A010718 -id:A010718 - cp's OEIS frontend

A195033 Multiples of 21 and of 20 interleaved: a(2n-1) = 21n, a(2n) = 20n.

21, 20, 42, 40, 63, 60, 84, 80, 105, 100, 126, 120, 147, 140, 168, 160, 189, 180, 210, 200, 231, 220, 252, 240, 273, 260, 294, 280, 315, 300, 336, 320, 357, 340, 378, 360, 399, 380, 420, 400, 441, 420, 462, 440, 483, 460, 504, 480, 525, 500, 546, 520, 567, 540

Offset: 1

Omar E. Pol, Sep 12 2011

First differences of A195034.

a(n) is also the length of the n-th edge of a square spiral in which the first two edges are the legs of the primitive Pythagorean triple [21, 20, 29]. Zero together with partial sums give A195034, the vertices of the spiral.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Ron Knott, Pythagorean triangles and Triples
Eric Weisstein's World of Mathematics, Pythagorean Triple
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A010693, A010718, A029578.

Cf. A195019, A195031, A195034, A195035.

&cat[[21*n,20*n]: n in [1..27]];  // Bruno Berselli, Sep 29 2011

Riffle[21*#, 20*#] & [Range[50]] (* Paolo Xausa, Mar 21 2024 *)

a(n)=(n+1)\2*if(n%2,21,20) \\ Charles R Greathouse IV, Oct 07 2015

From Bruno Berselli, Sep 29 2011:  (Start)
G.f.: x*(21+20*x)/((1-x)^2*(1+x)^2).
a(n) = A010693(n)*A010718(n)*A029578(n+1) = (41*n-(n+21)*(-1)^n+21)/4.
a(n) = 2*a(n-2) - a(n-4). (End)

More terms from Bruno Berselli, Sep 29 2011

A021136 Decimal expansion of 1/132.

Original entry on oeis.org

0, 0, 7, 5, 7, 5, 7, 5, 7, 5, 7, 5, 7, 5, 7, 5, 7, 5, 7, 5, 7, 5, 7, 5, 7, 5, 7, 5, 7, 5, 7, 5, 7, 5, 7, 5, 7, 5, 7, 5, 7, 5, 7, 5, 7, 5, 7, 5, 7, 5, 7, 5, 7, 5, 7, 5, 7, 5, 7, 5, 7, 5, 7, 5, 7, 5, 7, 5, 7, 5, 7, 5, 7, 5, 7, 5, 7, 5, 7, 5, 7, 5, 7, 5, 7, 5, 7, 5, 7, 5, 7, 5, 7, 5, 7, 5, 7, 5, 7

Offset: 0

N. J. A. Sloane

Multiplied by -1, this is zeta(-9), where zeta is the Riemann zeta function. - Alonso del Arte, Jan 13 2012

Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A010718.

RealDigits[1/132, 10, 100][[1]] (* Alonso del Arte, Jan 13 2012 *)

A176321 Decimal expansion of (35 + sqrt(1365))/14.

Original entry on oeis.org

5, 1, 3, 8, 9, 9, 3, 3, 1, 4, 5, 5, 8, 7, 3, 7, 9, 9, 9, 4, 0, 2, 5, 4, 3, 6, 8, 5, 6, 9, 9, 7, 9, 5, 8, 6, 1, 1, 7, 9, 7, 1, 2, 4, 4, 4, 5, 1, 2, 2, 5, 4, 1, 9, 6, 1, 7, 0, 7, 6, 0, 1, 3, 4, 8, 9, 2, 3, 3, 2, 9, 0, 4, 8, 0, 3, 6, 8, 5, 3, 3, 5, 8, 6, 3, 5, 9, 3, 1, 4, 7, 1, 8, 1, 7, 6, 0, 9, 2, 1, 7, 0, 0, 8, 9

Offset: 1

Klaus Brockhaus, Apr 15 2010

Continued fraction expansion of (35+sqrt(1365))/14 is A010718.

			5.13899331455873799940...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A176322 (decimal expansion of sqrt(1365)), A010718 (repeat 5, 7).

