A000351 -id:A000351 - cp's OEIS frontend

A230710 Values of x such that x^2 + y^2 = 5^n with x and y coprime and 0 < x < y.

1, 3, 2, 7, 38, 44, 29, 336, 718, 237, 2642, 10296, 8839, 16124, 108691, 164833, 24478, 922077, 2521451, 1476984, 6699319, 34182196, 35553398, 32125393, 306268562, 597551756, 130656229, 2465133864, 8701963882, 6890111163, 15949374758, 98248054847, 135250416961

Offset: 1

Colin Barker, Oct 28 2013

nonn

The corresponding y-values are in A230711.
For all non-coprime solutions (x,y) to the equation x^2 + y^2 = p^n, x and y are both divisible by the prime p.
Using de Moivre's Theorem (in essence), define (c,d)*(e,f) as (ce-df,cf+de). Then a(n) = min{|u(n)|, |v(n)|}, where (u(n),v(n)) = (2,1)^n = (2,1)*(2,1)^[n-1]. Proof: It can be readily seen that u^2(n) + v^2(n) = 5^n. To show that u(n) and v(n) are relatively prime, assume that x,y are relatively prime. Then (2,1)*(x,y) = (2x-y, x+2y). If a prime p were to divide both of 2x-y and x+2y, then p would divide 5y, so p=5. Now suppose x == 2 (mod 5) and y == 1 (mod 5). It can be seen that 2x-y == -2 (mod 5) and x+2y == -1 (mod 5). The reverse also holds. Because u(1)=2 and v(1)=1, the result follows inductively. - Richard Peterson, May 21 2021

			a(4)=7 because 7^2 + 24^2 = 625 = 5^4.

Zak Seidov, Table of n, a(n) for n = 1..200
Chris Busenhart, Lorenz Halbeisen, Norbert Hungerbühler, and Oliver Riesen, On primitive solutions of the Diophantine equation x^2+ y^2= M, Eidgenössische Technische Hochschule (ETH Zürich, Switzerland, 2020).

Cf. A000351, A230711.
Cf. A188948, A230622, A230644.
Cf. A139011.

Table[Select[PowersRepresentations[5^n, 2, 2], CoprimeQ[#[[1]], #[[2]]] &][[1,1]], {n, 33}] (* T. D. Noe, Nov 04 2013 *)

Typo in data fixed by Colin Barker, Nov 02 2013

A308621 Number of ways to write n as a(a+1)/2 + b(b+1)/2 + 2^c*5^(3d) with a,b,c,d nonnegative integers.

Original entry on oeis.org

1, 2, 2, 3, 3, 2, 3, 5, 2, 4, 4, 3, 3, 5, 3, 3, 6, 4, 4, 5, 2, 6, 5, 4, 4, 5, 2, 4, 6, 3, 4, 8, 5, 3, 5, 4, 5, 8, 5, 5, 4, 3, 5, 7, 4, 5, 8, 4, 2, 8, 2, 6, 7, 4, 3, 4, 6, 5, 8, 4, 4, 6, 5, 5, 5, 5, 6, 8, 4, 6, 7, 4, 6, 10, 4, 4, 7, 5, 2, 10, 4, 7, 7, 4, 8, 4, 4, 7, 8, 2, 4, 9, 5, 5, 9, 5, 5, 7, 5, 5

Offset: 1

Zhi-Wei Sun, Jun 11 2019

nonn

Conjecture: a(n) > 0 for all n > 0. Equivalently, any positive integer n can be written as a^2 + b*(b+1) + 2^c*5^(3d) with a,b,c,d nonnegative integers.
This was motivated by A308584, and we verified a(n) > 0 for all n = 1..2*10^8. Then, on the author's request, Giovanni Resta verified the above conjecture for n up to 10^10. G. Resta also noted that 7272774228 cannot be written as a*(a+1)/2 + b*(b+1)/2 + 2^c*250^d with a,b,c,d nonnegative integers.
See also A308566, A308584 and A308623 for similar conjectures.

			a(1) = 1 with 1 = 0*1/2 + 0*1/2 + 2^0*5^(3*0).
a(78210) = 1 with 78210 = 85*86/2 + 385*386/2 + 2*5^3.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000079, A000217, A000351, A308566, A308584, A308623.

