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7, 20, 33, 46, 59, 72, 85, 98, 111, 124, 137, 150, 163, 176, 189, 202, 215, 228, 241, 254, 267, 280, 293, 306, 319, 332, 345, 358, 371, 384, 397, 410, 423, 436, 449, 462, 475, 488, 501, 514, 527, 540, 553, 566, 579, 592, 605, 618, 631, 644, 657, 670, 683, 696, 709, 722, 735
COMMENTS
After 7 (which corresponds to n=0), all terms belong to A090767 because a(n) = 3*n*2*1 + 2*(n*2+2*1+n*1) + (n+2+1). This sequence is related to A152741 by the recurrence A152741(n+1) = (n+1)*a(n+1) - Sum_{k = 0..n} a(k). Any square mod 13 is one of 0, 1, 3, 4, 9, 10 or 12 ( A010376) but not 7, and for this reason there are no squares in the sequence. Likewise, any cube mod 13 is one of 0, 1, 5, 8 or 12, therefore no a(k) is a cube. The sum of the squares of any two terms of the sequence is also a term of the sequence, that is: a(h)^2 + a(k)^2 = a(h*(13*h+14) + k*(13*k+14) + 7). Therefore: a(h)^2 + a(k)^2 > a(a( h*(h+1) + k*(k+1) )) for h+k > 0.
The primes of the sequence are listed in A140371.
LINKS
Bruno Berselli, A description of the recursive method shown in the third comment: website Matem@ticamente (in Italian), 2008.
FORMULA
G.f.: (7 + 6*x)/(1 - x)^2.
Sum_{i=h..h+13*k} a(i) = a(h*(13*k + 1) + k*(169*k + 27)/2).
Sum_{i>=0} 1/a(i)^2 = 0.0257568950542502716970... = polygamma(1, 7/13)/13^2.
MATHEMATICA
13 Range[0, 60] + 7 (* or *) Range[7, 800, 13] (* or *) Table[13 n + 7, {n, 0, 60}]
PROG
(Magma) [13*n+7: n in [0..60]];
(Maxima) makelist(13*n+7, n, 0, 60);
(PARI) vector(60, n, n--; 13*n+7)
(Sage) [13*n+7 for n in (0..60)]
2, 15, 28, 41, 54, 67, 80, 93, 106, 119, 132, 145, 158, 171, 184, 197, 210, 223, 236, 249, 262, 275, 288, 301, 314, 327, 340, 353, 366, 379, 392, 405, 418, 431, 444, 457, 470, 483, 496, 509, 522, 535, 548, 561, 574, 587, 600, 613, 626, 639, 652, 665, 678, 691
COMMENTS
Any square mod 13 is one of 0, 1, 3, 4, 9, 10 or 12 ( A010376) but not 2, and for this reason there are no squares in the sequence. Likewise, any cube mod 13 is one of 0, 1, 5, 8 or 12, therefore no term is a cube. - Bruno Berselli, Feb 19 2016 Numbers k such that GCD(2*k^5+1, 3*k^3+2) > 1. This GCD is 13 if k == 2 (mod 13), or 1 otherwise. - Philippe Deléham, Jan 16 2024
PROG
(Magma) I:=[2, 15]; [n le 2 select I[n] else 2*Self(n-1)-1*Self(n-2): n in [1..60]]; // Vincenzo Librandi, Feb 25 2012
5, 18, 31, 44, 57, 70, 83, 96, 109, 122, 135, 148, 161, 174, 187, 200, 213, 226, 239, 252, 265, 278, 291, 304, 317, 330, 343, 356, 369, 382, 395, 408, 421, 434, 447, 460, 473, 486, 499, 512, 525, 538, 551, 564, 577, 590, 603, 616, 629, 642, 655, 668, 681, 694
COMMENTS
Any square mod 13 is one of 0, 1, 3, 4, 9, 10 or 12 ( A010376) but not 5, for this reason there are no squares in sequence. - Bruno Berselli, Feb 19 2016
FORMULA
G.f.: (5+8*x)/(1-x)^2.
