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 1-10 of 13 results. Next

A016814 a(n) = (4*n + 1)^2.

Original entry on oeis.org

1, 25, 81, 169, 289, 441, 625, 841, 1089, 1369, 1681, 2025, 2401, 2809, 3249, 3721, 4225, 4761, 5329, 5929, 6561, 7225, 7921, 8649, 9409, 10201, 11025, 11881, 12769, 13689, 14641, 15625, 16641, 17689, 18769, 19881, 21025, 22201, 23409, 24649, 25921, 27225, 28561, 29929
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

A bisection of A016754. Sequence arises from reading the line from 1, in the direction 1, 25, ..., in the square spiral whose vertices are the squares A000290. - Omar E. Pol, May 24 2008

Links

Crossrefs

Sequences of the form (m*n+1)^2: A000012 (m=0), A000290 (m=1), A016754 (m=2), A016778 (m-3), this sequence (m=4), A016862 (m=5), A016922 (m=6), A016994 (m=7), A017078 (m=8), A017174 (m=9), A017282 (m=10), A017402 (m=11), A017534 (m=12), A134934 (m=14).
Cf. A001539, A006752, A016742, A016802, A016813.
Cf. A016826, A016838, A017126, A068467, A087197.
Cf. A272399, A014105.

Programs

Formula

a(n) = a(n-1) + 32*n - 8, n > 0. - Vincenzo Librandi, Dec 15 2010
From George F. Johnson, Sep 28 2012: (Start)
G.f.: (1 + 22*x + 9*x^2)/(1 - x)^3.
a(n+1) = a(n) + 16 + 8*sqrt(a(n)).
a(n+1) = 2*a(n) - a(n-1) + 32 = 3*a(n) - 3*a(n-1) + a(n-2).
a(n-1)*a(n+1) = (a(n) - 16)^2 ; a(n+1) - a(n-1) = 16*sqrt(a(n)).
a(n) = A016754(2*n) = (A016813(n))^2. (End)
Sum_{n>=0} 1/a(n) = G/2 + Pi^2/16, where G is the Catalan constant (A006752). - Amiram Eldar, Jun 28 2020
Product_{n>=1} (1 - 1/a(n)) = 2*Gamma(5/4)^2/sqrt(Pi) = 2 * A068467^2 * A087197. - Amiram Eldar, Feb 01 2021
From G. C. Greubel, Dec 28 2022: (Start)
a(2*n) = A017078(n).
a(2*n+1) = A017126(n).
E.g.f.: (1 + 24*x + 16*x^2)*exp(x). (End)
a(n) = A272399(n+1) - A014105(n). - Leo Tavares, Dec 24 2023

A017282 a(n) = (10*n + 1)^2.

Original entry on oeis.org

1, 121, 441, 961, 1681, 2601, 3721, 5041, 6561, 8281, 10201, 12321, 14641, 17161, 19881, 22801, 25921, 29241, 32761, 36481, 40401, 44521, 48841, 53361, 58081, 63001, 68121, 73441, 78961, 84681, 90601, 96721, 103041, 109561, 116281, 123201
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Sequences of the form (m*n+1)^2: A000012 (m=0), A000290 (m=1), A016754 (m=2), A016778 (m=3), A016814 (m=4), A016862 (m=5), A016922 (m=6), A016994 (m=7), A017078 (m=8), A017174 (m=9), this sequence (m=10), A017402 (m=11), A017534 (m=12), A134934 (m=14).
Cf. A017281.

Programs

  • Magma
    [(10*n+1)^2: n in [0..35]]; // Vincenzo Librandi, Jul 30 2011
  • Mathematica
    (* Programs from Michael De Vlieger, Mar 30 2017 *)
Table[(10 n+1)^2, {n, 0, 35}]
FoldList[#1 + 200 #2 - 80 &, 1, Range@ 35]
CoefficientList[Series[(1+118x+81x^2)/(1-x)^3, {x,0,35}], x] (* End *)
LinearRecurrence[{3,-3,1},{1,121,441},40] (* Harvey P. Dale, Sep 21 2017 *)
  • PARI
    for(n=0, 35, print1((10*n+1)^2", ")); \\ Bruno Berselli, Jul 30 2011
  • SageMath
    [(10*n+1)^2 for n in range(51)] # G. C. Greubel, Dec 24 2022

Formula

G.f.: (1+118*x+81*x^2)/(1-x)^3. - Bruno Berselli, Jul 30 2011
a(n) = a(n-1) + 40*(5*n-2), n > 0; a(0)=1. - Miquel Cerda, Oct 30 2016
a(n) = A017281(n)^2. - Michel Marcus, Oct 30 2016
E.g.f.: (1 +120*x +100*x^2)*exp(x). - G. C. Greubel, Dec 24 2022

Extensions

More terms from Bruno Berselli, Jul 30 2011

A016994 a(n) = (7*n + 1)^2.

