A010692 -id:A010692 - cp's OEIS frontend

A168668 a(n) = n(2 + 5n).

0, 7, 24, 51, 88, 135, 192, 259, 336, 423, 520, 627, 744, 871, 1008, 1155, 1312, 1479, 1656, 1843, 2040, 2247, 2464, 2691, 2928, 3175, 3432, 3699, 3976, 4263, 4560, 4867, 5184, 5511, 5848, 6195, 6552, 6919, 7296, 7683, 8080, 8487, 8904, 9331, 9768, 10215, 10672

Offset: 0

Paul Curtz, Dec 02 2009

Appears on the main diagonal of the following table of terms of the Hydrogen series, A169603:
0,  3,  8, 15, 24,                        ... A005563
0,  7, 16,  1, 40,  55,                   ... A061039
0, 11, 24, 39, 56,   3,  96,              ... A061043
0, 15, 32, 51, 72,  95, 120,              ... A061047
0, 19, 40, 63, 88, 115, 144, 175, 208, 1, ...

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature(3,-3,1).

Cf. A000217, A000384, A001622, A010692, A131242, A254963 (comment).

Cf. A005563, A061039, A061043, A061047, A169603.

[n*(2+5*n): n in [0..50] ]; // Vincenzo Librandi, Aug 06 2011

A168668:=n->n*(2+5*n): seq(A168668(n), n=0..50); # Wesley Ivan Hurt, Mar 28 2015

f[n_]:=n*(2+5*n);f[Range[0,60]] (* Vladimir Joseph Stephan Orlovsky, Feb 05 2011*)
LinearRecurrence[{3,-3,1},{0,7,24},50] (* Harvey P. Dale, Sep 09 2021 *)

vector(50,n,n--;n*(2+5*n)) \\ Derek Orr, Jun 26 2015

G.f.: x*(7 + 3*x)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
First differences: a(n) - a(n-1) = 10*n-3.
Second differences: a(n) - 2*a(n-1) + a(n-2) = 10 = A010692(n).
a(n) = A131242(10n+6). - Philippe Deléham, Mar 27 2013
a(n) = A000384(n) + 6*A000217(n). - Luciano Ancora, Mar 28 2015
a(n) = A000217(n) + A000217(3*n). - Bruno Berselli, Jul 01 2016
E.g.f.: x*(7 + 5*x)*exp(x). - G. C. Greubel, Jul 29 2016
Sum_{n>=1} 1/a(n) = 5/4 - sqrt(1-2/sqrt(5))*Pi/4 + sqrt(5)*log(phi)/4 - 5*log(5)/8, where phi is the golden ratio (A001622). - Amiram Eldar, Sep 17 2023

Edited and extended by R. J. Mathar, Dec 05 2009

A040020 Continued fraction for sqrt(26).

Original entry on oeis.org

5, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10

Offset: 0

N. J. A. Sloane

			5.09901951359278483002822... = 5 + 1/(10 + 1/(10 + 1/(10 + 1/(10 + ...)))). - _Harry J. Smith_, Jun 03 2009

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 276.

Harry J. Smith, Table of n, a(n) for n = 0..20000
G. Xiao, Contfrac
Index entries for continued fractions for constants
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A010481 (decimal expansion), A041040/A041041 (convergents), A248253 (Egyptian fraction).

Cf. A040000, A010692.

numtheory[cfrac](sqrt(26), 100, 'quotients'); # obsolete code updated by Alois P. Heinz, Feb 24 2018

ContinuedFraction[Sqrt[26],300] (* Vladimir Joseph Stephan Orlovsky, Mar 05 2011 *)
PadRight[{5},120,{10}] (* Harvey P. Dale, Apr 22 2021 *)

contfrac(sqrt(26)) \\ Harry J. Smith, Jun 03 2009 [Edited by M. F. Hasler, Feb 24 2018]

A040020(n)=if(n,10,5) \\ M. F. Hasler, Feb 24 2018

From Elmo R. Oliveira, Feb 06 2024: (Start)
a(n) = 10 for n >= 1.
G.f.: 5*(1+x)/(1-x).
E.g.f.: 10*exp(x) - 5.
a(n) = 5*A040000(n). (End)

A010726 Period 2: repeat (6,10).

