A053606 -id:A053606 - cp's OEIS frontend

A107075 Centered square numbers that are also centered pentagonal numbers.

1, 181, 58141, 18721081, 6028129801, 1941039074701, 625008553923781, 201250813324382641, 64802136881897286481, 20866086825157601864101, 6718815155563865902953901, 2163437614004739663149291881

Offset: 1

Richard Choulet, Aug 30 2007, Sep 20 2007

The centered square numbers are n^2 + (n+1)^2 while the centered pentagonal numbers are (5*r^2 + 5*r + 2)/2. A number has both properties iff 5*(2*r+1)^2 = (4*n+2)^2 + 1. We solve the equation 5*Y^2 - 1 = X^2 whose solutions in positive integers are given by A075796 and A007805 respectively. The r values are 0,8,..., i.e., A053606. The n values define A119032.

Harvey P. Dale, Table of n, a(n) for n = 1..399
Index entries for linear recurrences with constant coefficients, signature (323,-323,1).

Cf. A007805, A053606, A075796, A119032.

a:= n-> (Matrix([181,1,1]). Matrix([[323,1,0], [ -323,0,1], [1,0,0]])^n)[1,3]: seq(a(n), n=1..20); # Alois P. Heinz, Aug 14 2008

LinearRecurrence[{323,-323,1},{1,181,58141},20] (* Harvey P. Dale, Nov 15 2018 *)

G.f.: (z*(1-142*z+z^2))/((1-z)*(1-322*z+z^2)).
a(n+2) = 322*a(n+1)-a(n)-140 with a(1)=1 and a(2)=181.
a(n+1) = 161*a(n)-70+18*(80*a(n)^2-70*a(n)+15)^0.5.
a(n) = (14+(9-4*sqrt(5))^(2*n-1)+(9+4*sqrt(5))^(2*n-1))/32. - Gerry Martens, Jun 06 2015

More terms from Alois P. Heinz, Aug 14 2008

A166259 Positive integers n such that a centered polygonal number nk(k+1)/2+1 is not a square for any k > 0.

Original entry on oeis.org

2, 18, 32, 50, 72, 98, 128, 162, 200, 242, 338, 392, 450, 512, 578, 648, 722, 882, 968, 1058, 1152, 1250, 1352, 1458, 1682, 1800, 1922, 2048, 2178, 2312, 2401, 2450, 2662, 2738, 2809, 2888, 3042, 3174, 3200, 3362, 3528, 3698, 3750, 4050, 4225, 4232, 4418, 4489, 4608, 4802

Offset: 1

Alexander Adamchuk, Oct 10 2009

nonn

Positive integers n such that A120744(n) = -1.

Eric Weisstein's World of Mathematics, Centered Polygonal Number.

Cf. A120744, A001542, A006451, A001921, A129444, A001570, A001652, A129556, A053606, A105038, A105040, A053141, A061278.

Edited and extended by Max Alekseyev, Jan 20 2010

A222393 Nonnegative integers m such that 18m(m+1)+1 is a square.

Original entry on oeis.org

0, 4, 12, 152, 424, 5180, 14420, 175984, 489872, 5978292, 16641244, 203085960, 565312440, 6898944364, 19203981732, 234361022432, 652370066464, 7961375818340, 22161378278060, 270452416801144, 752834491387592, 9187420795420572, 25574211328900084

Offset: 1

Bruno Berselli, Feb 19 2013

a(n+2)/a(n) tends to A156164.

a(n) is congruent to {0,2,4} (mod 5, 6 and 10).

Bruno Berselli, Table of n, a(n) for n = 1..500
Index entries for linear recurrences with constant coefficients, signature (1,34,-34,-1,1).

