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.

Showing 1-6 of 6 results.

A017437 a(n) = 11*n + 4.

Original entry on oeis.org

4, 15, 26, 37, 48, 59, 70, 81, 92, 103, 114, 125, 136, 147, 158, 169, 180, 191, 202, 213, 224, 235, 246, 257, 268, 279, 290, 301, 312, 323, 334, 345, 356, 367, 378, 389, 400, 411, 422, 433, 444, 455, 466, 477, 488, 499, 510, 521, 532, 543, 554, 565, 576, 587
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

These numbers do not occur in A000045 (Fibonacci numbers). - Arkadiusz Wesolowski, Jan 08 2012

Links

Crossrefs

Cf. A008593, A017425.
Powers of the form (11*n+4)^m: this sequence (m=1), A017438 (m=2), A017439 (m=3), A017440 (m=4), A017441 (m=5), A017442 (m=6), A017443 (m=7), A017444 (m=8), A017445 (m=9), A017446 (m=10), A017447 (m=11), A017448 (m=12).

Programs

Formula

a(0)=4, a(1)=15, a(n) = 2*a(n-1) - a(n-2). - Harvey P. Dale, May 19 2012
From G. C. Greubel, Sep 18 2019: (Start)
G.f.: (4 + 7*x)/(1-x)^2.
E.g.f.: (4 + 11*x)*exp(x). (End)

A226492 a(n) = n*(11*n-5)/2.

Original entry on oeis.org

0, 3, 17, 42, 78, 125, 183, 252, 332, 423, 525, 638, 762, 897, 1043, 1200, 1368, 1547, 1737, 1938, 2150, 2373, 2607, 2852, 3108, 3375, 3653, 3942, 4242, 4553, 4875, 5208, 5552, 5907, 6273, 6650, 7038, 7437, 7847, 8268, 8700, 9143, 9597, 10062, 10538, 11025, 11523
Offset: 0

Views

Author

Bruno Berselli, Jun 11 2013

Keywords

Comments

Sequences of numbers of the form n*(n*k - k + 6)/2:
. k from 0 to 10, respectively: A008585, A055998, A005563, A045943, A014105, A005475, A033428, A022264, A033991, A062741, A147874;
. k=11: a(n);
. k=12: A094159;
. k=13: 0, 3, 19, 48, 90, 145, 213, 294, 388, 495, 615, 748, 894, ...;
. k=14: 0, 3, 20, 51, 96, 155, 228, 315, 416, 531, 660, 803, 960, ...;
. k=15: A152773;
. k=16: A139272;
. k=17: 0, 3, 23, 60, 114, 185, 273, 378, 500, 639, 795, 968, ...;
. k=18: A152751;
. k=19: 0, 3, 25, 66, 126, 205, 303, 420, 556, 711, 885, 1078, ...;
. k=20: 0, 3, 26, 69, 132, 215, 318, 441, 584, 747, 930, 1133, ...;
. k=21: A152759;
. k=22: 0, 3, 28, 75, 144, 235, 348, 483, 640, 819, 1020, 1243, ...;
. k=23: 0, 3, 29, 78, 150, 245, 363, 504, 668, 855, 1065, 1298, ...;
. k=24: A152767;
. k=25: 0, 3, 31, 84, 162, 265, 393, 546, 724, 927, 1155, 1408, ...;
. k=26: 0, 3, 32, 87, 168, 275, 408, 567, 752, 963, 1200, 1463, ...;
. k=27: A153783;
. k=28: A195021;
. k=29: 0, 3, 35, 96, 186, 305, 453, 630, 836, 1071, 1335, 1628, ...;
. k=30: A153448;
. k=31: 0, 3, 37, 102, 198, 325, 483, 672, 892, 1143, 1425, 1738, ...;
. k=32: 0, 3, 38, 105, 204, 335, 498, 693, 920, 1179, 1470, 1793, ...;
. k=33: A153875.
Also:
a(n) - n = A180223(n);
a(n) + n = n*(11*n-3)/2 = 0, 4, 19, 45, 82, 130, 189, 259, ...;
a(n) - 2*n = A051865(n);
a(n) + 2*n = A022268(n);
a(n) - 3*n = A152740(n-1);
a(n) + 3*n = A022269(n);
a(n) - 4*n = n*(11*n-13)/2 = 0, -1, 9, 30, 62, 105, 159, 224, ...;
a(n) + 4*n = A254963(n);
a(n) - n*(n-1)/2 = A147874(n+1);
a(n) + n*(n-1)/2 = A094159(n) (case k=12);
a(n) - n*(n-1) = A062741(n) (see above, this is the case k=9);
a(n) + n*(n-1) = n*(13*n-7)/2 (case k=13);
a(n) - n*(n+1)/2 = A135706(n);
a(n) + n*(n+1)/2 = A033579(n);
a(n) - n*(n+1) = A051682(n);
a(n) + n*(n+1) = A186030(n);
a(n) - n^2 = A062708(n);
a(n) + n^2 = n*(13*n-5)/2 = 0, 4, 21, 51, 94, 150, 219, ..., etc.
Sum of reciprocals of a(n), for n > 0: 0.47118857003113149692081665034891...

