A006496 -id:A006496 - cp's OEIS frontend

A045873 a(n) = A006496(n)/2.

0, 1, 2, -1, -12, -19, 22, 139, 168, -359, -1558, -1321, 5148, 16901, 8062, -68381, -177072, -12239, 860882, 1782959, -738492, -10391779, -17091098, 17776699, 121008888, 153134281, -298775878, -1363223161, -1232566932

Offset: 0

N. J. A. Sloane

Partial sums of A006495. - Paul Barry, Mar 16 2006
This is the Lucas U(P=2,Q=5) sequence. - R. J. Mathar, Oct 24 2012
With different signs, 0, 1, -2, -1, 12, -19, -22, 139, -168, -359, 1558, ... we obtain the Lucas U(-2,5) sequence. - R. J. Mathar, Jan 08 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Ronald Orozco López, Deformed Differential Calculus on Generalized Fibonacci Polynomials, arXiv:2211.04450 [math.CO], 2022.
Wikipedia, Lucas sequence
Index entries for linear recurrences with constant coefficients, signature (2,-5).
Index entries for Lucas sequences

Cf. A006495, A006496, A084102, A088136, A088137, A088139, A094423.

a:=[0,1];; for n in [3..30] do a[n]:=2*a[n-1]-5*a[n-2]; od; a; # Muniru A Asiru, Oct 23 2018

I:=[0,1]; [n le 2 select I[n] else 2*Self(n-1) - 5*Self(n-2): n in [1..50]]; // G. C. Greubel, Oct 22 2018

seq(coeff(series(x/(1-2*x+5*x^2),x,n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Oct 23 2018

LinearRecurrence[{2,-5}, {0,1}, 40] (* G. C. Greubel, Jan 11 2024 *)

concat(0,Vec(1/(1-2*x+5*x^2)+O(x^99))) \\ Charles R Greathouse IV, Dec 22 2011

[lucas_number1(n,2,5) for n in range(0, 29)] # Zerinvary Lajos, Apr 23 2009

A045873=BinaryRecurrenceSequence(2,-5,0,1)
[A045873(n) for n in range(41)] # G. C. Greubel, Jan 11 2024

a(n)^2 = A094423(n).
From Paul Barry, Sep 20 2003: (Start)
O.g.f.: x/(1 - 2*x + 5*x^2).
E.g.f.: exp(x)*sin(2*x)/2.
a(n) = 2*a(n-1) - 5*a(n-2), a(0)=0, a(1)=1.
a(n) = ((1 + 2*i)^n - (1 - 2*i)^n)/(4*i), where i=sqrt(-1).
a(n) = Im{(1 + 2*i)^n/2}.
a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2k+1)*(-4)^k. (End)
a(n+1) = Sum_{k=0..n} binomial(k,n-k)*2^k*(-5/2)^(n-k). - Paul Barry, Mar 16 2006
G.f.: 1/(4*x - 1/G(0)) where G(k) = 1 - (k+1)/(1 - x/(x - (k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Dec 06 2012
G.f.: Q(0)*x/2, where Q(k) = 1 + 1/(1 - x*(4*k+2 - 5*x)/( x*(4*k+4 - 5*x) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 30 2013
a(n) = 5^((n-1)/2)*ChebyshevU(n-1, 1/sqrt(5)). - G. C. Greubel, Jan 11 2024

More terms from Paul Barry, Sep 20 2003

A000351 Powers of 5: a(n) = 5^n.

Original entry on oeis.org

1, 5, 25, 125, 625, 3125, 15625, 78125, 390625, 1953125, 9765625, 48828125, 244140625, 1220703125, 6103515625, 30517578125, 152587890625, 762939453125, 3814697265625, 19073486328125, 95367431640625, 476837158203125, 2384185791015625, 11920928955078125