TQ[n_]:=TQ[n]=IntegerQ[Sqrt[8n+1]];
tab={};Do[r=0;Do[If[TQ[n-5^(3k)*2^m-x(x+1)/2],r=r+1],{k,0,Log[5,n]/3},{m,0,Log[2,n/5^(3k)]},{x,0,(Sqrt[4(n-5^(3k)*2^m)+1]-1)/2}];tab=Append[tab,r],{n,1,100}];Print[tab]

A013599 a(n) = nextprime(5^n) - 5^n.

Original entry on oeis.org

1, 2, 4, 2, 6, 12, 4, 12, 22, 26, 4, 14, 58, 6, 12, 42, 24, 2, 12, 56, 48, 24, 18, 38, 58, 14, 12, 38, 34, 62, 28, 92, 214, 122, 102, 168, 136, 18, 48, 102, 108, 126, 18, 126, 76, 108, 22, 204, 52, 122, 96, 114, 94, 14, 52, 38, 58, 248, 64, 56, 16, 102, 106

Offset: 0

James Kilfiger (mapdn(AT)csv.warwick.ac.uk)

nonn

Zak Seidov, Table of n, a(n) for n = 0..500

Cf. A000351, A013632, A054321, A151800,

Maple
```
seq(nextprime(5^i)-5^i, i=0..100);
```

NextPrime[#]-#&/@(5^Range[0,70]) (* Harvey P. Dale, Sep 29 2011 *)

a(n) = nextprime(5^n+1) - 5^n; \\ Michel Marcus, Jun 14 2020

a(n) = A151800(5^n)-5^n = A054321(n)-5^n = A013632(5^n). - R. J. Mathar, Nov 28 2016

A013620 Triangle of coefficients in expansion of (2+3x)^n.

Original entry on oeis.org

1, 2, 3, 4, 12, 9, 8, 36, 54, 27, 16, 96, 216, 216, 81, 32, 240, 720, 1080, 810, 243, 64, 576, 2160, 4320, 4860, 2916, 729, 128, 1344, 6048, 15120, 22680, 20412, 10206, 2187, 256, 3072, 16128, 48384, 90720, 108864, 81648, 34992, 6561, 512

Offset: 0

N. J. A. Sloane

Row sums give A000351; central terms give A119309. - Reinhard Zumkeller, May 14 2006

			Triangle begins:
1;
2,3;
4,12,9;
8,36,54,27;
16,96,216,216,81;

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
Gábor Kallós, A generalization of Pascal’s triangle using powers of base numbers, Annales mathématiques Blaise Pascal, 13 no. 1 (2006), p. 1-15.
Index entries for triangles and arrays related to Pascal's triangle

Cf. A038220, A000079, A000244, A013613.

a013620 n k = a013620_tabl !! n !! k
a013620_row n = a013620_tabl !! n
a013620_tabl = iterate (\row ->
   zipWith (+) (map (* 2) (row ++ [0])) (map (* 3) ([0] ++ row))) [1]
-- Reinhard Zumkeller, May 26 2013, Apr 02 2011

Flatten[Table[Binomial[i, j] 2^(i-j) 3^j, {i, 0, 10}, {j, 0, i}]] (* Vincenzo Librandi, Apr 22 2014 *)

G.f.: 1 / [1 - x(2+3y)].

T(n,k) = A007318(n,k) * A036561(n,k). - Reinhard Zumkeller, May 14 2006

A025622 Numbers of form 5^i*6^j, with i, j >= 0.

Original entry on oeis.org

1, 5, 6, 25, 30, 36, 125, 150, 180, 216, 625, 750, 900, 1080, 1296, 3125, 3750, 4500, 5400, 6480, 7776, 15625, 18750, 22500, 27000, 32400, 38880, 46656, 78125, 93750, 112500, 135000, 162000, 194400, 233280, 279936, 390625, 468750, 562500, 675000

Offset: 1

David W. Wilson

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000351 (subset), A000400 (another subset), A025651 (exponent of 5).

n = 10^6; Flatten[Table[5^i*6^j, {i, 0, Log[5, n]}, {j, 0, Log[6, n/5^i]}]] // Sort (* Amiram Eldar, Sep 25 2020 *)

Sum_{n>=1} 1/a(n) = (5*6)/((5-1)*(6-1)) = 3/2. - Amiram Eldar, Sep 25 2020
a(n) ~ exp(sqrt(2*log(5)*log(6)*n)) / sqrt(30). - Vaclav Kotesovec, Sep 25 2020
a(n) = 5^A025651(n) *6^A025659(n). - R. J. Mathar, Jul 06 2025

A032122 Number of reversible strings with n beads of 5 colors.