SetDefaultRealField(RealField(120)); (35+Sqrt(1365))/14; // G. C. Greubel, Nov 26 2019

evalf( (35+sqrt(1365))/14, 120); # G. C. Greubel, Nov 26 2019

RealDigits[(35 + Sqrt[1365])/14, 10, 100][[1]] (* Vincenzo Librandi, Sep 24 2013 *)

default(realprecision, 120); (35+sqrt(1365))/14 \\ G. C. Greubel, Nov 26 2019

numerical_approx((35+sqrt(1365))/14, digits=120) # G. C. Greubel, Nov 26 2019

A260307 a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-6) - a(n-7) - a(n-8) + a(n-9) with a(0) - a(8) as shown below.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 8, 7, 10, 9, 13, 10, 15, 12, 17, 14, 20, 15, 22, 17, 24, 19, 27, 20, 29, 22, 31, 24, 34, 25, 36, 27, 38, 29, 41, 30, 43, 32, 45, 34, 48, 35, 50, 37, 52, 39, 55, 40, 57, 42, 59, 44, 62, 45, 64, 47, 66, 49, 69, 50, 71, 52, 73, 54, 76, 55, 78

Offset: 0

Paul Curtz, Nov 22 2015

A260708 difference table rows have the same nine-step recurrence:
0, 1, 3,  6, 10, 16, 21, 29, 36, 46, 55, 65, 78,  93, ...
1, 2, 3,  4,  6,  5,  8,  7, 10,  9, 13, 10, 15,  12, ...     = a(n)
1, 1, 1,  2, -1,  3, -1,  3, -1,  4, -3,  5, -3,   5, ...     = b(n)
0, 0, 1, -3,  4, -4,  4, -4,  5, -7,  8, -8,  8,  -8, ... (see A042965(n)).
(b(2n) + b(2n+1) = A052901(n+2).)

G. C. Greubel, Table of n, a(n) for n = 0..1500
Index entries for linear recurrences with constant coefficients, signature (0,1,0,0,0,1,0,-1).

Cf. A004767, A010718, A042965, A047212, A047282, A052901, A152467, A260160 (eight-step recurrence), A260699 (nine-step recurrence), A260708.

I:=[1,2,3,4,6,5,8,7];[n le 8 select I[n] else Self(n-2) + Self(n-6) - Self(n-8): n in [1..70]]; // Vincenzo Librandi, Dec 26 2015

RecurrenceTable[{a[n] == a[n-2] + a[n-6] - a[n-8], a[0]=1, a[1]=2, a[2]=3, a[3]=4, a[4]=6, a[5]=5, a[6]=8, a[7]=7}, a, {n,0,100}] (* G. C. Greubel, Nov 23 2015 *)

Vec((x^6+x^5+3*x^4+2*x^3+2*x^2+2*x+1)/((x-1)^2*(x+1)^2*(x^2-x+1)*(x^2+x+1)) + O(x^100)) \\ Colin Barker, Nov 22 2015

vector(100, n, n--; n + (-1)^n *((n+2)\6) + 1) \\ Altug Alkan, Nov 24 2015

a(2n) = A047282(n). a(2n+1) = A047212(n+1).
a(n) = A260708(n+1) - A260708(n).
a(n+6) = a(n) + period of length 2: repeat 7, 5.
a(2n) + a(2n+1) = 3 + 4*n.
a(n) = n + 1 + (-1)^n*A152467(n+2).
From Colin Barker, Nov 22 2015: (Start)
a(n) = a(n-2) + a(n-6) - a(n-8) for n>7.
G.f.: (x^6+x^5+3*x^4+2*x^3+2*x^2+2*x+1) / ((x-1)^2*(x+1)^2*(x^2-x+1)*(x^2+x+1)).
(End)

A175828 a(n) = (n(6n+1)+(n+2)*(-1)^n)/2.

Original entry on oeis.org

1, 2, 15, 26, 53, 74, 115, 146, 201, 242, 311, 362, 445, 506, 603, 674, 785, 866, 991, 1082, 1221, 1322, 1475, 1586, 1753, 1874, 2055, 2186, 2381, 2522, 2731, 2882, 3105, 3266, 3503, 3674, 3925, 4106, 4371, 4562, 4841, 5042, 5335, 5546, 5853, 6074

Offset: 0

Bruno Berselli, Sep 21 2010 - Sep 25 2010

a(n) == A068073(n) (mod 4).

a(h) == 0 (mod 11) for h = 11*(k-floor((k-1)/3))-2*(-1)^(k+floor(k/3)) (cf. A175833).

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

Cf. A068073, A010718, A142241, A175833.