a(n) = 2*a(n-1) - a(n-2). (End)
MATHEMATICA
CoefficientList[Series[(8 x + 5)/(1 - x)^2, {x, 0, 60}], x] (* Vincenzo Librandi, Feb 26 2012 *)
PROG
(Magma) I:=[5, 18]; [n le 2 select I[n] else 2*Self(n-1)-Self(n-2): n in [1..50]]; // Vincenzo Librandi, Feb 26 2012 (SageMath) [13*n+5 for n in range(61)] # G. C. Greubel, May 31 2024
11, 24, 37, 50, 63, 76, 89, 102, 115, 128, 141, 154, 167, 180, 193, 206, 219, 232, 245, 258, 271, 284, 297, 310, 323, 336, 349, 362, 375, 388, 401, 414, 427, 440, 453, 466, 479, 492, 505, 518, 531, 544, 557, 570, 583, 596, 609, 622, 635, 648, 661, 674, 687, 700, 713, 726, 739
COMMENTS
Any square mod 13 is one of 0, 1, 3, 4, 9, 10 or 12 ( A010376) but not 11, and for this reason there are no squares in the sequence. Likewise, any cube mod 13 is one of 0, 1, 5, 8 or 12, therefore no a(k) is a cube. Sequences of the type 13*n + k, for k = 0..12, without squares and cubes:
k = 11: this case.
The sum of the sixth powers of any two terms of the sequence is also a term of the sequence. Example: a(3)^6 + a(8)^6 = a(179129674278) = 2328685765625.
The primes of the sequence are listed in A140373.
FORMULA
G.f.: (11 + 2*x)/(1 - x)^2.
Sum_{i = h..h+13*k} a(i) = a(h*(13*k + 1) + k*(169*k + 35)/2).
Sum_{i >= 0} 1/a(i)^2 = .012486605016510955990... = polygamma(1, 11/13)/13^2.
MATHEMATICA
13 Range[0, 60] + 11
Range[11, 800, 13]
LinearRecurrence[{2, -1}, {11, 24}, 60] (* Harvey P. Dale, Jun 14 2023 *)
PROG
(PARI) vector(60, n, n--; 13*n+11)
(Sage) [13*n+11 for n in range(61)]
(Python) [13*n+11 for n in range(61)]
(Maxima) makelist(13*n+11, n, 0, 60);
(Magma) [13*n+11: n in [0..60]];
CROSSREFS
Similar sequences of the type k*n+k-2: A023443 (k=1), A005843 (k=2), A016777 (k=3), A016825 (k=4), A016885 (k=5), A016957 (k=6), A017041 (k=7), A017137 (k=8), A017245 (k=9), A017365 (k=10), A017497 (k=11), A017641 (k=12).
6, 19, 32, 45, 58, 71, 84, 97, 110, 123, 136, 149, 162, 175, 188, 201, 214, 227, 240, 253, 266, 279, 292, 305, 318, 331, 344, 357, 370, 383, 396, 409, 422, 435, 448, 461, 474, 487, 500, 513, 526, 539, 552, 565, 578, 591, 604, 617, 630, 643, 656, 669, 682
COMMENTS
These numbers appear in the G. E. Andrews paper, for example: see the abstract, formula (1.7), etc. Also "13n + 6" appears in the Folsom-Ono paper (see links).
Row 6 of triangle A151890 lists the first seven terms of this sequence. Any square mod 13 is one of 0, 1, 3, 4, 9, 10 or 12 ( A010376) but not 6, and for this reason there are no squares in the sequence. Likewise, any cube mod 13 is one of 0, 1, 5, 8 or 12, therefore no a(k) is a cube. - Bruno Berselli, Feb 19 2016
MATHEMATICA
LinearRecurrence[{2, -1}, {6, 19}, 60] (* Harvey P. Dale, May 12 2023 *)
PROG
(SageMath) [13*n+6 for n in range(61)] # G. C. Greubel, May 31 2024
4, 17, 30, 43, 56, 69, 82, 95, 108, 121, 134, 147, 160, 173, 186, 199, 212, 225, 238, 251, 264, 277, 290, 303, 316, 329, 342, 355, 368, 381, 394, 407, 420, 433, 446, 459, 472, 485, 498, 511, 524, 537, 550, 563, 576, 589, 602, 615, 628, 641, 654, 667, 680, 693, 706, 719
COMMENTS
Superhighway created by 'LQTL Ant' L90R90L45R45 from iteration 4 where the Ant moves in a 'Moore neighborhood' (nine cells), the L indicates a left turn, the R a right turn, and the numerical value is the size of the turn (in degrees) at each iteration.
REFERENCES
P. Sakar, "A Brief History of Cellular Automata," ACM Computing Surveys, vol. 32, pp. 80-107, 2000.
LINKS
James Propp, Further Ant-ics, Mathematical Intelligencer, 16 pp. 37-42, 1994.
FORMULA
G.f.: (4+9*x)/(1-x)^2.
