cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-20 of 20 results.

A155550 Triangle read by rows where T(m,n)=2*m*n + m + n - 6.

Original entry on oeis.org

-2, 1, 6, 4, 11, 18, 7, 16, 25, 34, 10, 21, 32, 43, 54, 13, 26, 39, 52, 65, 78, 16, 31, 46, 61, 76, 91, 106, 19, 36, 53, 70, 87, 104, 121, 138, 22, 41, 60, 79, 98, 117, 136, 155, 174, 25, 46, 67, 88, 109, 130, 151, 172, 193, 214, 28, 51, 74, 97, 120, 143, 166, 189, 212
Offset: 1

Views

Author

Vincenzo Librandi, Jan 24 2009

Keywords

Comments

Numbers n such that 2n+13 is not prime.
First column: A016777, second column: A016861, third column: A017029, fourth column: A017245. - Vincenzo Librandi, Nov 21 2012

Examples

			Triangle begins:
-2;
1,  6;
4,  11, 18;
7,  16, 25, 34;
10, 21, 32, 43, 54;
13, 26, 39, 52, 65,  78;
16, 31, 46, 61, 76,  91,  106;
19, 36, 53, 70, 87,  104, 121, 138;
22, 41, 60, 79, 98,  117, 136, 155, 174;
25, 46, 67, 88, 109, 130, 151, 172, 193, 214; etc.

Links

Crossrefs

Cf. A153082, A016777, A016861, A017029, A017245.

Programs

  • Magma
    [2*n*k + n + k - 6: k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 21 2012
  • Mathematica
    t[n_,k_]:= 2 n*k + n + k - 6; Table[t[n, k], {n, 15}, {k, n}]//Flatten (* Vincenzo Librandi, Nov 21 2012 *)

A301451 Numbers congruent to {1, 7} mod 9.

Original entry on oeis.org

1, 7, 10, 16, 19, 25, 28, 34, 37, 43, 46, 52, 55, 61, 64, 70, 73, 79, 82, 88, 91, 97, 100, 106, 109, 115, 118, 124, 127, 133, 136, 142, 145, 151, 154, 160, 163, 169, 172, 178, 181, 187, 190, 196, 199, 205, 208, 214, 217, 223, 226, 232, 235, 241, 244, 250, 253, 259, 262, 268
Offset: 1

Views

Author

Bruno Berselli, Mar 21 2018

Keywords

Comments

First bisection of A056991, second bisection of A242660.
The squares of the terms of A174396 are the squares of this sequence.

Links

Crossrefs

Cf. A274406: numbers congruent to {0, 8} mod 9.
Cf. A193910: numbers congruent to {2, 6} mod 9.
Subsequence of A016777, A026225, A029739, A033627, A047236, A047259, A055047, A055054, A056991, A153053, A187318, A242660.

Programs

  • GAP
    a := [1,7,10];; for n in [4..60] do a[n] := a[n-1] + a[n-2] - a[n-3]; od; a;
  • Magma
    &cat [[9*n+1, 9*n+7]: n in [0..40]];
  • Mathematica
    Table[2 (2 n - 1) + (2 n - 3 (1 - (-1)^n))/4, {n, 1, 60}]
{#+1,#+7}&/@(9*Range[0,30])//Flatten (* or *) LinearRecurrence[{1,1,-1},{1,7,10},60] (* Harvey P. Dale, Nov 08 2020 *)
  • PARI
    Vec(x*(1 + 6*x + 2*x^2) / ((1 - x)^2*(1 + x)) + O(x^60)) \\ Colin Barker, Mar 22 2018
  • Python
    [2*(2*n-1)+(2*n-3*(1-(-1)**n))/4 for n in range(1,70)]
  • Sage
    [n for n in (1..300) if n % 9 in (1,7)]

Formula

O.g.f.: x*(1 + 6*x + 2*x^2)/((1 + x)*(1 - x)^2).
E.g.f.: (3 + 8*exp(x) - 11*exp(2*x) + 18*x*exp(2*x))*exp(-x)/4.
a(n) = a(n-1) + a(n-2) - a(n-3).
a(n) = 2*(2*n - 1) + (2*n - 3*(1 - (-1)^n))/4. Therefore, for n even a(n) = (9*n - 4)/2, otherwise a(n) = (9*n - 7)/2.
a(2n+1) = A017173(n). a(2n) = A017245(n-1). - R. J. Mathar, Feb 28 2019

A017246 a(n) = (9*n + 7)^2.