Links

Crossrefs

Cf. sequences in Comments lines.
First differences are in A017425.
Cf. A033584, A180223.

Programs

  • Magma
    [n*(11*n-5)/2: n in [0..50]];
  • Magma
    I:=[0,3,17]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..46]]; // Vincenzo Librandi, Aug 18 2013
  • Mathematica
    Table[n (11 n - 5)/2, {n, 0, 50}]
CoefficientList[Series[x (3 + 8 x) / (1 - x)^3, {x, 0, 45}], x] (* Vincenzo Librandi, Aug 18 2013 *)
LinearRecurrence[{3,-3,1},{0,3,17},50] (* Harvey P. Dale, Jan 14 2019 *)
  • PARI
    a(n)=n*(11*n-5)/2 \\ Charles R Greathouse IV, Sep 24 2015

Formula

G.f.: x*(3+8*x)/(1-x)^3.
a(n) + a(-n) = A033584(n).
From Elmo R. Oliveira, Dec 27 2024: (Start)
E.g.f.: exp(x)*x*(6 + 11*x)/2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.
a(n) = n + A180223(n). (End)

A017413 a(n) = 11*n + 2.

Original entry on oeis.org

2, 13, 24, 35, 46, 57, 68, 79, 90, 101, 112, 123, 134, 145, 156, 167, 178, 189, 200, 211, 222, 233, 244, 255, 266, 277, 288, 299, 310, 321, 332, 343, 354, 365, 376, 387, 398, 409, 420, 431, 442, 453, 464, 475, 486, 497, 508, 519, 530, 541, 552, 563, 574, 585
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A008593, A017401, A017425.

Programs

Formula

From G. C. Greubel, Nov 11 2018: (Start)
a(n) = 2*a(n-1) - a(n-2).
G.f.: (2 + 9*x)/(1 - x)^2.
E.g.f.: (2 + 11*x)*exp(x). (End)

A017429 a(n) = (11*n+3)^5.

Original entry on oeis.org

243, 537824, 9765625, 60466176, 229345007, 656356768, 1564031349, 3276800000, 6240321451, 11040808032, 18424351793, 29316250624, 44840334375, 66338290976, 95388992557, 133827821568, 183765996899
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A000584, A017425.

Programs

  • Magma
    [(11*n+3)^5: n in [0..20]]; // Vincenzo Librandi, Apr 16 2017
  • Maple
    A017429:=n->(11*n+3)^5: seq(A017429(n), n=0..50); # Wesley Ivan Hurt, Apr 14 2017
  • Mathematica
    (* From Harvey P. Dale, Apr 30 2013: (Start) *)
(11*Range[0,20]+3)^5
LinearRecurrence[{6,-15,20,-15,6,-1},{243,537824,9765625,60466176,229345007,656356768},20] (* End *)
CoefficientList[Series[(243 + 536366*x + 6542326*x^2 + 9934926*x^3 + 2279491*x^4 + 32768*x^5)/(x - 1)^6, {x, 0, 50}], x] (* Indranil Ghosh, Apr 15 2017 *)
Table[(11 n + 3)^5, {n, 0, 30}] (* Vincenzo Librandi, Apr 16 2017 *)
  • PARI
    a(n) = (11*n + 3)^5; \\ Indranil Ghosh, Apr 15 2017

Formula

a(0)=243, a(1)=537824, a(2)=9765625, a(3)=60466176, a(4)=229345007, a(5)=656356768, a(n) = 6*a(n-1)-15*a(n-2)+20*a(n-3)-15*a(n-4)+6*a(n-5)-a(n-6). - Harvey P. Dale, Apr 30 2013
G.f.: (243 + 536366*x + 6542326*x^2 + 9934926*x^3 + 2279491*x^4 + 32768*x^5)/(x - 1)^6. - Indranil Ghosh, Apr 15 2017

A299647 Positive solutions to x^2 == -2 (mod 11).