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 5), L(1, 5), P(1, 5), T(1, 5). Essentially same as Pisot sequences E(5, 25), L(5, 25), P(5, 25), T(5, 25). See A008776 for definitions of Pisot sequences.
a(n) has leading digit 1 if and only if n = A067497 - 1. - Lekraj Beedassy, Jul 09 2002
With interpolated zeros 0, 1, 0, 5, 0, 25, ... (g.f.: x/(1 - 5*x^2)) second inverse binomial transform of Fibonacci(3n)/Fibonacci(3) (A001076). Binomial transform is A085449. - Paul Barry, Mar 14 2004
Sums of rows of the triangles in A013620 and A038220. - Reinhard Zumkeller, May 14 2006
Sum of coefficients of expansion of (1 + x + x^2 + x^3 + x^4)^n. a(n) is number of compositions of natural numbers into n parts less than 5. a(2) = 25 there are 25 compositions of natural numbers into 2 parts less than 5. - Adi Dani, Jun 22 2011
The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n >= 1, a(n) equals the number of 5-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011
Numbers n such that sigma(5n) = 5n + sigma(n). In fact we have this theorem: p is a prime if and only if all solutions of the equation sigma(p*x) = p*x + sigma(x) are powers of p. - Jahangeer Kholdi, Nov 23 2013
From Doug Bell, Jun 22 2015: (Start)
Empirical observation: Where n is an odd multiple of 3, let x = (a(n) + 1)/9 and let y be the decimal expansion of x/a(n); then y*(x+1)/x + 1 = y rotated to the left.
Example:
  a(3) = 125;
  x = (125 + 1)/9 = 14;
  y = 112, which is the decimal expansion of 14/125 = 0.112;
  112*(14 + 1)/14 + 1 = 121 = 112 rotated to the left.
(End)
a(n) is the number of n-digit integers that contain only odd digits (A014261). - Bernard Schott, Nov 12 2022
Number of pyramids in the Sierpinski fractal square-based pyramid at the n-th step, while A279511 gives the corresponding number of vertices (see IREM link with drawings). - Bernard Schott, Nov 29 2022

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=0..100
O. M. Cain, The Exceptional Selfcondensability of Powers of Five, arXiv:1910.13829 [math.HO], 2019.
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 270
IREM Paris-Nord, La pyramide de Sierpinski (in French).
Tanya Khovanova, Recursive Sequences
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
Yash Puri and Thomas Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Eric Weisstein's World of Mathematics, Box Fractal
Index entries for linear recurrences with constant coefficients, signature (5).

Cf. A009969 (even bisection), A013710 (odd bisection), A005054 (first differences), A003463 (partial sums).
Cf. A006495, A006496, A159991, A001021.
Cf. A001076, A008776, A013620, A038220, A067497, A085449, A014261.
Sierpinski fractal square-based pyramid: A020858 (Hausdorff dimension), A279511 (number of vertices), this sequence (number of pyramids).

a000351 = (5 ^)
a000351_list = iterate (* 5) 1  -- Reinhard Zumkeller, Oct 31 2012

[5^n : n in [0..30]]; // Wesley Ivan Hurt, Sep 27 2016

[ seq(5^n,n=0..30) ];
A000351:=-1/(-1+5*z); # Simon Plouffe in his 1992 dissertation

Table[5^n, {n, 0, 30}] (* Stefan Steinerberger, Apr 06 2006 *)
5^Range[0, 30] (* Harvey P. Dale, Aug 22 2011 *)

makelist(5^n,n,0,20); /* Martin Ettl, Dec 27 2012 */

a(n)=5^n \\ Charles R Greathouse IV, Jun 10 2011

def a(n): return 5**n
print([a(n) for n in range(24)]) # Michael S. Branicky, Nov 12 2022

(List.fill(50)(5: BigInt)).scanLeft(1: BigInt)( * ) // Alonso del Arte, May 31 2019

a(n) = 5^n.
a(0) = 1; a(n) = 5*a(n-1) for n > 0.
G.f.: 1/(1 - 5*x).
E.g.f.: exp(5*x).
a(n) = A006495(n)^2 + A006496(n)^2.
a(n) = A159991(n) / A001021(n). - Reinhard Zumkeller, May 02 2009
From Bernard Schott, Nov 12 2022: (Start)
Sum_{n>=0} 1/a(n) = 5/4.
Sum_{n>=0} (-1)^n/a(n) = 5/6. (End)
a(n) = Sum_{k=0..n} C(2*n+1,n-k)*A000045(2*k+1). - Vladimir Kruchinin, Jan 14 2025

A006495 Real part of (1 + 2*i)^n, where i is sqrt(-1).