Original entry on oeis.org

49, 256, 625, 1156, 1849, 2704, 3721, 4900, 6241, 7744, 9409, 11236, 13225, 15376, 17689, 20164, 22801, 25600, 28561, 31684, 34969, 38416, 42025, 45796, 49729, 53824, 58081, 62500, 67081, 71824
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A000290 (n^2), A017245 (9*n+7).

Programs

Formula

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=49, a(1)=256, a(2)=625. - Harvey P. Dale, Apr 24 2016
G.f.: -(49 + 109*x + 4*x^2)/(x-1)^3. - R. J. Mathar, Mar 20 2018

A177072 a(n) = (9*n+2)*(9*n+7).

Original entry on oeis.org

14, 176, 500, 986, 1634, 2444, 3416, 4550, 5846, 7304, 8924, 10706, 12650, 14756, 17024, 19454, 22046, 24800, 27716, 30794, 34034, 37436, 41000, 44726, 48614, 52664, 56876, 61250, 65786, 70484, 75344, 80366, 85550, 90896, 96404, 102074, 107906, 113900, 120056
Offset: 0

Views

Author

Vincenzo Librandi, May 31 2010

Keywords

Comments

Cf. comment of Reinhard Zumkeller in A177059: in general, (h*n+h-k)*(h*n+k) = h^2*A002061(n+1) + (h-k)*k - h^2; therefore a(n) = 81*A002061(n+1) - 67. - Bruno Berselli, Aug 24 2010

Links

Crossrefs

Cf. A002061, A017185, A017245, A177059.

Programs

  • Magma
    I:=[14, 176, 500]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]]; // Vincenzo Librandi, Apr 08 2013
  • Mathematica
    CoefficientList[Series[2(7 + 67 x + 7 x^2)/(1-x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Apr 08 2013 *)
Table[(9*n + 2)*(9*n + 7), {n, 0, 40}] (* Amiram Eldar, Feb 19 2023 *)
LinearRecurrence[{3,-3,1},{14,176,500},50] (* Harvey P. Dale, Jun 10 2023 *)
  • PARI
    a(n)=(9*n+2)*(9*n+7) \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = 162*n + a(n-1) with n > 0, a(0)=14.
From Vincenzo Librandi, Apr 08 2013: (Start)
G.f.: 2*(7+67*x+7*x^2)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
From Amiram Eldar, Feb 19 2023: (Start)
a(n) = A017185(n)*A017245(n).
Sum_{n>=0} 1/a(n) = cot(2*Pi/9)*Pi/45.
Product_{n>=0} (1 - 1/a(n)) = cosec(2*Pi/9)*cos(sqrt(29)*Pi/18).
Product_{n>=0} (1 + 1/a(n)) = cosec(2*Pi/9)*cos(sqrt(21)*Pi/18). (End)
E.g.f.: exp(x)*(14 + 81*x*(2 + x)). - Elmo R. Oliveira, Oct 18 2024

Extensions

Edited by N. J. A. Sloane, Jun 22 2010

A350522 a(n) = 18*n + 16.

Original entry on oeis.org

16, 34, 52, 70, 88, 106, 124, 142, 160, 178, 196, 214, 232, 250, 268, 286, 304, 322, 340, 358, 376, 394, 412, 430, 448, 466, 484, 502, 520, 538, 556, 574, 592, 610, 628, 646, 664, 682, 700, 718, 736, 754, 772, 790, 808, 826, 844, 862, 880, 898, 916, 934, 952, 970
Offset: 0

Views

Author

Omar E. Pol, Jan 03 2022

Keywords

Comments

Sixth column of A006370 (the Collatz or 3x+1 map) when it is interpreted as a rectangular array with six columns read by rows.