Original entry on oeis.org

1, 64, 225, 484, 841, 1296, 1849, 2500, 3249, 4096, 5041, 6084, 7225, 8464, 9801, 11236, 12769, 14400, 16129, 17956, 19881, 21904, 24025, 26244, 28561, 30976, 33489, 36100, 38809, 41616, 44521, 47524
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Sequences of the form (m*n+1)^2: A000012 (m=0), A000290 (m=1), A016754 (m=2), A016778 (m-3), A016814 (m=4), A016862 (m=5), A016922 (m=6), this sequence (m=7), A017078 (m=8), A017174 (m=9), A017282 (m=10), A017402 (m=11), A017534 (m=12), A134934 (m=14).

Programs

Formula

G.f.: (1 + 61*x + 36*x^2)/(1-x)^3. - Vincenzo Librandi, Jan 27 2013
From G. C. Greubel, Dec 28 2022: (Start)
a(2*n) = A134934(n).
a(2*n+1) = 4*A017030(n).
E.g.f.: (1 + 63*x + 49*x^2)*exp(x). (End)
Sum_{n>=0} 1/a(n) = Psi'(1/7)/49 = 1.027703498712483534.. - R. J. Mathar, May 07 2024

A017078 a(n) = (8*n + 1)^2.

Original entry on oeis.org

1, 81, 289, 625, 1089, 1681, 2401, 3249, 4225, 5329, 6561, 7921, 9409, 11025, 12769, 14641, 16641, 18769, 21025, 23409, 25921, 28561, 31329, 34225, 37249, 40401, 43681, 47089, 50625, 54289, 58081
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Sequences of the form (m*n+1)^2: A000012 (m=0), A000290 (m=1), A016754 (m=2), A016778 (m-3), A016814 (m=4), A016862 (m=5), A016922 (m=6), A016994 (m=7), this sequence (m=8), A017174 (m=9), A017282 (m=10), A017402 (m=11), A017534 (m=12), A134934 (m=14).

Programs

Formula

G.f.: (1 + 78*x + 49*x^2)/(1-x)^3. - R. J. Mathar, Mar 21 2016
From G. C. Greubel, Dec 28 2022: (Start)
a(2*n) = A016754(8*n).
E.g.f.: (1 + 80*x + 64*x^2)*exp(x). (End)
Sum_{n>=0} 1/a(n) = psi'(1/8)/64 = 1.02168958507793.. - R. J. Mathar, May 07 2024

A017174 a(n) = (9*n + 1)^2.

Original entry on oeis.org

1, 100, 361, 784, 1369, 2116, 3025, 4096, 5329, 6724, 8281, 10000, 11881, 13924, 16129, 18496, 21025, 23716, 26569, 29584, 32761, 36100, 39601, 43264, 47089, 51076, 55225, 59536, 64009, 68644
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Sequences of the form (m*n+1)^2: A000012 (m=0), A000290 (m=1), A016754 (m=2), A016778 (m-3), A016814 (m=4), A016862 (m=5), A016922 (m=6), A016994 (m=7), A017078 (m=8), this sequence (m=9), A017282 (m=10), A017402 (m=11), A017534 (m=12), A134934 (m=14).
Cf. A017222.

Programs

Formula

G.f.: x*(1 + 97*x + 64*x^2)/(1-x)^3. - Bruno Berselli, Aug 25 2011
From G. C. Greubel, Dec 28 2022: (Start)
a(2*n) = A016754(9*n).
a(2*n+1) = 4*A017222(n).
E.g.f.: (1 + 99*x + 81*x^2)*exp(x). (End)

A017402 a(n) = (11*n+1)^2.

Original entry on oeis.org

1, 144, 529, 1156, 2025, 3136, 4489, 6084, 7921, 10000, 12321, 14884, 17689, 20736, 24025, 27556, 31329, 35344, 39601, 44100, 48841, 53824, 59049, 64516, 70225, 76176, 82369, 88804, 95481, 102400
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Sequences of the form (m*n+1)^2: A000012 (m=0), A000290 (m=1), A016754 (m=2), A016778 (m=3), A016814 (m=4), A016862 (m=5), A016922 (m=6), A016994 (m=7), A017078 (m=8), A017174 (m=9), A017282 (m=10), this sequence (m=11), A017534 (m=12), A134934 (m=14).
Cf. A017534, A082043.