Original entry on oeis.org

1, 5, 15, 75, 325, 1625, 7875, 39375, 195625, 978125, 4884375, 24421875, 122078125, 610390625, 3051796875, 15258984375, 76294140625, 381470703125, 1907349609375, 9536748046875, 47683720703125

Offset: 0

Christian G. Bower

			For a(2)=15, the five achiral strings are AA, BB, CC, DD, and EE; the 10 (equivalent) chiral pairs are AB-BA, AC-CA, AD-DA, AE-EA, BC-CB, BD-DB, BE-EB, CD-DC, CE-EC, and DE-ED.

Vincenzo Librandi, Table of n, a(n) for n = 0..500
C. G. Bower, Transforms (2)
Index entries for linear recurrences with constant coefficients, signature (5,5,-25).

Column 5 of A277504.

Cf. A000351 (oriented), A032088(n>1) (chiral), A056451 (achiral).

I:=[1, 5, 15]; [n le 3 select I[n] else 5*Self(n-1)+ 5*Self(n-2)-25*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jan 31 2012

LinearRecurrence[{5, 5, -25}, {1, 5, 15}, 31] (* Vincenzo Librandi, Jan 31 2012 *)
k=5; Table[(k^n+k^Ceiling[n/2])/2,{n,0,30}] (*Robert A. Russell, Nov 25 2017*)

a(n)=(5^((n+1)\2)+5^n)/2 \\ Charles R Greathouse IV, Jan 31 2012

"BIK" (reversible, indistinct, unlabeled) transform of 5, 0, 0, 0...
a(n) = 1/2 * (5^n + 5^floor((n+1)/2)) = 5*A001447(n+1). - Ralf Stephan, Jul 07 2003
G.f.: (1-15*x^2) / ((1-5*x)*(1-5*x^2)). - Colin Barker, Jan 30 2012 [Adapted to offset 0 by Robert A. Russell, Nov 10 2018]
a(n) = 5*a(n-1) + 5*a(n-2) - 25*a(n-3). - Vincenzo Librandi, Jan 31 2012
a(n) = (A000351(n) + A056451(n)) / 2. - Robert A. Russell, Nov 10 2018

a(0)=1 prepended by Robert A. Russell, Nov 10 2018

A038220 Triangle whose (i,j)-th entry is binomial(i,j)3^(i-j)2^j.

Original entry on oeis.org

1, 3, 2, 9, 12, 4, 27, 54, 36, 8, 81, 216, 216, 96, 16, 243, 810, 1080, 720, 240, 32, 729, 2916, 4860, 4320, 2160, 576, 64, 2187, 10206, 20412, 22680, 15120, 6048, 1344, 128, 6561, 34992, 81648, 108864, 90720, 48384, 16128, 3072, 256

Offset: 0

N. J. A. Sloane

Row sums give A000351; central terms give A119309. - Reinhard Zumkeller, May 14 2006

Triangle of coefficients in expansion of (3 + 2x)^n, where n is a nonnegative integer. - Zagros Lalo, Jul 23 2018

			Triangle begins:
   1;
   3,   2;
   9,  12,   4;
  27,  54,  36,   8;
  81, 216, 216,  96,  16;
  ...

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 44, 48

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
B. N. Cyvin et al., Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, Match, No. 34 (Oct 1996), pp. 109-121.
Index entries for triangles and arrays related to Pascal's triangle

Cf. A013620, A000079, A000244, A013613, A038221.