Original entry on oeis.org

6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10, 6, 10

Offset: 0

N. J. A. Sloane

From Klaus Brockhaus, Dec 10 2009: (Start)
Interleaving of A010722 and A010692.
Also continued fraction expansion of 3 + 4*sqrt(15)/5.
Binomial transform of 6 followed by A122803 without initial terms 1,-2.
Inverse binomial transform of A171494. (End)

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

Equals 2*A010703. Cf. A010722 (all 6's sequence), A010692 (all 10's sequence), A122803 (powers of -2), A171494. - Klaus Brockhaus, Dec 10 2009

&cat[ [6, 10]: n in [1..42] ]; // Klaus Brockhaus, Dec 10 2009

LinearRecurrence[{0,1},{6,10},90] (* or *) PadRight[{},90,{6,10}] (* Harvey P. Dale, Mar 07 2015 *)

a(n)=6+n%2*4 \\ Charles R Greathouse IV, Dec 21 2011

a(n) = -2*(-1)^n + 8. - Paolo P. Lava, Oct 27 2006
From Klaus Brockhaus, Dec 10 2009: (Start)
a(n) = a(n-2) for n > 1; a(0) = 6, a(1) = 10.
G.f.: 2*(3+5*x)/((1-x)*(1+x)). (End)

A176537 Decimal expansion of 5 + sqrt(26).

Original entry on oeis.org

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

Offset: 2

Klaus Brockhaus, Apr 24 2010

Continued fraction expansion of 5 + sqrt(26) is A010692.

This is the shape of a 10-extension rectangle; see A188640 for definitions. - Clark Kimberling, Apr 09 2011

			5+sqrt(26) = 10.09901951359278483002...

Daniel Starodubtsev, Table of n, a(n) for n = 2..10000
Wikipedia, Metallic mean
Index entries for algebraic numbers, degree 2.

Cf. A010481 (decimal expansion of sqrt(26)), A010692 (all 10's sequence).

r=10; t = (r + (4+r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]
RealDigits[5+Sqrt[26],10,120][[1]] (* Harvey P. Dale, Jun 24 2013 *)

5+sqrt(26) \\ Michel Marcus, Jul 23 2018

a(n) = A010481(n-2) for n > 3.
Equals exp(arcsinh(5)), since arcsinh(x) = log(x + sqrt(x^2 + 1)). - Stanislav Sykora, Nov 01 2013
Equals limit_{n->infinity} S(n, 2*sqrt(2*13))/ S(n-1, 2*sqrt(2*13)), with the S-Chebyshev polynomilas (see A049310). - Wolfdieter Lang, Nov 15 2023

A023009 Number of partitions of n into parts of 10 kinds.

Original entry on oeis.org

1, 10, 65, 330, 1430, 5512, 19415, 63570, 195910, 573430, 1605340, 4322110, 11240645, 28341730, 69488650, 166096270, 387890625, 886698670, 1987322415, 4373271870, 9461022285, 20144164040, 42254620785, 87398226990, 178396331100, 359618772656, 716409453320

Offset: 0

David W. Wilson

nonn

a(n) is Euler transform of A010692. - Alois P. Heinz, Oct 17 2008

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)
Index entries for expansions of Product_{k >= 1} (1-x^k)^m
N. J. A. Sloane, Transforms

Cf. 10th column of A144064. - Alois P. Heinz, Oct 17 2008

with(numtheory): a:= proc(n) option remember; `if`(n=0, 1, add(add(d*10, d=divisors(j)) *a(n-j), j=1..n)/n) end: seq(a(n), n=0..40); # Alois P. Heinz, Oct 17 2008

nmax=50; CoefficientList[Series[Product[1/(1-x^k)^10,{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Feb 28 2015 *)

G.f.: Product_{m>=1} 1/(1-x^m)^10.
a(n) ~ 5^(11/4) * exp(2 * Pi * sqrt(5*n/3)) / (64 * 3^(11/4) * n^(13/4)). - Vaclav Kotesovec, Feb 28 2015
a(0) = 1, a(n) = (10/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Mar 27 2017
G.f.: exp(10*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018

A317633 Numbers congruent to {1, 7, 9} mod 10.

Original entry on oeis.org

1, 7, 9, 11, 17, 19, 21, 27, 29, 31, 37, 39, 41, 47, 49, 51, 57, 59, 61, 67, 69, 71, 77, 79, 81, 87, 89, 91, 97, 99, 101, 107, 109, 111, 117, 119, 121, 127, 129, 131, 137, 139, 141, 147, 149, 151, 157, 159, 161, 167, 169

Offset: 1

Paul Curtz, Aug 02 2018

When multiplied by 10, one gets the numbers ending in "dix" in French (10, 70, 90, 110, ...).