E.g.f.: (4 + 13*x)*exp(x).
a(n) = 2*a(n-1) - a(n-2) for n >= 2. (End)
PROG
(SageMath) [13*n+4 for n in range(61)] # G. C. Greubel, May 31 2024
Numbers k having at least one prime factor p such that p^2 divides 2^(k-1) - 1.
+10
2
1093, 3511, 398945, 796797, 1194649, 1592501, 1990353, 2388205, 2786057, 3183909, 3581761, 3979613, 4377465, 4775317, 5173169, 5571021, 5968873, 6165316, 6366725, 6764577, 7162429, 7560281, 7958133, 8355985, 8753837, 9151689, 9549541, 9947393, 10345245
COMMENTS
Sequence is infinite since if k is a term then also k^m is a term, for every m >= 2.
What is the smallest number in this sequence which is not of the form 13*n + 1?
Complete factorizations of the first 15 terms:
a(1) = 1093
a(2) = 3511
a(3) = 5 * 73 * 1093
a(4) = 3^6 * 1093
a(5) = 1093^2
a(6) = 31 * 47 * 1093
a(7) = 3 * 607 * 1093
a(8) = 5 * 19 * 23 * 1093
a(9) = 1093 * 2549
a(10) = 3 * 971 * 1093
a(11) = 29 * 113 * 1093
a(12) = 11 * 331 * 1093
a(13) = 3^2 * 5 * 89 * 1093
a(14) = 17 * 257 * 1093
a(15) = 1093 * 4733
These are the numbers k for which gcd(k^2, 2^(k-1)-1) is not squarefree. However, numbers k such that gcd(k^2, 2^(k-1)-1) > k are a proper subset of them. Are there infinitely many such numbers? See A331021. - Amiram Eldar and Thomas Ordowski, Jan 06 2020
PROG
(Magma) lst:=[]; for n in [2..10345245] do f:=Factorization(n); if not IsNull([x: x in [1..#f] | Modexp(2, n-1, f[x][1]^2) eq 1]) then Append(~lst, n); end if; end for; lst;
Array T(n,k) = binomial(n, 2) + k*n + 1 read by antidiagonals.
+10
1
1, 1, 1, 1, 2, 2, 1, 3, 4, 4, 1, 4, 6, 7, 7, 1, 5, 8, 10, 11, 11, 1, 6, 10, 13, 15, 16, 16, 1, 7, 12, 16, 19, 21, 22, 22, 1, 8, 14, 19, 23, 26, 28, 29, 29, 1, 9, 16, 22, 27, 31, 34, 36, 37, 37, 1, 10, 18, 25, 31, 36, 40, 43, 45, 46, 46, 1, 11, 20, 28, 35, 41
COMMENTS
Columns are linear recurrence sequences with signature (3,-3,1).
8*T(n,k) + A166147(k-1) are squares. Columns k are binomial transforms of [1, k, 1, 0, 0, 0, ...].
REFERENCES
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley, 1994.
FORMULA
G.f.: (3*x^2*y - 3*x*y + y - 2*x^2 + 2*x - 1)/((x - 1)^3*(y - 1)^2).
E.g.f.: (1/2)*(2*x*y + x^2 + 2)*exp(y + x).
T(n,k) = 3*T(n-1,k) - 3*T(n-2,k) + T(n-3,k), with T(0,k) = 1, T(1,k) = k + 1 and T(2,k) = 2*k + 2.
T(n,k) = T(n-1,k) + n + k - 1.
T(n,k) = T(n,k-1) + n, with T(n,0) = 1.
T(n,10) = A051942(n+10) + 1, n > 0. T(n,n+2) = T(n+1,n) = A005448(n+1). T(2*n,n+4) = A186353(4*n+1) (conjectured). T(2*n,n+5) = A184103(8*n+1) (conjectured). T(n+2,2*n) = A249013(5*n+4) (conjectured). T(3*n,n) = A025742(3*n-1), n > 1 (conjectured). T(n*(n+1),n) = A211905(n+1), n > 0 (conjectured). floor(T(n,n/2)) = A267682(n) (conjectured). floor(T(n,n/3)) = A025742(n-1), n > 0 (conjectured). floor(T(n,n/4)) = A263807(n-1), n > 0 (conjectured). ceiling(T(n,2^n)/n) = A134522(n), n > 0 (conjectured). ceiling(T(n,n/2+n)/n) = A051755(n+1) (conjectured). floor(T(n,n)/n) = A133223(n), n > 0 (conjectured). ceiling(T(n,n)/n) = A007494(n), n > 0. ceiling(T(n,n^2)/n) = A171769(n), n > 0. ceiling(T(2*n,n^2)/n) = A046092(n), n > 0. ceiling(T(2*n,2^n)/n) = A131520(n+2), n > 0.
