A220082 -id:A220082 - cp's OEIS frontend

A132356 a(2k) = k(10k+2), a(2k+1) = 10k^2 + 18k + 8, with k >= 0.

0, 8, 12, 36, 44, 84, 96, 152, 168, 240, 260, 348, 372, 476, 504, 624, 656, 792, 828, 980, 1020, 1188, 1232, 1416, 1464, 1664, 1716, 1932, 1988, 2220, 2280, 2528, 2592, 2856, 2924, 3204, 3276, 3572, 3648, 3960, 4040, 4368, 4452, 4796, 4884, 5244, 5336, 5712

Offset: 0

Mohamed Bouhamida, Nov 08 2007

X values of solutions to the equation 10*X^3 + X^2 = Y^2.
Polygonal number connection: 2*H_n + 6S_n, where H_n is the n-th hexagonal number and S_n is the n-th square number. This is the base formula that is expanded upon to achieve the full series. See contributing formula below. - William A. Tedeschi, Sep 12 2010
Equivalently, numbers of the form 2*h*(5*h+1), where h = 0, -1, 1, -2, 2, -3, 3, -4, 4, ... . - Bruno Berselli, Feb 02 2017

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A054000, A056220, A087475, A028347, A046092, A132209.
Cf. numbers m such that k*m+1 is a square: A005563 (k=1), A046092 (k=2), A001082 (k=3), A002378 (k=4), A036666 (k=5), A062717 (k=6), A132354 (k=7), A000217 (k=8), A132355 (k=9), A219257 (k=11), A152749 (k=12), A219389 (k=13), A219390 (k=14), A204221 (k=15), A074378 (k=16), A219394 (k=17), A219395 (k=18), A219396 (k=19), A219190 (k=20), A219391 (k=21), A219392 (k=22), A219393 (k=23), A001318 (k=24), A219259 (k=25), A217441 (k=26), A219258 (k=27), A219191 (k=28).
Cf. A220082 (numbers k such that 10*k-1 is a square).

CoefficientList[Series[4*x*(2*x^2 + x + 2)/((1 - x)^3*(1 + x)^2), {x, 0, 50}], x] (* G. C. Greubel, Jun 12 2017 *)
LinearRecurrence[{1,2,-2,-1,1},{0,8,12,36,44},50] (* Harvey P. Dale, Dec 15 2023 *)

my(x='x+O('x^50)); concat([0], Vec(4*x*(2*x^2+x+2)/((1-x)^3*(1+x)^2))) \\ G. C. Greubel, Jun 12 2017

a(n) = n^2 + n + 6*((n+1)\2)^2 \\ Charles R Greathouse IV, Sep 11 2022

G.f.: 4*x*(2*x^2+x+2)/((1-x)^3*(1+x)^2). - R. J. Mathar, Apr 07 2008
a(n) = 10*x^2 - 2*x, where x = floor(n/2)*(-1)^n for n >= 1. - William A. Tedeschi, Sep 12 2010
a(n) = ((2*n+1-(-1)^n)*(10*(2*n+1)-2*(-1)^n))/16. - Luce ETIENNE, Sep 13 2014
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 4. - Chai Wah Wu, May 24 2016
Sum_{n>=1} 1/a(n) = 5/2 - sqrt(1+2/sqrt(5))*Pi/2. - Amiram Eldar, Mar 15 2022
a(n) = n^2 + n + 6*ceiling(n/2)^2. - Ridouane Oudra, Aug 06 2022

More terms from Max Alekseyev, Nov 13 2009

A212570 Number of (w,x,y,z) with all terms in {1,...,n} and |w-x|=|x-y|+|y-z|.

Original entry on oeis.org

0, 1, 6, 23, 52, 105, 178, 287, 424, 609, 830, 1111, 1436, 1833, 2282, 2815, 3408, 4097, 4854, 5719, 6660, 7721, 8866, 10143, 11512, 13025, 14638, 16407, 18284, 20329, 22490, 24831, 27296, 29953, 32742, 35735, 38868, 42217, 45714, 49439

Offset: 0

Clark Kimberling, May 22 2012

For a guide to related sequences, see A211795.

Apart from the first term, partial sums of A220082. [Bruno Berselli, Dec 05 2012]

Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A211795.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[Abs[w - x] == Abs[x - y] + Abs[y - z], s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]]   (* A212570 *)
LinearRecurrence[{2,1,-4,1,2,-1},{0,1,6,23,52,105},40] (* Harvey P. Dale, Oct 02 2021 *)

a(n) = 2a(n-1)+a(n-2)-4a(n-3)+a(n-4)+2a(n-5)-a(n-6).

a(n) = n*(-1-3*(-1)^n+10*n^2)/12. G.f.: x*(x^4+4*x^3+10*x^2+4*x+1)/((x-1)^4*(x+1)^2). [Colin Barker, Oct 04 2012]

A274545 Values of k such that 5k-1 and 10k-1 are both perfect squares.

Original entry on oeis.org

1, 29, 33293, 1130977, 1305146305, 44336554445, 51164345409437, 1738081606216033, 2005744667435597089, 68136275082544365341, 78629202401645931667661, 2671078254047822603875969, 3082421990543579145800043553, 104711609647046466634601365517

Offset: 1

Colin Barker, Jun 27 2016

Intersection of A062317 and A220082.

			29 is in the sequence because 5*29-1 = 144 = 12^2, and 10*29-1 = 289 = 17^2.

Colin Barker, Table of n, a(n) for n = 1..400
Index entries for linear recurrences with constant coefficients, signature (1,39202,-39202,-1,1).

Cf. A062317, A220082, A274544.

Rest@ CoefficientList[Series[x (1 + 28 x- 5938 x^2 + 28 x^3 + x^4) / ((1 - x) (1 - 198 x + x^2) (1 + 198 x + x^2)), {x, 0, 17}], x] (* Michael De Vlieger, Jun 27 2016 *)

Vec(x*(1+x)*(1-6*x+x^2)/((1-x)*(1-34*x+x^2)*(1+x+x^2)) + O(x^20))

isok(n) = issquare(5*n-1) && issquare(10*n-1); \\ Michel Marcus, Jun 28 2016

a(n) = a(n-1) + 39202*a(n-2) - 39202*a(n-3) - a(n-4) + a(n-5) for n>5.

G.f.: x*(1 + 28*x - 5938*x^2 + 28*x^3 + x^4)/((1 - x)*(1 - 198*x + x^2)*(1 + 198*x + x^2)).

cp's OEIS Frontend

A132356 a(2k) = k(10k+2), a(2k+1) = 10k^2 + 18k + 8, with k >= 0.

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A212570 Number of (w,x,y,z) with all terms in {1,...,n} and |w-x|=|x-y|+|y-z|.

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A274545 Values of k such that 5k-1 and 10k-1 are both perfect squares.

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cp's OEIS Frontend

A132356 a(2*k) = k*(10*k+2), a(2*k+1) = 10*k^2 + 18*k + 8, with k >= 0.

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A212570 Number of (w,x,y,z) with all terms in {1,...,n} and |w-x|=|x-y|+|y-z|.

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A274545 Values of k such that 5*k-1 and 10*k-1 are both perfect squares.

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A132356 a(2k) = k(10k+2), a(2k+1) = 10k^2 + 18k + 8, with k >= 0.

A274545 Values of k such that 5k-1 and 10k-1 are both perfect squares.