A005667 -id:A005667 - cp's OEIS frontend

A191347 Array read by antidiagonals: ((floor(sqrt(n)) + sqrt(n))^k + (floor(sqrt(n)) - sqrt(n))^k)/2 for columns k >= 0 and rows n >= 0.

1, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 4, 3, 1, 1, 0, 8, 7, 4, 2, 1, 0, 16, 17, 10, 8, 2, 1, 0, 32, 41, 28, 32, 9, 2, 1, 0, 64, 99, 76, 128, 38, 10, 2, 1, 0, 128, 239, 208, 512, 161, 44, 11, 2, 1, 0, 256, 577, 568, 2048, 682, 196, 50, 12, 3, 1

Offset: 0

Charles L. Hohn, May 31 2011

			1, 0,  0,   0,    0,    0,     0,      0,       0,        0,        0, ...
1, 1,  2,   4,    8,   16,    32,     64,     128,      256,      512, ...
1, 1,  3,   7,   17,   41,    99,    239,     577,     1393,     3363, ...
1, 1,  4,  10,   28,   76,   208,    568,    1552,     4240,    11584, ...
1, 2,  8,  32,  128,  512,  2048,   8192,   32768,   131072,   524288, ...
1, 2,  9,  38,  161,  682,  2889,  12238,   51841,   219602,   930249, ...
1, 2, 10,  44,  196,  872,  3880,  17264,   76816,   341792,  1520800, ...
1, 2, 11,  50,  233, 1082,  5027,  23354,  108497,   504050,  2341691, ...
1, 2, 12,  56,  272, 1312,  6336,  30592,  147712,   713216,  3443712, ...
1, 3, 18, 108,  648, 3888, 23328, 139968,  839808,  5038848, 30233088, ...
1, 3, 19, 117,  721, 4443, 27379, 168717, 1039681,  6406803, 39480499, ...
1, 3, 20, 126,  796, 5028, 31760, 200616, 1267216,  8004528, 50561600, ...
1, 3, 21, 135,  873, 5643, 36477, 235791, 1524177,  9852435, 63687141, ...
1, 3, 22, 144,  952, 6288, 41536, 274368, 1812352, 11971584, 79078912, ...
1, 3, 23, 153, 1033, 6963, 46943, 316473, 2133553, 14383683, 96969863, ...
...

Row 1 is A000007, row 2 is A011782, row 3 is A001333, row 4 is A026150, row 5 is A081294, row 6 is A001077, row 7 is A084059, row 8 is A108851, row 9 is A084128, row 10 is A081341, row 11 is A005667, row 13 is A141041.
Row 3*2 is A002203, row 4*2 is A080040, row 5*2 is A155543, row 6*2 is A014448, row 8*2 is A080042, row 9*2 is A170931, row 11*2 is A085447.
Cf. A191348 which uses ceiling() in place of floor().

T(n, k) = if (n==0, k==0, my(x=sqrtint(n)); sum(i=0, (k+1)\2, binomial(k, 2*i)*x^(k-2*i)*n^i));
matrix(9,9, n, k, T(n-1,k-1)) \\ Michel Marcus, Aug 22 2019

T(n, k) = if (k==0, 1, if (k==1, sqrtint(n), T(n,k-2)*(n-T(n,1)^2) + T(n,k-1)*T(n,1)*2));
matrix(9, 9, n, k, T(n-1, k-1)) \\ Charles L. Hohn, Aug 22 2019

For each row n>=0 let T(n,0)=1 and T(n,1)=floor(sqrt(n)), then for each column k>=2: T(n,k)=T(n,k-2)*(n-T(n,1)^2) + T(n,k-1)*T(n,1)*2. - Charles L. Hohn, Aug 22 2019

T(n, k) = Sum_{i=0..floor((k+1)/2)} binomial(k, 2*i)*floor(sqrt(n))^(k-2*i)*n^i for n > 0, with T(0, 0) = 1 and T(0, k) = 0 for k > 0. - Michel Marcus, Aug 23 2019

A382209 Numbers k such that 10+k and 10*k are perfect squares.

Original entry on oeis.org

90, 136890, 197402490, 284654260890, 410471246808090, 591899253243012090, 853518312705176632890, 1230772815021611461622490, 1774773545742851022483004890, 2559222222188376152809031436090, 3690396669622092669499600847844090, 5321549438372835441042271613559748890

Offset: 1

Emilio Martín, Mar 18 2025

The limit of a(n+1)/a(n) is 1441.99930651839... = 721+228*sqrt(10) = (19+6*sqrt(10))^2.