Programs

Formula

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, May 05 2014
From G. C. Greubel, Dec 24 2022: (Start)
G.f.: (1 + 141*x + 100*x^2)/(1-x)^3.
E.g.f.: (1 + 143*x + 121*x^2)*exp(x). (End)

A017534 a(n) = (12*n + 1)^2.

Original entry on oeis.org

1, 169, 625, 1369, 2401, 3721, 5329, 7225, 9409, 11881, 14641, 17689, 21025, 24649, 28561, 32761, 37249, 42025, 47089, 52441, 58081, 64009, 70225, 76729, 83521, 90601, 97969, 105625, 113569, 121801
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Sequences of the form (m*n+1)^2: A000012 (m=0), A000290 (m=1), A016754 (m=2), A016778 (m=3), A016814 (m=4), A016862 (m=5), A016922 (m=6), A016994 (m=7), A017078 (m=8), A017174 (m=9), A017282 (m=10), A017402 (m=11), this sequence (m=12), A134934 (m=14).
Cf. A082043.

Programs

  • Magma
    I:=[1, 169, 625]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]]; // Vincenzo Librandi, Jul 07 2012
  • Mathematica
    CoefficientList[Series[(1+166*x+121*x^2)/(1-x)^3,{x,0,50}],x] (* Vincenzo Librandi, Jul 07 2012 *)
LinearRecurrence[{3,-3,1},{1,169,625},30] (* Harvey P. Dale, Feb 27 2023 *)
  • PARI
    a(n)=(12*n+1)^2 \\ Charles R Greathouse IV, Jun 17 2017
  • SageMath
    [(12*n+1)^2 for n in range(51)] # G. C. Greubel, Dec 24 2022

Formula

G.f.: (1 + 166*x + 121*x^2 )/(1-x)^3. - R. J. Mathar, Mar 10 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Jul 07 2012
E.g.f.: (1 + 168*x + 144*x^2)*exp(x). - G. C. Greubel, Dec 24 2022

A082043 Square array, A(n, k) = (k*n)^2 + 2*k*n + 1, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 9, 9, 1, 1, 16, 25, 16, 1, 1, 25, 49, 49, 25, 1, 1, 36, 81, 100, 81, 36, 1, 1, 49, 121, 169, 169, 121, 49, 1, 1, 64, 169, 256, 289, 256, 169, 64, 1, 1, 81, 225, 361, 441, 441, 361, 225, 81, 1, 1, 100, 289, 484, 625, 676, 625, 484, 289, 100, 1
Offset: 0

Views

Author

Paul Barry, Apr 03 2003

Keywords

Examples

			Array, A(n, k), begins as:
  1,   1,   1,    1,    1,    1,    1,    1,     1, ... A000012;
  1,   4,   9,   16,   25,   36,   49,   64,    81, ... A000290;
  1,   9,  25,   49,   81,  121,  169,  225,   289, ... A016754;
  1,  16,  49,  100,  169,  256,  361,  484,   625, ... A016778;
  1,  25,  81,  169,  289,  441,  625,  841,  1089, ... A016814;
  1,  36, 121,  256,  441,  676,  961, 1296,  1681, ... A016862;
  1,  49, 169,  361,  625,  961, 1369, 1849,  2401, ... A016922;
  1,  64, 225,  484,  841, 1296, 1849, 2500,  3249, ... A016994;
  1,  81, 289,  625, 1089, 1681, 2401, 3249,  4225, ... A017078;
  1, 100, 361,  784, 1369, 2116, 3025, 4096,  5329, ... A017174;
  1, 121, 441,  961, 1681, 2601, 3721, 5041,  6561, ... A017282;
  1, 144, 529, 1156, 2025, 3136, 4489, 6084,  7921, ... A017402;
  1, 169, 625, 1369, 2401, 3721, 5329, 7225,  9409, ... A017534;
  1, 196, 729, 1600, 2809, 4356, 6241, 8464, 11025, ... ;
Antidiagonals, T(n, k), begin as:
  1;
  1,   1;
  1,   4,   1;
  1,   9,   9,   1;
  1,  16,  25,  16,   1;
  1,  25,  49,  49,  25,   1;
  1,  36,  81, 100,  81,  36,   1;
  1,  49, 121, 169, 169, 121,  49,   1;
  1,  64, 169, 256, 289, 256, 169,  64,   1;
  1,  81, 225, 361, 441, 441, 361, 225,  81,   1;
  1, 100, 289, 484, 625, 676, 625, 484, 289, 100,  1;

Links

Crossrefs

Rows include A000290, A016754, A016778, A016814, A016862, A016922, A016994, A017078, A017174, A017282, A017402, A017534.
Diagonals include A000583, A058031, A062938, A082044 (main diagonal).
Diagonal sums (row sums if viewed as number triangle) are A082045.
Cf. A082039, A082045, A082046, A082105.