Original entry on oeis.org

1, 1, -3, -11, -7, 41, 117, 29, -527, -1199, 237, 6469, 11753, -8839, -76443, -108691, 164833, 873121, 922077, -2521451, -9653287, -6699319, 34867797, 103232189, 32125393, -451910159, -1064447283, 130656229, 5583548873

Offset: 0

N. J. A. Sloane

Row sums of the Euler related triangle A117411. Partial sums are A006495. - Paul Barry, Mar 16 2006
Binomial transform of [1, 0, -4, 0, 16, 0, -64, 0, 256, 0, ...], i.e. powers of -4 with interpolated zeros. - Philippe Deléham, Dec 02 2008
The absolute values of these numbers are the odd numbers y such that x^2 + y^2 = 5^n with x and y coprime. See A098122. - T. D. Noe, Apr 14 2011
Pisano period lengths: 1, 1, 8, 1, 4, 8, 48, 4, 24, 4, 60, 8, 12, 48, 8, 8, 16, 24, 90, 4, ... - R. J. Mathar, Aug 10 2012
Multiplied by a signed sequence of 2's we obtain 2, -2, -6, 22, -14, -82, 234, -58, -1054, 2398, 474, -12938, ..., the Lucas V(-2,5) sequence. - R. J. Mathar, Jan 08 2013

			1 + x - 3*x^2 - 11*x^3 - 7*x^4 + 41*x^5 + 117*x^6 + 29*x^7 - 527*x^8 + ...

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

Vincenzo Librandi, Table of n, a(n) for n = 0..200 (a(88) onwards corrected by Sean A. Irvine, Apr 29 2019)
Beata Bajorska-Harapińska, Barbara Smoleń, and Roman Wituła, On Quaternion Equivalents for Quasi-Fibonacci Numbers, Shortly Quaternaccis, Advances in Applied Clifford Algebras (2019) Vol. 29, 54.
G. Berzsenyi, Gaussian Fibonacci numbers, Fib. Quart., 15 (1977), 233-236.
Wikipedia, Lucas sequence.
Index entries for Lucas sequences.
Index entries for Gaussian integers and primes.
Index entries for linear recurrences with constant coefficients, signature (2,-5).

Cf. A006496, A045873 (partial sums).

A006495:=func< n | Integers()!Real((1+2*Sqrt(-1))^n) >; [ A006495(n): n in [0..30] ]; // Klaus Brockhaus, Feb 04 2011

a := n -> hypergeom([1/2 - n/2, -n/2], [1/2], -4):
seq(simplify(a(n)), n=0..28); # Peter Luschny, Jul 26 2020

Table[Re[(1+2I)^n],{n,0,29}] (* Giovanni Resta, Mar 28 2006 *)

{a(n) = local(A); n++; if( n<1, 0, A = vector(n); A[1] = 1; for( k=2, n, A[k] = (4*k + 1) * A[k-1] - 8 * sum( j=1, k-1, A[j] * A[k-j])); A[n])} /* Michael Somos, Jul 23 2011 */

a(n) = real( (1 + 2*I)^n ) \\ Charles R Greathouse IV, Nov 21 2014

{a(n) = my(A=1);
A = sum(m=0, n+1, (1 + (-1)^m*I)^m * x^m / (1 - (-1)^m*I*x +x*O(x^n))^(m+1) ); polcoeff(A, n)} \\ Paul D. Hanna, Mar 09 2019