a038220 n k = a038220_tabl !! n !! k
a038220_row n = a038220_tabl !! n
a038220_tabl = iterate (\row ->
   zipWith (+) (map (* 3) (row ++ [0])) (map (* 2) ([0] ++ row))) [1]
-- Reinhard Zumkeller, May 26 2013, Apr 02 2011

t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, 3 t[n - 1, k] + 2 t[n - 1, k - 1]]; Table[t[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Zagros Lalo, Jul 23 2018 *)
Table[CoefficientList[ Expand[(3 + 2x)^n], x], {n, 0, 9}] // Flatten  (* Zagros Lalo, Jul 23 2018 *)
Table[CoefficientList[Binomial[i, j] *3^(i - j)*2^j, x], {i, 0, 9}, {j, 0, i}] // Flatten (* Zagros Lalo, Jul 23 2018 *)

T(i,j)=binomial(i,j)*3^(i-j)*2^j \\ Charles R Greathouse IV, Jul 19 2016

T(n,k) = A007318(n,k) * A036561(n,k). - Reinhard Zumkeller, May 14 2006
G.f.: 1/(1 - 3*x - 2*x*y). - Ilya Gutkovskiy, Apr 21 2017
T(0,0) = 1; T(n,k) = 3 T(n-1,k) + 2 T(n-1,k-1) for k = 0...n; T(n,k)=0 for n or k < 0. - Zagros Lalo, Jul 23 2018

A049303 Numbers k such that k is a substring of 5^k.

Original entry on oeis.org

2, 5, 6, 7, 9, 19, 25, 32, 34, 36, 41, 54, 55, 56, 59, 62, 64, 67, 69, 70, 71, 75, 80, 81, 82, 84, 86, 87, 89, 92, 93, 95, 96, 111, 115, 125, 128, 140, 163, 166, 177, 178, 189, 192, 205, 212, 219, 221, 226, 233, 236, 242, 258, 259, 267, 294, 303, 309, 323, 327, 329

Offset: 1

David W. Wilson

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000351, A032740, A049301, A049302, A049304, A049305, A049306, A049307.

ss5nQ[n_]:=Module[{len=IntegerLength[n]},MemberQ[Partition[ IntegerDigits[ 5^n], len,1],IntegerDigits[n]]]; Select[Range[400],ss5nQ] (* Harvey P. Dale, Jan 06 2013 *)
Select[Range[350],SequenceCount[IntegerDigits[5^#],IntegerDigits[#]]>0&] (* Harvey P. Dale, Dec 27 2024 *)

A066770 a(n) = 5^nsin(2narctan(1/2)) or numerator of tan(2n*arctan(1/2)).

Original entry on oeis.org

4, 24, 44, -336, -3116, -10296, 16124, 354144, 1721764, 1476984, -34182196, -242017776, -597551756, 2465133864, 29729597084, 116749235904, -42744511676, -3175197967656, -17982575014036, -28515500892816, 278471369994004, 2383715742284424, 7340510203856444

Offset: 1

Barbara Haas Margolius, (b.margolius(AT)csuohio.edu), Jan 17 2002

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 430-433.

J. M. Borwein and R. Girgensohn, Addition theorems and binary expansions, Canadian J. Math. 47 (1995) 262-273.
E. Eckert, The group of primitive Pythagorean triangles, Mathematics Magazine 57 (1984) 22-27.
Steven R. Finch, Plouffe's Constant [Broken link]
Steven R. Finch, Plouffe's Constant [From the Wayback machine]
Simon Plouffe, The Computation of Certain Numbers Using a Ruler and Compass, J. Integer Seqs. Vol. 1 (1998), #98.1.3.
Index entries for linear recurrences with constant coefficients, signature (6,-25).

Cf. A066771, A000351 powers of 5 and also hypotenuse of right triangle with legs given by A066770 and A066771.

Note that A066770, A066771 and A000351 are primitive Pythagorean triples with hypotenuse 5^n. The offset of A000351 is zero, but the offset is 1 for A066770, A066771.

a[1] := 4/3; for n from 1 to 40 do a[n+1] := (4/3+a[n])/(1-4/3*a[n]):od: seq(abs(numer(a[n])), n=1..40);# a[n]=tan(2n arctan(1/2))

Table[ 5^n*Sin[2*n*ArcCot[2]] // Simplify, {n, 1, 23}] (* Jean-François Alcover, Mar 04 2013 *)