			G.f. = x + 7*x^2 + 9*x^3+ 11*x^4 + 17*x^5 + 19*x^6 + 21*x^7 + 27*x^8 + ... - _Michael Somos_, Aug 19 2018

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A008592, A010692.

[n: n in [0..170]|n mod 10 in {1, 7, 9}]; // Vincenzo Librandi, Aug 05 2018

Table[2 n + 4 Floor[(n + 1)/3] - 1, {n, 1, 60}] (* Bruno Berselli, Jul 02 2018 *)
Select[Range[0, 250], MemberQ[{1, 7, 9}, Mod[#, 10]]&] (* Vincenzo Librandi, Aug 05 2018 *)
CoefficientList[ Series[(x^3 + 2x^2 + 6x + 1)/((x - 1)^2 (x^2 + x + 1)), {x, 0, 60}], x] (* or *)
LinearRecurrence[{1, 0, 1, -1}, {1, 7, 9, 11}, 61] (* Robert G. Wilson v, Aug 08 2018 *)

x='x+O('x^60); Vec(x*(1+6*x+2*x^2+x^3)/((1-x)^2*(1+x+x^2))) \\ G. C. Greubel, Aug 08 2018

a(n) = a(n-3) + 10, a(1) = 1, a(2) = 7, a(3) = 9.
From Bruno Berselli, Jul 02 2018: (Start)
G.f.: x*(1 + 6*x + 2*x^2 + x^3)/((1 - x)^2*(1 + x + x^2)).
a(n) = 2*n + 4*floor((n+1)/3) - 1. (End)

Definition from Jianing Song, Aug 02 2018

A166577 Inverse binomial transform of A166517.

Original entry on oeis.org

1, 4, -5, 10, -20, 40, -80, 160, -320, 640, -1280, 2560, -5120, 10240, -20480, 40960, -81920, 163840, -327680, 655360, -1310720, 2621440, -5242880, 10485760, -20971520, 41943040, -83886080, 167772160, -335544320, 671088640, -1342177280, 2684354560, -5368709120

Offset: 0

Paul Curtz, Oct 17 2009

The definition assumes that the offset of A166517 is changed to 0.
A166517 mod 9 yields a periodic sequence with period 1, 5, 4, 8, 7, 2.
This set of numbers in the period appears also in A153130, A146501, and A166304.

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

Cf. A010716, A010692, A010859.

Join[{1,4},NestList[-2#&,-5,40]] (* Harvey P. Dale, Aug 02 2012 *)
Join[{1, 4}, LinearRecurrence[{-2}, {-5}, 48]] (* G. C. Greubel, May 17 2016 *)

a(n) = -2*a(n-1), n>2.
a(n) = (-1)^(n+1)*A020714(n-2), n>1.
From Colin Barker, Jan 07 2013: (Start)
a(n) = -5*(-1)^n*2^(n-2) for n>1.
G.f.: (3*x^2+6*x+1)/(2*x+1). (End)
E.g.f.: (9/4) + (3/2)*x - (5/4)*exp(-2*x). - Alejandro J. Becerra Jr., Feb 15 2021

Edited, comments not concerning this sequence removed, and extended by R. J. Mathar, Oct 21 2009

A317095 a(n) = 40*n.

Original entry on oeis.org

0, 40, 80, 120, 160, 200, 240, 280, 320, 360, 400, 440, 480, 520, 560, 600, 640, 680, 720, 760, 800, 840, 880, 920, 960, 1000, 1040, 1080, 1120, 1160, 1200, 1240, 1280, 1320, 1360, 1400, 1440, 1480, 1520, 1560, 1600, 1640, 1680, 1720, 1760, 1800, 1840, 1880

Offset: 0

Felix Fröhlich, Sep 07 2018

a(n) is equal to the freshwater zone below sea level for a water table of elevation n above sea level in a simplified freshwater-saltwater interface in a coastal water-table aquifer (cf. Barlow, 2003, p. 14, eq. (2) and p. 15, Fig. B-1 and B-2).
From Bruno Berselli, Sep 10 2018: (Start)
After 0, subsequence of A065607: 1/a(n)^2 + 1/(30*n)^2 = 1/(24*n)^2, with n > 0 and a(n) > 30*n.
Also, all positive terms belong to A049094: 2^(40*n)-1 = 1024^(4*n)-1 and (25*41-1)^(4*n)-1 is divisible by 25. (End)

P. M. Barlow, Ground Water in Freshwater-Saltwater Environments of the Atlantic Coast, Circular 1262 (2003).
Wikipedia, Aquifer
Wikipedia, Saltwater intrusion
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A005843, A007395, A008586, A008592, A008602, A010692, A049094, A065607.
Row n = 40 of A004247. Intersection of A008587 and A008590.
After 0, subsequence of A005101.