[lucas_number2(n,2,5)/2 for n in range(0,30)] # Zerinvary Lajos, Jul 08 2008

a(n) = (1/2)*((1+2*i)^n + (1-2*i)^n). - Benoit Cloitre, Oct 28 2002
From Paul Barry, Mar 16 2006: (Start)
G.f.: (1-x)/(1 - 2*x + 5*x^2);
a(n) = 2*a(n-1) - 5*a(n-2);
a(n) = 5^(n/2)*cos(n*atan(1/3) + Pi*n/4);
a(n) = Sum_{k=0..n} Sum_{j=0..n-k} C(n,k-j)*C(j,n-k)*(-4)^(n-k). (End)
A000351(n) = a(n)^2 + A006496(n)^2. - Fabrice Baubet (intih(AT)free.fr), May 28 2007
a(n) = upper left and lower right terms of the 2 X 2 matrix [1,-2; 2,1]^n. - Gary W. Adamson, Mar 28 2008
a(n) = Sum_{k=0..n} A124182(n,k)*(-5)^(n-k). - Philippe Deléham, Nov 01 2008
a(n) = Sum_{k=0..n} A098158(n,k)*(-4)^(n-k). - Philippe Deléham, Nov 14 2008
a(n) = (4*n+5)*a(n-1) - 8*Sum_{k=1..n} a(k-1)*a(n-k) if n > 0. - Michael Somos, Jul 23 2011
E.g.f.: exp(x)*cos(2*x). - Sergei N. Gladkovskii, Jul 22 2012
a(n) = 5^(n/2) * cos(n*arctan(2)). - Sergei N. Gladkovskii, Aug 13 2012
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(4*k+1)/(x*(4*k+5) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013
From Paul D. Hanna, Mar 09 2019: (Start)
G.f.: Sum_{n>=0} (1 + (-1)^n*i)^n * x^n / (1 - (-1)^n*i*x)^(n+1).
G.f.: Sum_{n>=0} (1 - (-1)^n*i)^n * x^n / (1 + (-1)^n*i*x)^(n+1).
(End)
a(n) = hypergeom([1/2 - n/2, -n/2], [1/2], -4). - Peter Luschny, Jul 26 2020

Signs from Christian G. Bower, Nov 15 1998

Corrected by Giovanni Resta, Mar 28 2006

A139011 Real part of (2 + i)^n, where i = sqrt(-1).

Original entry on oeis.org

1, 2, 3, 2, -7, -38, -117, -278, -527, -718, -237, 2642, 11753, 33802, 76443, 136762, 164833, -24478, -922077, -3565918, -9653287, -20783558, -34867797, -35553398, 32125393, 306268562, 1064447283, 2726446322, 5583548873, 8701963882

Offset: 0

Gary W. Adamson, Apr 05 2008

Imaginary part of (2 + i)^n gives A099456.
Irrespective of signs, odd-indexed terms of A006496 interleaved with even-indexed signs of A006495.
Binomial transform of A146559, second binomial transform of A056594. - Philippe Deléham, Dec 02 2008

			1 + 2*x + 3*x^2 + 2*x^3 - 7*x^4 - 38*x^5 - 117*x^6 - 278*x^7 - 527*x^8 + ...
a(5) = -38 since (2 + i)^5 = (-38 + 41*i).
a(5) = -38 since [2,-1; 1,2]^5 = [ -38,-41; 41,-38], where 41 = A099456(5).
a(5) = -38 = A006496(5).

Seiichi Manyama, Table of n, a(n) for n = 0..2862 (first 201 terms from Vincenzo Librandi)
Beata Bajorska-Harapińska, Barbara Smoleń, and Roman Wituła, On Quaternion Equivalents for Quasi-Fibonacci Numbers, Shortly Quaternaccis, Advances in Applied Clifford Algebras (2019) Vol. 29, 54.
Index entries for linear recurrences with constant coefficients, signature (4,-5).

Cf. A099456, A006495, A006496, A056594, A146559 (inv bin. transf.).

[ Integers()!Real((2+Sqrt(-1))^n): n in [0..29] ];  // Bruno Berselli, Apr 26 2011

restart: G(x):=exp(x)^2*cos(x): f[0]:=G(x): for n from 1 to 54 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=1..29 ); # Zerinvary Lajos, Apr 06 2009