Links

Crossrefs

Bisection of A017245.
Cf. A006370, A008588, A008600, A016969, A017257, A062728, A239129.

Programs

  • GAP
    List([0..53], n-> 18*n+16)
  • Magma
    [18*n+16: n in [0..53]];
  • Maple
    seq(18*n+16, n=0..53);
  • Mathematica
    Table[18n+16, {n, 0, 53}]
  • Maxima
    makelist(18*n+16, n, 0, 53);
  • PARI
    a(n)=18*n+16
  • Python
    [18*n+16 for n in range(53)]

Formula

a(n) = A239129(n+1) - 1.
From Stefano Spezia, Jan 04 2022: (Start)
O.g.f.: 2*(8 + x)/(1 - x)^2.
E.g.f.: 2*exp(x)*(8 + 9*x).
a(n) = 2*a(n-1) - a(n-2) for n > 1. (End)
a(n) = 3*A008588(n+1) - 2. - Leo Tavares, Sep 14 2022
From Elmo R. Oliveira, Apr 12 2024: (Start)
a(n) = 2*A017257(n) = A006370(A016969(n)).
a(n) = 2*(A062728(n+1) - A062728(n)). (End)

A354939 Row 9 of A354940: Numbers k for which A345992(k) = 9, divided by 9.

Original entry on oeis.org

5, 7, 10, 14, 16, 19, 23, 25, 28, 32, 37, 41, 43, 46, 50, 59, 61, 64, 68, 73, 79, 82, 86, 91, 97, 100, 109, 113, 118, 122, 127, 131, 136, 145, 149, 151, 158, 163, 167, 169, 172, 181, 185, 194, 199, 212, 221, 223, 226, 235, 239, 241, 244, 253, 257, 262, 271, 277, 289, 293, 298, 302, 307, 311, 313, 316, 325, 331, 334
Offset: 1

Views

Author

Antti Karttunen, Jun 15 2022

Keywords

Comments

Apparently, all terms are either of the form 9k+1 (in A017173), or 9k+5 (in A017221), or 9k+7 (in A017245).

Crossrefs

Row 9 of A354940.
Cf. A017173, A017221, A017245, A344005, A345992.

Programs

  • Mathematica
    q[n_] := Module[{m = 1}, While[!Divisible[m*(m + 1), 9*n], m++]; GCD[9*n, m] == 9]; Select[Range[335], q] (* Amiram Eldar, Jun 17 2022 *)
  • PARI
    A354939(n) = A354940sq(9,n); \\ See the program in A354940.

A031383 a(n) = prime(9*n - 2).

Original entry on oeis.org

17, 53, 97, 139, 191, 239, 283, 349, 401, 457, 509, 577, 631, 683, 751, 821, 877, 941, 1009, 1061, 1117, 1193, 1259, 1307, 1409, 1459, 1523, 1583, 1637, 1721, 1787, 1871, 1933, 2003, 2081, 2137, 2221, 2287, 2351, 2411, 2477, 2579, 2659
Offset: 1

Views

Author

Jeff Burch

Keywords

Examples

			a(1) = 17 because 9 * 1 - 2 = 7 and the 7th prime is 17.
a(2) = 53 because 9 * 2 - 2 = 16 and the 16th prime is 53.

Links

Crossrefs

Cf. A017245.

Programs

A099048 Number of 5 X n 0-1 matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (01;0), (10;0) and (11;0).

Original entry on oeis.org

32, 50, 68, 86, 104, 122, 140, 158, 176, 194, 212, 230, 248, 266, 284, 302, 320, 338, 356, 374, 392, 410, 428, 446, 464, 482, 500, 518, 536, 554, 572, 590, 608, 626, 644, 662, 680, 698, 716, 734, 752, 770, 788, 806, 824, 842, 860, 878, 896, 914, 932, 950, 968, 986
Offset: 1

Views

Author

Sergey Kitaev, Nov 13 2004

Keywords

Comments

An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1
Also, temperatures in Fahrenheit that convert to Celsius as nonnegative multiples of 10. - J. Lowell, Jul 28 2007

Links

Crossrefs

Cf. A017245.

