A134934 -id:A134934 - cp's OEIS frontend

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

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

N. J. A. Sloane

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

G. C. Greubel, Table of n, a(n) for n = 0..10000 (terms 0..200 from Ivan Panchenko).
Leo Tavares, Illustration: Square trapeziums
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), 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.

[(4*n+1)^2: n in [0..40]]; // G. C. Greubel, Dec 28 2022

A016814:=n->(4*n+1)^2; seq(A016814(k), k=0..100); # Wesley Ivan Hurt, Nov 02 2013

(4*Range[0,40] +1)^2 (* or *) LinearRecurrence[{3,-3,1}, {1,25,81}, 40] (* Harvey P. Dale, Nov 20 2012 *)
Accumulate[32Range[0, 47] - 8] + 9 (* Alonso del Arte, Aug 19 2017 *)

a(n)=(4*n+1)^2 \\ Charles R Greathouse IV, Oct 07 2015

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

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

A016862 a(n) = (5*n + 1)^2.

Original entry on oeis.org

1, 36, 121, 256, 441, 676, 961, 1296, 1681, 2116, 2601, 3136, 3721, 4356, 5041, 5776, 6561, 7396, 8281, 9216, 10201, 11236, 12321, 13456, 14641, 15876, 17161, 18496, 19881, 21316, 22801, 24336, 25921

Offset: 0

N. J. A. Sloane

Ivan Panchenko, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Polygamma Function.
Wikipedia, Polygamma Function.
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), this sequence (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).

[(5*n+1)^2: n in [0..40]]; // G. C. Greubel, Dec 28 2022

(5*Range[0,40]+1)^2 (* or *) LinearRecurrence[{3,-3,1},{1,36,121},40] (* Harvey P. Dale, Jul 11 2012 *)

a(n)=(5*n+1)^2 \\ Charles R Greathouse IV, Jun 17 2017

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

a(n) = 3*a(n-1) -3*a(n-2) +a(n-3). - Harvey P. Dale, Jul 11 2012
Sum_{n>=0} 1/a(n) = polygamma(1, 1/5)/25 = 1.050695088216... - Amiram Eldar, Oct 02 2020
G.f.: (1 +33*x +16*x^2)/(1-x)^3. - Wesley Ivan Hurt, Oct 02 2020
From G. C. Greubel, Dec 28 2022: (Start)
a(2*n) = A017282(n).
a(2*n+1) = 4*A016886(n).
E.g.f.: (1 + 35*x + 25*x^2)*exp(x). (End)

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

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), A017174 (m=9), this sequence (m=10), A017402 (m=11), A017534 (m=12), A134934 (m=14).

Cf. A017281.

[(10*n+1)^2: n in [0..35]]; // Vincenzo Librandi, Jul 30 2011

(* 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 *)

for(n=0, 35, print1((10*n+1)^2", ")); \\ Bruno Berselli, Jul 30 2011

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

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

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

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), this sequence (m=7), A017078 (m=8), A017174 (m=9), A017282 (m=10), A017402 (m=11), A017534 (m=12), A134934 (m=14).

[(7*n+1)^2: n in [0..40]]; // Vincenzo Librandi, Jul 13 2011

(7Range[0,50]+1)^2  (* Harvey P. Dale, Mar 05 2011 *)
CoefficientList[Series[(1+61*x+36*x^2)/(1-x)^3, {x,0,40}], x] (* Vincenzo Librandi, Jan 27 2013 *)

a(n)=(7*n+1)^2 \\ Charles R Greathouse IV, Jun 17 2017

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

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

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), this sequence (m=8), A017174 (m=9), A017282 (m=10), A017402 (m=11), A017534 (m=12), A134934 (m=14).

[(8*n+1)^2: n in [0..40]]; // Vincenzo Librandi, Jul 11 2011

(8*Range[0,40] +1)^2 (* G. C. Greubel, Dec 28 2022 *)

a(n)=(8*n+1)^2 \\ Charles R Greathouse IV, Jun 17 2017

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

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

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

A347533 Array A(n,k) where A(n,0) = n and A(n,k) = (k*n + 1)^2 - A(n,k-1), n > 0, read by ascending antidiagonals.

