for h in range(1, 256) if isok(h)])

A100832 Amenable numbers: n such that there exists a multiset of integers (s(1), ..., s(n)) whose size, sum and product are all n.

Original entry on oeis.org

1, 5, 8, 9, 12, 13, 16, 17, 20, 21, 24, 25, 28, 29, 32, 33, 36, 37, 40, 41, 44, 45, 48, 49, 52, 53, 56, 57, 60, 61, 64, 65, 68, 69, 72, 73, 76, 77, 80, 81, 84, 85, 88, 89, 92, 93, 96, 97, 100, 101, 104, 105, 108, 109, 112, 113, 116, 117, 120, 121, 124, 125, 128, 129, 132

Offset: 1

Lekraj Beedassy, Jan 07 2005

nonn

Positive numbers k == 0 or 1 (mod 4), excluding k=4.
Essentially the same as A042948 (except 4 is not in this sequence).
The set {s(i)} is closed under multiplication. - Lekraj Beedassy, Jan 21 2005

			5 and 8, for instance, are in the sequence because we have 5 = 1-1+1-1+5 = 1*(-1)*1*(-1)*5 and 8 = 1-1+1-1+1+1+2+4 = 1*(-1)*1*(-1)*1*1*2*4.

O. P. Lossers, Solution to problem 10454: Amenable Numbers, Amer. Math. Monthly Vol. 105 No. 4 April 1998.
Eric Weisstein's World of Mathematics, Amenable Number
Wikipedia, Amenable number
Index entries for linear recurrences with constant coefficients, signature (1, 1, -1).

Cf. A014601, A042948.

From Colin Barker, Jan 26 2012: (Start)
a(n) = a(n-1) + a(n-2) - a(n-3), n > 4.
G.f.: x*(1+3*x)*(1+x-x^2)/(1-x-x^2+x^3). (End)

More terms from David W. Wilson, Jan 24 2005

cp's OEIS Frontend

A114383 Length of all-prime chain of prime(n) + successive even triangular numbers.

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Maple

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A137595 Integers k such that the run lengths of identical digits in their binary expansion are palindromic after the rightmost bit is duplicated.

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Python

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A139691 Discriminants of the normalized general quintic polynomials with nonnegative coefficients.

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A154708 Numbers a such that b and c exist with b <= a < c and a*(a+1) + b^2 = c^2.

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A225478 Triangle read by rows, 4^k*s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.

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A305494 Let s(D) = Sum_{(a,b,c)} j((-b+sqrt(D))/(2*a)) where (a,b,c) is taken over all the primitive reduced binary quadratic forms a*x^2+b*xy+c*y^2 with b^2-4*ac = D. This sequence is s(D) as D runs through the numbers -3, -4, -7, -8, -11, -12, ... .

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A354522 Square array A(n, k), n, k >= 0, read by antidiagonals; A(n, k) = g(f(n) + f(k)) where f denotes A001057 and g denotes its inverse.

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PARI

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A305494 Let s(D) = Sum_{(a,b,c)} j((-b+sqrt(D))/(2a)) where (a,b,c) is taken over all the primitive reduced binary quadratic forms ax^2+bxy+cy^2 with b^2-4*ac = D. This sequence is s(D) as D runs through the numbers -3, -4, -7, -8, -11, -12, ... .