PARI
```
a(n)=imag((2+I)^(2*n))
```

G.f.: 4*x/(1-6*x+25*x^2). - Ralf Stephan, Jun 12 2003
a(n) = 5^n*sin(2*n*arctan(1/2)). A recursive formula for T(n) = tan(2*n*arctan(1/2)) is T(n+1) = (4/3+T(n))/(1-4/3*T(n)). Unsigned a(n) is the absolute value of numerator of T(n).
a(n) is the imaginary part of (2+I)^(2*n) = Sum_{k=0..n} 2^(2*n-2*k-1)*(-1)^k*binomial(2*n, 2*k+1). - Benoit Cloitre, Aug 03 2002
a(n) = 6*a(n-1)-25*a(n-2), n>2. - Gary Detlefs, Dec 11 2010
a(n) = 5^n*sin(n*x), where x = arcsin(4/5) = 0.927295218.. . - Gary Detlefs, Dec 11 2010

A081143 5th binomial transform of (0,0,0,1,0,0,0,0,......).

Original entry on oeis.org

0, 0, 0, 1, 20, 250, 2500, 21875, 175000, 1312500, 9375000, 64453125, 429687500, 2792968750, 17773437500, 111083984375, 683593750000, 4150390625000, 24902343750000, 147857666015625, 869750976562500, 5073547363281250

Offset: 0

Paul Barry, Mar 08 2003

Starting at 1, four-fold convolution of A000351 (powers of 5).

With a different offset, number of n-permutations (n=4)of 6 objects u, v, w, z, x, y with repetition allowed, containing exactly three u's. Example: a(4)=20 because we have uuuv, uuvu, uvuu, vuuu, uuuw, uuwu, uwuu, wuuu, uuuz, uuzu, uzuu, zuuu, uuux, uuxu, uxuu, xuuu, uuuy, uuyu, uyuu and yuuu. - Zerinvary Lajos, Jun 03 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (20,-150,500,-625).

Cf. A038846, A036216, A001789, A081144, A081135.

[5^(n-3) * Binomial(n, 3): n in [0..25]]; // Vincenzo Librandi, Aug 06 2013

seq(binomial(n,3)*5^(n-3), n=0..25); # Zerinvary Lajos, Jun 03 2008

CoefficientList[Series[x^3/(1-5x)^4, {x, 0, 30}], x] (* Vincenzo Librandi, Aug 06 2013 *)
LinearRecurrence[{20,-150,500,-625}, {0,0,0,1}, 30] (* Harvey P. Dale, Dec 24 2015 *)

vector(31, n, my(m=n-1); 5^(m-3)*binomial(m,3)) \\ G. C. Greubel, Mar 05 2020

[lucas_number2(n, 5, 0)*binomial(n,3)/5^3 for n in range(0, 22)] # Zerinvary Lajos, Mar 12 2009

a(n) = 20*a(n-1) - 150*a(n-2) + 500*a(n-3) - 625*a(n-4), with a(0)=a(1)=a(2)=0, a(3)=1.
a(n) = 5^(n-3)*binomial(n,3).
G.f.: x^3/(1-5*x)^4.
E.g.f.: x^3*exp(x)/6. - G. C. Greubel, Mar 05 2020
From Amiram Eldar, Jan 04 2022: (Start)
Sum_{n>=3} 1/a(n) = 240*log(5/4) - 105/2.
Sum_{n>=3} (-1)^(n+1)/a(n) = 540*log(6/4) - 195/2. (End)

cp's OEIS Frontend

A230710 Values of x such that x^2 + y^2 = 5^n with x and y coprime and 0 < x < y.

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A308621 Number of ways to write n as a*(a+1)/2 + b*(b+1)/2 + 2^c*5^(3d) with a,b,c,d nonnegative integers.

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A013599 a(n) = nextprime(5^n) - 5^n.

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A013620 Triangle of coefficients in expansion of (2+3x)^n.

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A025622 Numbers of form 5^i*6^j, with i, j >= 0.

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Mathematica

Formula

A032122 Number of reversible strings with n beads of 5 colors.

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Magma

Mathematica

PARI

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A038220 Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.

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A308621 Number of ways to write n as a(a+1)/2 + b(b+1)/2 + 2^c*5^(3d) with a,b,c,d nonnegative integers.

A038220 Triangle whose (i,j)-th entry is binomial(i,j)3^(i-j)2^j.

A066770 a(n) = 5^nsin(2narctan(1/2)) or numerator of tan(2n*arctan(1/2)).