Original entry on oeis.org

1, 2, 3, 3, 7, 6, 4, 13, 18, 10, 5, 21, 36, 31, 15, 6, 31, 60, 64, 50, 21, 7, 43, 90, 109, 105, 71, 28, 8, 57, 126, 166, 180, 151, 98, 36, 9, 73, 168, 235, 275, 261, 210, 127, 45, 10, 91, 216, 316, 390, 401, 364, 274, 162, 55, 11, 111, 270, 409, 525, 571, 560, 477, 351, 199, 66

Offset: 1

Lamine Ngom, Sep 05 2021

A(n,k) is also the distance from A(n, k-1) to the earliest square greater than 3*A(n,k-1) - A(n,k-2).

In column k, every term is the arithmetic mean of its neighbors minus A000217(k).

			Array, A(n, k), begins:
  1  3   6  10  15   21   28   36   45 ... A000217;
  2  7  18  31  50   71   98  127  162 ... A195605;
  3 13  36  64 105  151  210  274  351 ...
  4 21  60 109 180  261  364  477  612 ...
  5 31  90 166 275  401  560  736  945 ...
  6 43 126 235 390  571  798 1051 1350 ...
  7 57 168 316 525  771 1078 1422 1827 ...
  8 73 216 409 680 1001 1400 1849 2376 ...
  9 91 270 514 855 1261 1764 2332 2997 ...
Antidiagonals, T(n, k), begin as:
   1;
   2,  3;
   3,  7,   6;
   4, 13,  18,  10;
   5, 21,  36,  31,  15;
   6, 31,  60,  64,  50,  21;
   7, 43,  90, 109, 105,  71,  28;
   8, 57, 126, 166, 180, 151,  98,  36;
   9, 73, 168, 235, 275, 261, 210, 127,  45;
  10, 91, 216, 316, 390, 401, 364, 274, 162,  55;

G. C. Greubel, Antidiagonals n = 1..50, flattened

Family of sequences (k*n + 1)^2: A016754 (k=2), A016778 (k=3), A016814 (k=4), A016862 (k=5), A016922 (k=6), A016994 (k=7), A017078 (k=8), A017174 (k=9), A017282 (k=10), A017402 (k=11), A017534 (k=12), A134934 (k=14).

Cf. A000027, A000217, A002061, A028896, A085473, A195605.

A347533:= func< n,k | (1/2)*((k*(n-k)+1)*((k+1)*(n-k)+1) +(-1)^k*(n-k- 1)) >;
[A347533(n,k): k in [0..n-1], n in [1..13]]; // G. C. Greubel, Dec 25 2022

A[n_, 0]:= n; A[n_, k_]:= (k*n+1)^2 -A[n,k-1]; Table[Function[n, A[n, k]][m-k+1], {m,0,10}, {k,0,m}]//Flatten (* Michael De Vlieger, Oct 27 2021 *)

def A347533(n,k): return (1/2)*((k*(n-k)+1)*((k+1)*(n-k)+1) +(-1)^k*(n-k- 1))
flatten([[A347533(n,k) for k in range(n)] for n in range(1,14)]) # G. C. Greubel, Dec 25 2022

A(n,k) = A000217(k)*n^2 + k*n + 1, for k odd.
A(n,k) = A000217(k)*n^2 + (k+1)*n = (k+1)*x*(k*n/2 + 1), for k even.
A(n,k) = (A(n,k-1) + A(n,k+1) + k*(k+1))/2, for any k.
A(n, 0) = A000027(n).
A(n, 1) = A002061(n+1).
A(n, 2) = A028896(n).
A(n, 3) = A085473(n).
From G. C. Greubel, Dec 25 2022: (Start)
A(n, k) = (1/2)*( (k*n+1)*(k*n+n+1) + (-1)^k*(n-1) ).
T(n, k) = (1/2)*( (k*(n-k)+1)*((k+1)*(n-k)+1) + (-1)^k*(n-k-1) ).
Sum_{k=0..n-1} T(n, k) = (1/120)*(2*n^5 + 5*n^4 + 20*n^3 + 25*n^2 + 98*n - 15*(1-(-1)^n)). (End)

cp's OEIS Frontend

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

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A016862 a(n) = (5*n + 1)^2.

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

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

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

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

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

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