Cf. nonnegative integers n such that k*n*(n+1)+1 is a square: A001652 (k=2), A001921 (k=3), A001477 (k=4), A053606 (k=5), A105038 (k=6), A105040 (k=7), A053141 (k=8), A222390 (k=10), A105838 (k=11), A061278 (k=12), A104240 (k=13); A105063 (k=17), this sequence (k=18), A101180 (k=19), A077259 (k=20) [incomplete list].

m:=22; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(4*(1+x)^2/((1-x)*(1-6*x+x^2)*(1+6*x+x^2))));

I:=[0,4,12,152,424]; [n le 5 select I[n] else Self(n-1)+34*Self(n-2)-34*Self(n-3)-Self(n-4)+Self(n-5): n in [1..25]]; // Vincenzo Librandi, Aug 18 2013

LinearRecurrence[{1, 34, -34, -1, 1}, {0, 4, 12, 152, 424}, 23]
CoefficientList[Series[4 x (1 + x)^2 / ((1 - x) (1 - 6 x + x^2) (1 + 6 x + x^2)), {x, 0, 25}], x] (* Vincenzo Librandi, Aug 18 2013 *)

makelist(expand(-1/2+((3+sqrt(2)*(-1)^n)*(3-2*sqrt(2))^(2*floor(n/2))+(3-sqrt(2)*(-1)^n)*(3+2*sqrt(2))^(2*floor(n/2)))/12), n, 1, 23);

x='x+O('x^30); concat([0], Vec(4*x*(1+x)^2/((1-x)*(1-6*x+x^2)*(1+6*x+x^2)))) \\ G. C. Greubel, Jul 15 2018

G.f.: 4*x*(1+x)^2/((1-x)*(1-6*x+x^2)*(1+6*x+x^2)).
a(n) = a(-n+1) = a(n-1)+34*a(n-2)-34*a(n-3)-a(n-4)+a(n-5).
a(n) = -1/2+((3+t*(-1)^n)*(3-2*t)^(2*floor(n/2))+(3-t*(-1)^n)*(3+2*t)^(2*floor(n/2)))/12, where t=sqrt(2).

A145856 Least number k>1 such that centered n-gonal number n*k(k-1)/2+1 is a perfect square, or 0 if no such k exists.

Original entry on oeis.org

3, 0, 2, 4, 3, 8, 16, 2, 17, 9, 15, 5, 6, 16, 2, 3, 6, 0, 7, 4, 3, 40, 7, 2, 22, 8, 111, 4, 16, 8, 16, 0, 3, 9, 2, 5, 990, 9, 15, 3, 46, 16, 10, 5, 6, 336, 10, 2, 30, 0, 31, 16, 11, 416, 7, 3, 11, 33, 55, 4, 78, 56, 2, 6, 3, 8, 47751, 12, 16, 24, 48, 0, 49, 25, 17, 13, 6, 9, 2640, 2, 6721

Offset: 1

Alexander Adamchuk, Oct 22 2008

nonn

Jonathan Vos Post, When Centered Polygonal Numbers are Perfect Squares, submitted to Mathematics Magazine, 4 May 2004, manuscript no. 04-1165, unpublished, available upon request. - Jonathan Vos Post, Oct 25 2008

Eric Weisstein's World of Mathematics, Centered Polygonal Numbers
Index entries for sequences related to centered polygonal numbers

Cf. A120744, A001542, A166259, A006451, A001921, A129444, A001570, A001652, A129556, A053606, A105038, A105040, A053141, A061278.

a(n) = 0 for n in A166259.

a(n) = A120744(n) + 1. - Alexander Adamchuk, Oct 10 2009

Edited by Max Alekseyev, Jan 23 2010

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A107075 Centered square numbers that are also centered pentagonal numbers.

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A166259 Positive integers n such that a centered polygonal number n*k*(k+1)/2+1 is not a square for any k > 0.

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A222393 Nonnegative integers m such that 18*m*(m+1)+1 is a square.

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A145856 Least number k>1 such that centered n-gonal number n*k(k-1)/2+1 is a perfect square, or 0 if no such k exists.

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A166259 Positive integers n such that a centered polygonal number nk(k+1)/2+1 is not a square for any k > 0.

A222393 Nonnegative integers m such that 18m(m+1)+1 is a square.