Original entry on oeis.org

3, 8, 14, 19, 25, 30, 36, 41, 47, 52, 58, 63, 69, 74, 80, 85, 91, 96, 102, 107, 113, 118, 124, 129, 135, 140, 146, 151, 157, 162, 168, 173, 179, 184, 190, 195, 201, 206, 212, 217, 223, 228, 234, 239, 245, 250, 256, 261, 267, 272, 278, 283, 289, 294, 300, 305, 311, 316
Offset: 1

Views

Author

Bruno Berselli, Mar 06 2018

Keywords

Comments

Positive numbers congruent to {3, 8} mod 11.
Equivalently, interleaving of A017425 and A017485.

Links

Crossrefs

Subsequence of A106252, A279000.
Cf. A017425, A017485.
Cf. A017497: positive solutions to x == -2 (mod 11).
Cf. A017437: positive solutions to x^3 == -2 (mod 11).
Nonnegative solutions to x^2 == -2 (mod j): A005843 (j=2), A001651 (j=3), A047235 (j=6), A156638 (j=9), this sequence (j=11).

Programs

  • GAP
    List([1..60], n -> 5*n-2+(2*n-(-1)^n-3)/4);
  • Julia
    [(11(2n-1)-(-1)^n)>>2 for n in 1:60] # Peter Luschny, Mar 07 2018
  • Magma
    [5*n-2+(2*n-(-1)^n-3)/4: n in [1..60]];
  • Mathematica
    Table[5 n - 2 + (2 n - (-1)^n - 3)/4, {n, 1, 60}]
CoefficientList[ Series[(3 + 5x + 3x^2)/((x - 1)^2 (x + 1)), {x, 0, 57}], x] (* or *)
LinearRecurrence[{1, 1, -1}, {3, 8, 14}, 58] (* Robert G. Wilson v, Mar 08 2018 *)
  • Maxima
    makelist(5*n-2+(2*n-(-1)^n-3)/4, n, 1, 60);
  • PARI
    vector(60, n, nn; 5*n-2+(2*n-(-1)^n-3)/4)
  • Python
    [5*n-2+(2*n-(-1)**n-3)/4 for n in range(1, 60)]
  • Sage
    [5*n-2+(2*n-(-1)^n-3)/4 for n in (1..60)]

Formula

O.g.f.: x*(3 + 5*x + 3*x^2)/((1 + x)*(1 - x)^2).
E.g.f.: (-1 + 12*exp(x) - 11*exp(2*x) + 22*x*exp(2*x))*exp(-x)/4.
a(n) = -a(-n+1) = a(n-1) + a(n-2) - a(n-3).
a(n) = 5*n - 2 + (2*n - (-1)^n - 3)/4.
a(n) = 4*n - 1 + floor((n - 1)/2) + floor((3*n - 1)/3).
a(n+k) - a(n) = 11*k/2 + (1 - (-1)^k)*(-1)^n/4.
a(n+k) + a(n) = 11*(2*n + k - 1)/2 - (1 + (-1)^k)*(-1)^n/4.
E.g.f.: 3 + ((22*x - 11)*exp(x) - exp(-x))/4. - David Lovler, Aug 08 2022
Sum_{n>=1} (-1)^(n+1)/a(n) = tan(5*Pi/22)*Pi/11. - Amiram Eldar, Feb 27 2023
From Amiram Eldar, Nov 23 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = cosec(3*Pi/22)/2.
Product_{n>=1} (1 + (-1)^n/a(n)) = sec(5*Pi/22)*sin(2*Pi/11). (End)

A017427 a(n) = (11*n+3)^3.

Original entry on oeis.org

27, 2744, 15625, 46656, 103823, 195112, 328509, 512000, 753571, 1061208, 1442897, 1906624, 2460375, 3112136, 3869893, 4741632, 5735339, 6859000, 8120601, 9528128, 11089567, 12812904, 14706125
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A000578, A017425.

Programs

  • Mathematica
    (11Range[0,30]+3)^3 (* or *) LinearRecurrence[{4,-6,4,-1},{27,2744,15625,46656},30] (* Harvey P. Dale, Apr 01 2012 *)

Formula

a(n) = 4*a(n-1)-6*a(n-2)+ 4*a(n-3)- a(n-4). - Harvey P. Dale, Apr 01 2012
a(n) = A000578(A017425(n))). - Michel Marcus, Aug 25 2025
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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