Programs

Formula

a(n) = 18*n + 14.
a(n) = 2*A017245(n).
From Elmo R. Oliveira, Jul 01 2025: (Start)
G.f.: 2*x*(16-7*x)/(1-x)^2.
E.g.f.: 2*(exp(x)*(9*x + 7) - 7).
a(n) = 2*a(n-1) - a(n-2) for n > 2. (End)

Extensions

More terms from Robert G. Wilson v, Nov 16 2004

A168411 a(n) = 7 + 9*floor((n-1)/2).

Original entry on oeis.org

7, 7, 16, 16, 25, 25, 34, 34, 43, 43, 52, 52, 61, 61, 70, 70, 79, 79, 88, 88, 97, 97, 106, 106, 115, 115, 124, 124, 133, 133, 142, 142, 151, 151, 160, 160, 169, 169, 178, 178, 187, 187, 196, 196, 205, 205, 214, 214, 223, 223, 232, 232, 241, 241, 250, 250, 259
Offset: 1

Views

Author

Vincenzo Librandi, Nov 25 2009

Keywords

Links

Crossrefs

Cf. A017245.

Programs

  • Magma
    [7+9*Floor((n-1)/2): n in [1..70]]; // Vincenzo Librandi, Sep 19 2013
  • Mathematica
    Table[7 + 9 Floor[(n - 1)/2], {n, 70}] (* or *) CoefficientList[Series[(7 + 2 x^2)/((1 + x) (x - 1)^2), {x, 0, 70}], x] (* Vincenzo Librandi, Sep 19 2013 *)

Formula

a(n) = 9*n - a(n-1) - 4, with n>1, a(1)=7.
G.f.: x*(7 + 2*x^2) / ( (1+x)*(x-1)^2 ). - R. J. Mathar, Jul 10 2011
a(n) = a(n-1) +a(n-2) -a(n-3). - Vincenzo Librandi, Sep 19 2013
E.g.f.: (1/4)*(-9 + 8*exp(x) + (18*x + 1)*exp(2*x))*exp(-x). - G. C. Greubel, Jul 22 2016

Extensions

New definition by Vincenzo Librandi, Sep 19 2013

A301628 Numbers not divisible by 2, 3 or 5 (A007775) with digital root 7.

Original entry on oeis.org

7, 43, 61, 79, 97, 133, 151, 169, 187, 223, 241, 259, 277, 313, 331, 349, 367, 403, 421, 439, 457, 493, 511, 529, 547, 583, 601, 619, 637, 673, 691, 709, 727, 763, 781, 799, 817, 853, 871, 889, 907, 943, 961, 979, 997, 1033, 1051, 1069, 1087, 1123
Offset: 1

Views

Author

Gary Croft, Mar 24 2018

Keywords

Comments

Numbers == {7, 43, 61, 79} mod 90 with additive sum sequence 7{+36+18+18+18} {repeat ...}. Includes all prime numbers > 5 with digital root 7.

Examples

			7+36=43; 43+18=61; 61+18=79; 79+18=97; 97+36=133.

Links

Crossrefs

Intersection of A007775 and A017245.

Programs

  • GAP
    Filtered(Filtered([1..1200],n->n mod 2 <> 0 and n mod 3 <> 0 and n mod 5 <> 0),i->i-9*Int((i-1)/9)=7); # Muniru A Asiru, Apr 22 2018
  • PARI
    Vec(x*(7 + 36*x + 18*x^2 + 18*x^3 + 11*x^4) / ((1 - x)^2*(1 + x)*(1 + x^2)) + O(x^40)) \\ Colin Barker, Sep 21 2019

Formula

Numbers == {7, 43, 61, 79} mod 90.
From Colin Barker, Sep 21 2019: (Start)
G.f.: x*(7 + 36*x + 18*x^2 + 18*x^3 + 11*x^4) / ((1 - x)^2*(1 + x)*(1 + x^2)).
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
(End)
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