Table[40 n, {n, 0, 50}] (* or *)
LinearRecurrence[{2, -1}, {0, 40}, 50] (* or *)
CoefficientList[Series[40*x/(1 - x)^2, {x, 0, 50}], x] (* Stefano Spezia, Sep 07 2018 *)

PARI
```
a(n) = 40*n
```

a(n) = if(n==0, 0, if(n==1, 40, 2*a(n-1)-a(n-2)))

PARI
```
concat(0, Vec(40*x/(1-x)^2 + O(x^60)))
```

O.g.f.: 40*x/(1 - x)^2.
E.g.f.: 40*x*exp(x). - Bruno Berselli, Sep 10 2018
a(n) = 2*a(n - 1) - a(n - 2) for n > 1. - Stefano Spezia, Sep 07 2018
a(n) = A008586(A008592(n)) = 4*A008592(n).
a(n) = A010692(n)*A008586(n) = 10*A008586(n).
a(n) = A008602(A005843(n)) = 20*A005843(n).
a(n) = A007395(n)*A008602(n) = 2*A008602(n).

A140724 Period 10: 1, 5, 9, 7, 7, 9, 5, 1, 3, 3 repeated.

Original entry on oeis.org

1, 5, 9, 7, 7, 9, 5, 1, 3, 3, 1, 5, 9, 7, 7, 9, 5, 1, 3, 3, 1, 5, 9, 7, 7, 9, 5, 1, 3, 3, 1, 5, 9, 7, 7, 9, 5, 1, 3, 3, 1, 5, 9, 7, 7, 9, 5, 1, 3, 3, 1, 5, 9, 7, 7, 9, 5, 1, 3, 3, 1, 5, 9, 7, 7, 9, 5, 1, 3, 3, 1, 5, 9, 7, 7, 9, 5, 1, 3, 3, 1, 5, 9, 7, 7, 9, 5, 1, 3, 3, 1, 5, 9, 7, 7, 9, 5, 1, 3, 3, 1, 5, 9, 7, 7

Offset: 0

Paul Curtz, Jul 12 2008

The last digit of A108981(n).
Also the continued fraction of (290003+sqrt(240183699293))/652402.
Also the decimal expansion of 13073/81819.
The period contains each of the 5 odd digits twice.

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

PadRight[{},120,{1,5,9,7,7,9,5,1,3,3}] (* Harvey P. Dale, Apr 23 2020 *)

13073/81819. \\ Charles R Greathouse IV, Jul 21 2015

a(n)+a(n+5) = 10 = A010692(n).
a(n) = a(n+10) .
a(10*k+9+i) = a(10*k+18-i) (palindromic).
a(n) = 2*a(n-1)-2*a(n-2)+2*a(n-3)-2*a(n-4)+a(n-5). G.f.: -(1+3*x+x^2-3*x^3+3*x^4)/ ((x-1) * (x^4-x^3+x^2-x+1)).

Edited by R. J. Mathar, Sep 07 2009

A324471 a(n) = 10 mod n.

Original entry on oeis.org

0, 0, 1, 2, 0, 4, 3, 2, 1, 0, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10

Offset: 1

N. J. A. Sloane, Mar 07 2019, following a suggestion from Charles Kusniec

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

Cf. A010692, A010879, A048158, A051127, A090976, A324472.

Mod[10,Range[100]] (* Paolo Xausa, Nov 14 2023 *)

From Elmo R. Oliveira, Aug 03 2024: (Start)
G.f.: x^3*(1 + x - 2*x^2 + 4*x^3 - x^4 - x^5 - x^6 - x^7 + 10*x^8)/(1 - x).
a(n) = 10 for n > 10. (End)

cp's OEIS Frontend

A168668 a(n) = n*(2 + 5*n).

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A040020 Continued fraction for sqrt(26).

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A010726 Period 2: repeat (6,10).

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A176537 Decimal expansion of 5 + sqrt(26).

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A023009 Number of partitions of n into parts of 10 kinds.

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A317633 Numbers congruent to {1, 7, 9} mod 10.

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A166577 Inverse binomial transform of A166517.

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A168668 a(n) = n(2 + 5n).