[(n*(6*n+1)+(n+2)*(-1)^n)/2: n in [0..50]];

I:=[1,2,15,26,53]; [n le 5 select I[n] else Self(n-1)+2*Self(n-2)-2*Self(n-3)-Self(n-4)+Self(n-5): n in [1..50]]; // Vincenzo Librandi, Aug 19 2013

Table[(n (6 n + 1) + (n + 2) (-1)^n)/2, {n, 0, 50}]
CoefficientList[Series[(1 + x + 11 x^2 + 9 x^3 + 2 x^4) / ((1 + x)^2 (1 - x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 19 2013 *)
LinearRecurrence[{1,2,-2,-1,1},{1,2,15,26,53},70] (* Harvey P. Dale, Jul 03 2019 *)

G.f.: (1+x+11*x^2+9*x^3+2*x^4)/((1+x)^2*(1-x)^3).
a(n)-a(n-1)-2*a(n-2)+2*a(n-3)+a(n-4)-a(n-5) = 0 for n>4.
a(n)-a(n-2)-(a(n-1)-a(n-3)) = 2*A010718(n-1) for n>2.
a(n)-a(n-2)+(a(n-1)-a(n-3)) = A142241(n-2) for n>2.

A176437 Decimal expansion of (35+sqrt(1365))/10.

Original entry on oeis.org

7, 1, 9, 4, 5, 9, 0, 6, 4, 0, 3, 8, 2, 2, 3, 3, 1, 9, 9, 1, 6, 3, 5, 6, 1, 1, 5, 9, 9, 7, 9, 7, 1, 4, 2, 0, 5, 6, 5, 1, 5, 9, 7, 4, 2, 2, 3, 1, 7, 1, 5, 5, 8, 7, 4, 6, 3, 9, 0, 6, 4, 1, 8, 8, 8, 4, 9, 2, 6, 6, 0, 6, 6, 7, 2, 5, 1, 5, 9, 4, 7, 0, 2, 0, 9, 0, 3, 0, 4, 0, 6, 0, 5, 4, 4, 6, 5, 2, 9, 0, 3, 8, 1, 2, 5

Offset: 1

Klaus Brockhaus, Apr 19 2010

Continued fraction expansion of (35+sqrt(1365))/10 is A010718 preceded by 7.

			(35+sqrt(1365))/10 = 7.19459064038223319916...

Cf. A176322 (decimal expansion of sqrt(1365)), A010718 (repeat 5, 7).

RealDigits[(35+Sqrt[1365])/10,10,120][[1]] (* Harvey P. Dale, Dec 18 2012 *)

A225539 Numbers n where 2^n and n have the same digital root.

Original entry on oeis.org

5, 16, 23, 34, 41, 52, 59, 70, 77, 88, 95, 106, 113, 124, 131, 142, 149, 160, 167, 178, 185, 196, 203, 214, 221, 232, 239, 250, 257, 268, 275, 286, 293, 304, 311, 322, 329, 340, 347, 358, 365, 376, 383, 394, 401, 412, 419, 430, 437, 448

Offset: 1

Marcus Hedbring, May 17 2013

The digital roots of n have a cycle length of 9 (A010888) and the digital roots of 2^n have a cycle length of 6 (A153130). Therefore, if n is a term so is n+18.

The only values of the digital roots of a(n) are 5 and 7 (A010718).

			For n=23, the digital root of n is 5. 2^n equals 8388608 so the digital root of 2^n is 5 as well.

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

Cf. A010888, A153130, A010718.

digitalRoot[n_] :=  Module[{r = n}, While[r > 9, r = Total[IntegerDigits[ r]]]; r]; Select[Range[448], digitalRoot[2^#] == digitalRoot[#] &] (* T. D. Noe, May 19 2013 *)
LinearRecurrence[{1,1,-1},{5,16,23},60] (* Harvey P. Dale, Dec 29 2018 *)

forstep(n=16,500,[7,11],print1(n", ")) \\ Charles R Greathouse IV, May 19 2013

a(n) = 9*n - 3 + (-1)^n.
a(n) = a(n-1) + 7 (odd n), a(n) = a(n-1) + 11 (even n) with a(1) = 5.
G.f. x*(5 + 11*x + 2*x^2) / ((1-x)^2 * (1+x)). - Joerg Arndt, May 17 2013

cp's OEIS Frontend

A195033 Multiples of 21 and of 20 interleaved: a(2n-1) = 21n, a(2n) = 20n.

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A021136 Decimal expansion of 1/132.

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A176321 Decimal expansion of (35 + sqrt(1365))/14.

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A260307 a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-6) - a(n-7) - a(n-8) + a(n-9) with a(0) - a(8) as shown below.

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A175828 a(n) = (n*(6*n+1)+(n+2)*(-1)^n)/2.

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A176437 Decimal expansion of (35+sqrt(1365))/10.

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A225539 Numbers n where 2^n and n have the same digital root.

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A175828 a(n) = (n(6n+1)+(n+2)*(-1)^n)/2.