Re[(2+I)^Range[0,30]] (* or *) LinearRecurrence[{4,-5},{1,2},30] (* Harvey P. Dale, Nov 02 2022 *)

a(n) = real((2 + I)^n) /* Michael Somos, Dec 26 2009 */

Vec((1 - 2*x) / (1 - 4*x + 5*x^2) + O(x^30)) \\ Colin Barker, Sep 22 2017

[lucas_number2(n,4,5)/2 for n in range(0,31)] # Zerinvary Lajos, Jul 08 2008

Real part of (2 + i)^n, i^2 = -1.
Term (1,1) of matrix [2,-1; 1,2]^n.
(a(n))^2 + (A099456(n))^2 = 5^n.
From R. J. Mathar, Apr 06 2008: (Start)
O.g.f.: (1-2x) /(1-4x+5x^2).
a(n) = 4*a(n-1) - 5*a(n-2) = 2*A099456(n-1) - 5*A099456(n-2). (End)
E.g.f.: exp(x)^2*cos(x). - Zerinvary Lajos, Apr 06 2009
a(-n) = a(n) / 5^n. - Michael Somos, Dec 26 2010
a(n) = Sum_{k=0..n} A098158(n,k)*2^(2k-n)*(-1)^(n-k). - Philippe Deléham, Dec 02 2008
2*a(n) - a(n+1) = A099456(n-1) for n>0. First differences are (up to sign) A118444. - Paul Curtz, Apr 25 2011
a(n) = Sum_{k=0..n} A201730(n,k)*(-2)^k. - Philippe Deléham, Dec 06 2011
a(n) = Sum_{k=0..floor(n/2)} (-1)^k*2^(n-2*k)*binomial(n,2*k). - Gerry Martens, Sep 18 2022

Cross-reference corrected by Franklin T. Adams-Watters, Jan 06 2009
Added a(0)=1 by Michael Somos, Dec 26 2010
Edited by Franklin T. Adams-Watters, Apr 10 2011

A098122 Let (A,B)=(a(2n),a(2n+1)), then (A,B) is (even,odd), gcd(A,B)=1 and A^2 + B^2 = 5^n. Note: a(0)=0.

Original entry on oeis.org

0, 1, 2, 1, 4, 3, 2, 11, 24, 7, 38, 41, 44, 117, 278, 29, 336, 527, 718, 1199, 3116, 237, 2642, 6469, 10296, 11753, 33802, 8839, 16124, 76443, 136762, 108691, 354144, 164833, 24478, 873121, 1721764, 922077, 3565918, 2521451, 1476984, 9653287

Offset: 0

James R. Buddenhagen, Sep 24 2004

nonn

(a(4*n),a(4*n+1)) are legs of the unique Pythagorean right triangle with hypotenuse 5^n and relatively prime legs.

			(a(2*3),a(2*3+1)) = (2,11) because (2,11) are (even,odd), relatively prime and 2^2 + 11^2 = 5^3. There is just one such pair.

Jacobi, C. G. J. (1829) Fundamenta Nova Theoriae Functionum Ellipticarum, Regiomonti, Sumptibus fratrum Borntraeger; reprinted in Jacobi, C. G. J. (1881-1891) Gesammelte Werke (Reimer, Berlin), Vol. 1, pp. 49-239 [reprinted (1969) by Chelsea, New York; now distributed by Am. Mathematical Soc., Providence, RI].

Elias M. Stein and Rami Shakarchi, Complex Analysis, Ch. 10.
XIAO Gang, Two Squares (a section of WWW Interactive Multipurpose Server)
Eric Weisstein's World of Mathematics, Sum of Squares Function

Cf. A006495, A006496 (the odd and even numbers separately).

A292495 Triangle read by rows: T(n,k) = (-2)T(n-1,k-1) + T(n,k-1) with T(2m,0) = 0 and T(2*m+1,0) = (-1)^m.

Original entry on oeis.org

0, 1, 1, 0, -2, -4, -1, -1, 3, 11, 0, 2, 4, -2, -24, 1, 1, -3, -11, -7, 41, 0, -2, -4, 2, 24, 38, -44, -1, -1, 3, 11, 7, -41, -117, -29, 0, 2, 4, -2, -24, -38, 44, 278, 336, 1, 1, -3, -11, -7, 41, 117, 29, -527, -1199, 0, -2, -4, 2, 24, 38, -44, -278, -336, 718

Offset: 0

Seiichi Manyama, Sep 22 2017

			First few rows are:
   0;
   1,  1;
   0, -2, -4;
  -1, -1,  3,  11;
   0,  2,  4,  -2, -24;
   1,  1, -3, -11,  -7,  41;
   0, -2, -4,   2,  24,  38,  -44;
  -1, -1,  3,  11,   7, -41, -117, -29;
   0,  2,  4,  -2, -24, -38,   44, 278, 336.

