A016814 -id:A016814 - cp's OEIS frontend

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

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

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

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.

[(9*n+1)^2: n in [0..40]]; // Vincenzo Librandi, Aug 25 2011

(9*Range[0,40] +1)^2 (* G. C. Greubel, Dec 28 2022 *)
LinearRecurrence[{3,-3,1},{1,100,361},50] (* Harvey P. Dale, Feb 25 2024 *)

a(n)=(9*n+1)^2 \\ Charles R Greathouse IV, Jun 17 2017

[(9*n+1)^2 for n in range(41)] # G. C. Greubel, Dec 28 2022

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

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

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.

[(11*n+1)^2: n in [0..50]]; // G. C. Greubel, Dec 24 2022

(11*Range[0,30]+1)^2 (* or *) LinearRecurrence[{3,-3,1},{1,144,529},30] (* Harvey P. Dale, May 05 2014 *)

a(n)=(11*n+1)^2 \\ Charles R Greathouse IV, Jun 17 2017

[(11*n+1)^2 for n in range(51)] # G. C. Greubel, Dec 24 2022

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

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

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.

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

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 *)

a(n)=(12*n+1)^2 \\ Charles R Greathouse IV, Jun 17 2017

[(12*n+1)^2 for n in range(51)] # G. C. Greubel, Dec 24 2022

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) = (kn)^2 + 2k*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

Paul Barry, Apr 03 2003

			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;

G. C. Greubel, Antidiagonals n = 0..50, flattened

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.

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

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 *)

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

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

Hans Isdahl, Jan 26 2008

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

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

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.

[(14*n+1)^2: n in [0..50]]; // Vincenzo Librandi, Sep 06 2011

(14*Range[0,50]+1)^2 (* G. C. Greubel, Dec 24 2022 *)

a(n)=(14*n+1)^2 \\ Charles R Greathouse IV, Jun 17 2017

[(14*n+1)^2 for n in range(51)] # G. C. Greubel, Dec 24 2022

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

A017222 a(n) = (9*n + 5)^2.

Original entry on oeis.org

25, 196, 529, 1024, 1681, 2500, 3481, 4624, 5929, 7396, 9025, 10816, 12769, 14884, 17161, 19600, 22201, 24964, 27889, 30976, 34225, 37636, 41209, 44944, 48841, 52900, 57121, 61504, 66049, 70756

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Sequences of the form (m*n+5)^2: A010864 (m=0), A000290 (m=1), A016754 (m=2), A016790 (m=3), A016814 (m=4), A016850 (m=5), A016970 (m=6), A017042 (m=7), A017126 (m=8), this sequence (m=9), A017330 (m=10), A017450 (m=11), A017582 (m=12).

Cf. A017221, A017246.

[(9*n+5)^2: n in [0..35]]; // Vincenzo Librandi, Jul 24 2011

(9Range[0,30]+5)^2 (* or *) LinearRecurrence[{3,-3,1},{25,196,529},30] (* Harvey P. Dale, May 22 2012 *)

a(n)=(9*n+5)^2 \\ Charles R Greathouse IV, Jun 17 2017

[(9*n+5)^2 for n in range(41)] # G. C. Greubel, Dec 29 2022

a(n) = A017221(n)^2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, May 22 2012
G.f.: (25 + 121*x + 16*x^2)/(1-x)^3. - R. J. Mathar, Mar 20 2018
From G. C. Greubel, Dec 29 2022: (Start)
a(2*n+1) = 4*A017246(n).
a(n) = a(n-1) + 9*(18*n + 1).
E.g.f.: (25 + 171*x + 81*x^2)*exp(x). (End)

A017090 a(n) = (8*n + 2)^2.

Original entry on oeis.org

4, 100, 324, 676, 1156, 1764, 2500, 3364, 4356, 5476, 6724, 8100, 9604, 11236, 12996, 14884, 16900, 19044, 21316, 23716, 26244, 28900, 31684, 34596, 37636, 40804, 44100, 47524, 51076, 54756, 58564, 62500, 66564, 70756, 75076, 79524, 84100, 88804, 93636, 98596

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A006752, A016814, A017089 (8n+2), A000290 (n^2).

[(8*n+2)^2: n in [0..35]]; // Vincenzo Librandi, Jul 12 2011

Table[(8*n + 2)^2, {n, 0, 40}] (* Amiram Eldar, Apr 24 2023 *)

a(n)=(8*n+2)^2 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: -4*(1 + 22*x + 9*x^2)/(x-1)^3. - R. J. Mathar, Jul 14 2016
From Amiram Eldar, Apr 24 2023: (Start)
a(n) = A017089(n)^2.
a(n) = 2^2*A016814(n).
Sum_{n>=0} 1/a(n) = Pi^2/64 + G/8, where G is Catalan's constant (A006752). (End)

A016820 a(n) = (4*n + 1)^8.

Original entry on oeis.org

1, 390625, 43046721, 815730721, 6975757441, 37822859361, 152587890625, 500246412961, 1406408618241, 3512479453921, 7984925229121, 16815125390625, 33232930569601, 62259690411361, 111429157112001, 191707312997281, 318644812890625, 513798374428641

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9, -36, 84, -126, 126, -84, 36, -9, 1).

Cf. A001016, A016813, A016814.