If 10*A158490(n) is a perfect square, then A158490(n) is a term.

			90 is a term because 10+90=100 is a square and 10*90=900 is a square.
(3,1) is a solution to x^2 - 10*y^2 = -1, from which a(n) = 100*y^2-10 = 10*x^2 = 90.

Emilio Martín, Table of n, a(n) for n = 1..100
Wikipedia, Negative Pell equation (in German)
Wikipedia, Pell's equation
Index entries for linear recurrences with constant coefficients, signature (1443,-1443,1).

Subsequence of A158490.
Cf. A097315, A005667, A173127, A081071.
Cf. A383734 = 2*A008843 (2+k and 2*k are squares).
Cf. 5*A075796^2 (5+k and 5*k are squares).
Cf. 5*A081071 (20+k and 20*k are squares).
Cf. A245226 (m such that k+m and k*m are squares).

CoefficientList[Series[ 90*(1 + 78*x + x^2)/((1 - x)*(1 - 1442*x + x^2)),{x,0,11}],x] (* or *) LinearRecurrence[{1443,-1443,1},{90,136890,197402490},12] (* James C. McMahon, May 08 2025 *)

from itertools import islice
def A382209_gen(): # generator of terms
    x, y = 30, 10
    while True:
        yield x**2//10
        x, y = x*19+y*60, x*6+y*19
A382209_list = list(islice(A382209_gen(),30)) # Chai Wah Wu, Apr 24 2025

a(n) = 10 * ((1/2) * (3+sqrt(10))^(2*n-1) + (1/2) * (3-sqrt(10))^(2*n-1))^2.
a(n) = 10 * (sinh((2n-1) * arcsinh(3)))^2.
a(n) = 10 * A173127(n)^2 = 100 * A097315(n)^2 - 10 (negative Pell's equation solutions).
a(n+2) = 1442 * a(n+1) - a(n) + 7200.
G.f.: 90*(1 + 78*x + x^2)/((1 - x)*(1 - 1442*x + x^2)). - Stefano Spezia, Apr 24 2025

A005673 a(n) = F(n+2) - 2^[ (n+1)/2 ] - 2^[ n/2 ] + 1.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 6, 11, 24, 42, 81, 138, 250, 419, 732, 1214, 2073, 3414, 5742, 9411, 15664, 25586, 42273, 68882, 113202, 184131, 301428, 489654, 799273, 1297118, 2112774, 3426275, 5571816, 9030858, 14668209, 23764602, 38563882, 62459555, 101285580, 164007278

Offset: 0

N. J. A. Sloane

R. K. Guy, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. K. Guy, Letter to N. J. A. Sloane, 1987
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (2,2,-5,0,2).

Cf. A000045.

A005673:=-z**4/(z-1)/(z**2+z-1)/(-1+2*z**2); # Conjectured by Simon Plouffe in his 1992 dissertation.

LinearRecurrence[{2,2,-5,0,2},{0,0,0,0,1},40] (* Harvey P. Dale, Apr 22 2024 *)

G.f.: x^4/((1-x)*(1-x-x^2)*(1-2x^2)); a(n) = 2*a(n-1)+2*a(n-2)-5*a(n-3)+2*a(n-5); a(n+1) = Sum_{k=0..n} (2^floor(k/2)-1)*F(n-k). - Paul Barry, Jul 28 2004

A089926 a(n) = 12*a(n-1) + a(n-2), a(0)=1, a(1)=6.

Original entry on oeis.org

1, 6, 73, 882, 10657, 128766, 1555849, 18798954, 227143297, 2744518518, 33161365513, 400680904674, 4841332221601, 58496667563886, 706801342988233, 8540112783422682, 103188154744060417, 1246797969712147686

Offset: 0

Paul Barry, Nov 15 2003

The family of recurrences a(n) = 2*k*a(n-1) + a(n-2), a(0)=1, a(1)=k has solution a(n) = ((k+sqrt(k^2+1))^n + (k-sqrt(k^2+1))^n)/2; a(n) = Sum_{j=0..floor(n/2)} C(n,2k)*(k^2+1)^jk^(n-2j); a(n) = T(n,ki)*(-i)^n; e.g.f. exp(kx)*cosh(sqrt(k^2+1)*x).

Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (12,1).

Cf. A088320, A088317, A005667, A001077.

Essentially the same as A041060.

E.g.f.: exp(6x)*cosh(sqrt(37)x);
a(n) = ((6+sqrt(37))^n + (6-sqrt(37))^n)/2;
a(n) = Sum_{k=0..floor(n/2)} C(n, 2k)*37^k*6^(n-2k).
a(n) = T(n, 6i)*(-i)^n with T(n, x) Chebyshev's polynomials of the first kind (see A053120) and i^2 = -1.
G.f.: (1-6x)/(1-12*x-x^2). - Philippe Deléham, Nov 21 2008

A116559 Expansion of g.f. x(1+x+2x^2+2x^3+5x^4+5x^5-3x^6+2x^7-x^8-x^9)/(1-6x^6-x^12).

Original entry on oeis.org

0, 1, 1, 2, 2, 5, 5, 3, 8, 11, 11, 30, 30, 19, 49, 68, 68, 185, 185, 117, 302, 419, 419, 1140, 1140, 721, 1861, 2582, 2582, 7025, 7025, 4443, 11468, 15911, 15911, 43290, 43290, 27379, 70669, 98048, 98048, 266765, 266765, 168717, 435482, 604199, 604199, 1643880, 1643880, 1039681

Offset: 0

Roger L. Bagula, Mar 17 2006

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,6,0,0,0,0,0,1).

CoefficientList[Series[x*(1 + x + 2*x^2 + 2*x^3 + 5*x^4 + 5*x^5 - 3*x^6 + 2*x^7 - x^8 - x^9)/(1 - 6*x^6 - x^12), {x, 0, 50}], x] (* G. C. Greubel, Sep 20 2017 *)

x='x+O('x^50); Vec(x*(1 + x + 2*x^2 + 2*x^3 + 5*x^4 + 5*x^5 - 3*x^6 + 2*x^7 - x^8 - x^9)/(1 - 6*x^6 - x^12)) \\ G. C. Greubel, Sep 20 2017

From R. J. Mathar, Nov 28 2008: (Start)
a(n) = 6*a(n-6) + a(n-12).
G.f.: x*(1+x+2*x^2+2*x^3+5*x^4+5*x^5-3*x^6+2*x^7-x^8-x^9)/(1-6*x^6-x^12).
a(6n+1) = A005667(n). (End)

More terms added by G. C. Greubel, Sep 20 2017

Better name using given g.f. from Joerg Arndt, Oct 26 2024

A005671 Nearest integer to tan(n)^2.

Original entry on oeis.org

0, 2, 5, 0, 1, 11, 0, 1, 46, 0, 0, 51054, 0, 0, 52, 1, 0, 12, 1, 0, 5, 2, 0, 3, 5, 0, 1, 11, 0, 1, 41, 0, 0, 5672, 0, 0, 60, 1, 0, 13, 1, 0, 5, 2, 0, 3, 4, 0, 1, 10, 0, 1, 37, 0, 0, 2042, 0, 0, 69, 1, 0, 14, 1, 0, 6, 2, 0, 3, 4, 0, 1

Offset: 0

N. J. A. Sloane

nonn

R. K. Guy, Letter to N. J. A. Sloane, 1987

Round[Tan[Range[0,70]]^2] (* Harvey P. Dale, Aug 01 2016 *)

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A191347 Array read by antidiagonals: ((floor(sqrt(n)) + sqrt(n))^k + (floor(sqrt(n)) - sqrt(n))^k)/2 for columns k >= 0 and rows n >= 0.

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A382209 Numbers k such that 10+k and 10*k are perfect squares.

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A005673 a(n) = F(n+2) - 2^[ (n+1)/2 ] - 2^[ n/2 ] + 1.

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A089926 a(n) = 12*a(n-1) + a(n-2), a(0)=1, a(1)=6.

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A116559 Expansion of g.f. x*(1+x+2*x^2+2*x^3+5*x^4+5*x^5-3*x^6+2*x^7-x^8-x^9)/(1-6*x^6-x^12).

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A005671 Nearest integer to tan(n)^2.

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Mathematica

A116559 Expansion of g.f. x(1+x+2x^2+2x^3+5x^4+5x^5-3x^6+2x^7-x^8-x^9)/(1-6x^6-x^12).