EXAMPLE
The array T(n,k) begins
1 1 1 1 1 1 1 1 1 1 1 1 1 ... A0000121 2 3 4 5 6 7 8 9 10 11 12 13 ... A0000272 4 6 8 10 12 14 16 18 20 22 24 26 ... A0058434 7 10 13 16 19 22 25 28 31 34 37 40 ... A0167777 11 15 19 23 27 31 35 39 43 47 51 55 ... A00476711 16 21 26 31 36 41 46 51 56 61 66 71 ... A01686116 22 28 34 40 46 52 58 64 70 76 82 88 ... A01695722 29 36 43 50 57 64 71 78 85 92 99 106 ... A01699329 37 45 53 61 69 77 85 93 101 109 117 125 ... A00477037 46 55 64 73 82 91 100 109 118 127 136 145 ... A01717346 56 66 76 86 96 106 116 126 136 146 156 166 ... A01734156 67 78 89 100 111 122 133 144 155 166 177 188 ... A01740167 79 91 103 115 127 139 151 163 175 187 199 211 ... A01760579 92 105 118 131 144 157 170 183 196 209 222 235 ... A190991...
The inverse binomial transforms of the columns are
1 1 1 1 1 1 1 1 1 1 1 1 1 ...
0 1 2 3 4 5 6 7 8 9 10 11 12 ...
1 1 1 1 1 1 1 1 1 1 1 1 1 ...
0 0 0 0 0 0 0 0 0 0 0 0 0 ...
0 0 0 0 0 0 0 0 0 0 0 0 0 ...
0 0 0 0 0 0 0 0 0 0 0 0 0 ...
...
T(k,n-k) = A087401(n,k) + 1 as triangle 1
1 1
1 2 2
1 3 4 4
1 4 6 7 7
1 5 8 10 11 11
1 6 10 13 15 16 16
1 7 12 16 19 21 22 22
1 8 14 19 23 26 28 29 29
1 9 16 22 27 31 34 36 37 37
1 10 18 25 31 36 40 43 45 46 46
...
MAPLE
T := (n, k) -> binomial(n, 2) + k*n + 1;
for n from 0 to 20 do seq(T(n, k), k = 0 .. 20) od;
MATHEMATICA
Table[With[{n = m - k}, Binomial[n, 2] + k n + 1], {m, 0, 11}, {k, m, 0, -1}] // Flatten (* Michael De Vlieger, Apr 21 2018 *)
PROG
(Maxima)
T(n, k) := binomial(n, 2)+ k*n + 1$
for n:0 thru 20 do
print(makelist(T(n, k), k, 0, 20));
(PARI) T(n, k) = binomial(n, 2) + k*n + 1;
tabl(nn) = for (n=0, nn, for (k=0, nn, print1(T(n, k), ", ")); print); \\ Michel Marcus, May 17 2018
Square array T(n,k) read by antidiagonals upwards: T(n,0)=1; T(n,1) = n+1; T(n,2) = 2n+1, T(n,k>2) = T(n,k-1) - T(n,k-2) - T(n,k-3).
+10
0
1, 1, 1, 1, 2, 1, 1, 3, 3, -1, 1, 4, 5, 0, -3, 1, 5, 7, 1, -5, -3, 1, 6, 9, 2, -7, -8, 1, 1, 7, 11, 3, -9, -13, -3, 7, 1, 8, 13, 4, -11, -18, -7, 10, 9, 1, 9, 15, 5, -13, -23, -11, 13, 21, 1, 1, 10, 17, 6, -15, -28, -15, 16, 33, 14, -15
EXAMPLE
Array starts:
1 1 1 -1 -3 -3 1 7 9 1 -15 -25
1 2 3 0 -5 -8 -3 10 21 14 -17 -52
1 3 5 1 -7 -13 -7 13 33 27 -19 -79
1 4 7 2 -9 -18 -11 16 45 40 -21 -106
1 5 9 3 -11 -23 -15 19 57 53 -23 -133
1 6 11 4 -13 -28 -19 22 69 66 -25 -160
1 7 13 5 -15 -33 -23 25 81 79 -27 -187
MATHEMATICA
T[n_, k_]:= T[n, k]= If[k<3, k*n+1, T[n, k-1] - T[n, k-2] - T[n, k-3]];
Table[T[n-k, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, May 26 2020 *)
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