A010720 -id:A010720 - cp's OEIS frontend

A176324 Decimal expansion of (15+7*sqrt(5))/6.

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

Offset: 1

Klaus Brockhaus, Apr 15 2010

Continued fraction expansion of (15+7*sqrt(5))/6 is A010720.

Volume of a triangular hebesphenorotunda (Johnson solid J_92) with unit edges. - Paolo Xausa, Aug 02 2025

			5.10874597374975464581...

G. C. Greubel, Table of n, a(n) for n = 1..1000
Wikipedia, Triangular hebesphenorotunda.

Cf. A002163 (decimal expansion of sqrt(5)), A010720 (repeat 5, 9).

SetDefaultRealField(RealField(100)); (15+7*Sqrt(5))/6; // G. C. Greubel, Dec 05 2019

evalf( (15+7*sqrt(5))/6, 100); # G. C. Greubel, Dec 05 2019

RealDigits[(15+7Sqrt[5])/6,10,120][[1]]  (* Harvey P. Dale, Apr 18 2011 *)

default(realprecision, 100); (15+7*sqrt(5))/6 \\ G. C. Greubel, Dec 05 2019

numerical_approx((15+7*sqrt(5))/6, digits=100) # G. C. Greubel, Dec 05 2019

A306285 Numbers congruent to 4 or 21 mod 26.

Original entry on oeis.org

4, 21, 30, 47, 56, 73, 82, 99, 108, 125, 134, 151, 160, 177, 186, 203, 212, 229, 238, 255, 264, 281, 290, 307, 316, 333, 342, 359, 368, 385, 394, 411, 420, 437, 446, 463, 472, 489, 498, 515, 524, 541, 550, 567, 576, 593, 602, 619, 628, 645, 654, 671, 680, 697, 706, 723, 732, 749, 758, 775, 784, 801, 810, 827, 836, 853, 862

Offset: 1

Davis Smith, Feb 03 2019

A007310(a(n)+1) is always a multiple of 13.
a(n) mod 6 follows the following pattern: 4,3,0,5,2,1,4,3,0,5,2,1 and so on.
a(n) mod 4 = A010873(n).
A020639(A007310(a(n)+1)) = 5 when n is congruent to 2 or 9 (mod 10) (n is a term in A273669). It equals 7 when n is congruent to 3 or 12 (mod 14) but not congruent to 2 or 9 (mod 10). It equals 11 when n is congruent to 4 or 19 (mod 22) but not congruent to 2 or 9 (mod 10) and not congruent to 3 or 12 (mod 14). Otherwise, it is 13.

Davis Smith, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A007310, A010720, A020639, A010873, A273669.

Maple
```
seq(seq(26*i+j, j=[4, 21]), i=0..200);
```

Select[Range[200], MemberQ[{4, 21}, Mod[#, 26]] &]

for(n=1, 1000, if((n%26==4) || (n%26==21), print1(n, ", ")))

Vec(x*(4 + 17*x + 5*x^2) / ((1 - x)^2*(1 + x)) + O(x^40)) \\ Colin Barker, Feb 08 2019

a(n) = 13*n - A010720(n+1).
From Colin Barker, Feb 08 2019: (Start)
G.f.: x*(4 + 17*x + 5*x^2) / ((1 - x)^2*(1 + x)).
a(n) = 13*n - 5 for n even.
a(n) = 13*n - 9 for n odd.
a(n) = a(n-1) + a(n-2) - a(n-3) for n>3. (End)
E.g.f.: 5 + (13*x - 7)*exp(x) + 2*exp(-x). - David Lovler, Sep 09 2022

A121511 a(n) = a(n-1) + a(n-4) - a(n-5).

Original entry on oeis.org

1, 5, 13, 25, 5, 9, 17, 29, 9, 13, 21, 33, 13, 17, 25, 37, 17, 21, 29, 41, 21, 25, 33, 45, 25, 29, 37, 49, 29, 33, 41, 53, 33, 37, 45, 57, 37, 41, 49, 61, 41, 45, 53, 65, 45, 49, 57, 69, 49, 53, 61, 73, 53, 57, 65, 77, 57, 61, 69, 81, 61, 65, 73, 85, 65, 69, 77

Offset: 1

Roger L. Bagula, Sep 07 2006

This sequence gives a linearly increasing triangle shaped form on plotting.

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

Cf. A001844.

LinearRecurrence[{1,0,0,1,-1},{1,5,13,25,5},67] (* James C. McMahon, Oct 19 2024 *)

Vec(-x*(-1-4*x-8*x^2-12*x^3+21*x^4)/((1+x)*(x^2+1)*(x-1)^2) + O(x^75)) \\ Jinyuan Wang, Jun 07 2020

G.f.: -x*(-1-4*x-8*x^2-12*x^3+21*x^4)/((1+x)*(x^2+1)*(x-1)^2).

a(n) = n + 17/2 + 7*(-1)^n/2 + A057077(n+1)*A010720(n-1).

Definition replaced by recurrence - The Assoc. Editors of the OEIS, Oct 14 2009

More terms from Jinyuan Wang, Jun 07 2020

A176518 Decimal expansion of 3(15+7sqrt(5))/10.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 23 2010

Continued fraction expansion of (45+21*sqrt(5))/10 is A010720 preceded by 9.

Quadratic number with denominator 5 and minimal polynomial 5x^2 - 45x - 9. - Charles R Greathouse IV, Apr 25 2016

			(45+21*sqrt(5))/10 = 9.19574275274955836245...

Cf. A002163 (decimal expansion of sqrt(5)), A010720(repeat 5, 9).

RealDigits[(45+21*Sqrt[5])/10,10,120][[1]] (* Harvey P. Dale, Aug 24 2012 *)

(21*sqrt(5)+45)/10 \\ Charles R Greathouse IV, Apr 25 2016

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A176324 Decimal expansion of (15+7*sqrt(5))/6.

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A306285 Numbers congruent to 4 or 21 mod 26.

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A121511 a(n) = a(n-1) + a(n-4) - a(n-5).

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A176518 Decimal expansion of 3*(15+7*sqrt(5))/10.

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A176518 Decimal expansion of 3(15+7sqrt(5))/10.