Programs

  • Magma
    A082043:= func< n,k | (k*(n-k))^2 +2*k*(n-k) +1 >;
[A082043(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Dec 24 2022
  • Mathematica
    T[n_, k_]:= (k*(n-k))^2 +2*k*(n-k) +1;
Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 24 2022 *)
  • SageMath
    def A082043(n,k): return (k*(n-k))^2 +2*k*(n-k) +1
flatten([[A082043(n,k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Dec 24 2022

Formula

A(n, k) = (k*n)^2 + 2*k*n + 1 (square array).
T(n, k) = (k*(n-k))^2 + 2*k*(n-k) + 1 (number triangle).
A(k, n) = A(n, k).
T(n, n-k) = T(n, k).
A(n, n) = T(2*n, n) = A082044(n).
A(n, n-1) = T(2*n+1, n-1) = A058031(n), n >= 1.
A(n, n-2) = T(2*(n-1), n) = A000583(n-1), n >= 2.
A(n, n-3) = T(2*n-3, n) = A062938(n-3), n >= 3.
Sum_{k=0..n} T(n, k) = A082045(n) (diagonal sums).
Sum_{k=0..n} (-1)^k * T(n, k) = (1/4)*(1+(-1)^n)*(2 - 3*n). - G. C. Greubel, Dec 24 2022

A134934 a(n) = (14*n+1)^2.

Original entry on oeis.org

1, 225, 841, 1849, 3249, 5041, 7225, 9801, 12769, 16129, 19881, 24025, 28561, 33489, 38809, 44521, 50625, 57121, 64009, 71289, 78961, 87025, 95481, 104329, 113569, 123201, 133225, 143641, 154449, 165649, 177241, 189225, 201601, 214369, 227529, 241081
Offset: 0

Views

Author

Hans Isdahl, Jan 26 2008

Keywords

Comments

Number of rats in population after n years, starting with one rat at year 0 (see A016754 for more details).

Links

Crossrefs

Sequences of the form (m*n+1)^2: A000012 (m=0), A000290 (m=1), A016754 (m=2), A016778 (m=3), A016814 (m=4), A016862 (m=5), A016922 (m=6), A016994 (m=7), A017078 (m=8), A017174 (m=9), A017282 (m=10), A017402 (m=11), A017534 (m=12), this sequence (m=14).
Cf. A016754.

Programs

Formula

O.g.f.: (1+222*x+169*x^2)/(1-x)^3 = 169/(1-x) - 560/(1-x)^2 + 392/(1-x)^3. - R. J. Mathar, Jan 31 2008
a(n) = A016754(7*n).
E.g.f.: (1 + 224*x + 196*x^2)*exp(x). - G. C. Greubel, Dec 24 2022

A016863 a(n) = (5*n + 1)^3.

Original entry on oeis.org

1, 216, 1331, 4096, 9261, 17576, 29791, 46656, 68921, 97336, 132651, 175616, 226981, 287496, 357911, 438976, 531441, 636056, 753571, 884736, 1030301, 1191016, 1367631, 1560896, 1771561, 2000376, 2248091, 2515456, 2803221, 3112136, 3442951, 3796416, 4173281, 4574296
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Examples

			a(0) = (5*0 + 1)^3 = 1.

Links

Crossrefs

Cf. A000578, A016861, A016862, A016864.

Programs

  • Mathematica
    Table[(5*n + 1)^3, {n, 0, 25}] (* Amiram Eldar, Oct 02 2020 *)

Formula

Sum_{n>=0} 1/a(n) = -polygamma(2, 1/5)/250. - Amiram Eldar, Oct 02 2020
From Wesley Ivan Hurt, Oct 02 2020: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
G.f.: (1+212*x+473*x^2+64*x^3)/(-1+x)^4. (End)
From Stefano Spezia, Nov 01 2024: (Start)
a(n) = A000578(A016861(n)).
E.g.f.: exp(x)*(1 + 215*x + 450*x^2 + 125*x^3). (End)
Showing 1-10 of 13 results. Next

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

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