[(4*n+1)^8: n in [0..20]]; // Vincenzo Librandi, Aug 28 2017

A016820:=n->(4*n+1)^8: seq(A016820(n), n=0..30); # Wesley Ivan Hurt, Jan 22 2017

(4 * Range[0, 20] + 1)^8 (* or *) LinearRecurrence[{9, -36, 84, -126, 126, -84, 36, -9, 1}, {1, 390625, 43046721, 815730721, 6975757441, 37822859361, 152587890625, 500246412961, 1406408618241}, 20] (* Harvey P. Dale, Jul 11 2013 *)

a(0) = 1, a(1) = 390625, a(2) = 43046721, a(3) = 815730721, a(4) = 6975757441, a(5) = 37822859361, a(6) = 152587890625, a(7) = 500246412961, a(8) = 1406408618241, a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9). - Harvey P. Dale, Jul 11 2013

a(n) = A001016(A016813(n)). - Michel Marcus, Jan 23 2017

A017558 a(n) = (12*n + 3)^2.

Original entry on oeis.org

9, 225, 729, 1521, 2601, 3969, 5625, 7569, 9801, 12321, 15129, 18225, 21609, 25281, 29241, 33489, 38025, 42849, 47961, 53361, 59049, 65025, 71289, 77841, 84681, 91809, 99225, 106929, 114921, 123201

Offset: 0

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

(12*Range[0,30]+3)^2 (* or *) LinearRecurrence[{3,-3,1},{9,225,729},30] (* Harvey P. Dale, Aug 04 2015 *)

a(n)=(12*n+3)^2 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: 9*(1 + 22*x + 9*x^2)/(1-x)^3.
a(n) = 9*A016814(n).
a(n) = a(n-1) + 288*n - 72, n >= 1. - Vincenzo Librandi, Mar 20 2011

A227168 a(n) = gcd(2n, n(n+1)/2)^2.

Original entry on oeis.org

1, 1, 36, 4, 25, 9, 196, 16, 81, 25, 484, 36, 169, 49, 900, 64, 289, 81, 1444, 100, 441, 121, 2116, 144, 625, 169, 2916, 196, 841, 225, 3844, 256, 1089, 289, 4900, 324, 1369, 361, 6084, 400

Offset: 1

Paul Curtz, Jul 03 2013

a(n) is defined as A062828(n)^2 for n >= 1. If we extend the sequence to n=0 and negative n by use of the recurrence that relates a(n) to a(n+12), a(n+8) and a(n+4), we obtain a(0)=0, a(-1)=4 and a(-n) = A176743(n-2)^2 for n >= 2.
Define c(n) = a(n+2) - a(n-2) for c >= 0. Because a(n) is a shuffle of three interleaved 2nd-order polynomials, c(n) is a shuffle of three interleaved 1st-order polynomials: c(n) = 4* A062828(n)*(periodically repeated 1, 8, 1, 1).
The sequence a(n) is case p=0 of the family A062828(n)*A062828(n+p):
0, 1, 1, 36,  4, 25,  9, 196, ... = a(n).
0, 1, 6, 12, 10, 15, 42,  56, ... = A130658(n)*A000217(n) = A177002(n-1)*A064038(n+1).
0, 6, 2, 30,  6, 70, 12, 126, ... = 2*A198148(n)
0, 2, 5, 18, 28, 20, 27,  70, ... = A177002(n+2)*A160050(n+1) = A014695(n+2)*A000096(n).

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,3,0,0,0,-3,0,0,0,1).

Cf. A062828, A016814, A017138, A005565, A006752, A061037, A061038.

[GCD(2*n, n*(n+1)/2)^2: n in [1..50]]; // G. C. Greubel, Sep 20 2018

A227168 := proc(n)
    A062828(n)^2 ;
end proc: # R. J. Mathar, Jul 25 2013

a[n_] := GCD[2*n, n*(n + 1)/2]^2; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Jul 03 2013 *)

a(n)=if(n%2, n*if(n%4>2, 2, 1), n/2)^2 \\ Charles R Greathouse IV, Jul 07 2013

a(n) = A062828(n)^2.
a(4n) = (4*n+1)^2; a(2n+1) = (n+1)^2; a(4n+2) = 4*(4*n+3)^2.
a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12).
a(n) * (period 4: repeat 4, 1, 1, 4) = A061038(n).
A005565(n-3) = a(n+1) * A061037(n). - Corrected by R. J. Mathar, Jul 25 2013
a(n) = A130658(n-1)^2 * A181318(n). - Corrected by R. J. Mathar, Aug 01 2013
G.f.: -x*(1 + x + 36*x^2 + 4*x^3 + 22*x^4 + 6*x^5 + 88*x^6 + 4*x^7 + 9*x^8 + x^9 + 4*x^10) / ( (x-1)^3*(1+x)^3*(x^2+1)^3 ). - R. J. Mathar, Jul 20 2013
Sum_{n>=1} 1/a(n) = 47*Pi^2/192 + 3*G/8, where G is Catalan's constant (A006752). - Amiram Eldar, Aug 21 2022

cp's OEIS Frontend

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

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

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

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

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

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A017222 a(n) = (9*n + 5)^2.

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A017090 a(n) = (8*n + 2)^2.

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

A227168 a(n) = gcd(2n, n(n+1)/2)^2.