Seiichi Manyama, Rows n = 0..139, flattened

The diagonal of the triangle is related to A099456.
The next diagonal of the triangle is related to A139011.
Cf. A006495, A006496.
T(n,k) = b*T(n-1,k-1) + T(n,k-1): A292789 (b=-3), this sequence (b=-2), A117918 and A228405 (b=1), A227418 (b=2), A292466 (b=4).

T(n+1,n)^2 + T(n,n)^2 = 5^n.

A349196 a(n) is the Y-coordinate of the n-th point of the R5 dragon curve; A349195 gives X-coordinates.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 3, 3, 2, 2, 3, 3, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 4, 4, 5, 5, 4, 4, 3, 3, 2, 2, 3, 3, 2, 2, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, -1, -1, -2, -2, -1, -1, -2, -2, -1, -1, 0, 0, -1, -1, 0, 0, -1, -1, -2, -2, -3, -3, -2, -2, -3, -3

Offset: 0

Rémy Sigrist, Nov 10 2021

sign

The R5 dragon curve can be represented using an L-system.

			The R5 dragon curve starts as follows:
         +-----+
       24|   25
         |
         |
         +-----+     +-----+     +-----+
       23    22|   11|   10|    7|    6|
               |     |     |     |     |
             21|   12|    9|    8|     |
         +-----+-----+-----+-----+-----+
       20|   17|   16|   13|    4|    5
         |     |     |     |     |
         |     |     |     |     |
         +-----+     +-----+     +-----+
       19    18    15    14     3     2|
                                       |
                                       |
                                 +-----+
                                0     1
- so a(0) = a(1) = 0,
     a(2) = a(3) = a(14) = a(15) = a(18) = a(19) = 1.

Rémy Sigrist, Table of n, a(n) for n = 0..3125
Kevin Ryde, Iterations of the R5 Dragon Curve
Rémy Sigrist, Colored representation of the first 1 + 5^9 points of the R5 dragon curve (where the hue is function of the number of steps from the origin)
Rémy Sigrist, PARI program for A349196
Index entries for sequences related to coordinates of 2D curves

Cf. A006496, A349008, A349009, A349010, A349195.

PARI
```
See Links section.
```

a(5^k) = A006496(k) for any k >= 0.

A121705 Triangle read by rows: 5^n expressed as the sum of two squares.

Original entry on oeis.org

0, 1, 1, 2, 0, 5, 3, 4, 2, 11, 5, 10, 0, 25, 7, 24, 15, 20, 10, 55, 25, 50, 38, 41, 0, 125, 35, 120, 44, 117, 75, 100, 29, 278, 50, 275, 125, 250, 190, 205, 0, 625, 175, 600, 220, 585, 336, 527, 375, 500, 145, 1390, 250, 1375, 625, 1250, 718, 1199, 950, 1025, 0, 3125

Offset: 0

Zak Seidov, Sep 10 2006

			5^n expressed as the sum of two squares: 5^n = x^2 + y^2, 0 <= x < y.
Number of solutions for n=0,1,...: a(n)=1,1,2,2,3,3,4,4,5,5,6,6,...
Triangle of solutions for n=0,1,...:
  {x,y}
  {{0,1}},
  {{1,2}},
  {{0,5},{3,4}},
  {{2,11},{5,10}},
  {{0,25},{7,24},{15,20}},
  {{10,55},{25,50},{38,41}},
  {{0,125},{35,120},{44,117},{75,100}},
  {{29,278},{50,275},{125,250},{190,205}},
  {{0,625},{175,600},{220,585},{336,527},{375,500}},
  {{145,1390},{250,1375},{625,1250},{718,1199},{950,1025}},
  {{0,3125},{237,3116},{875,3000},{1100,2925},{1680,2635},{1875,2500}},
  {{725,6950},{1250,6875},{2642,6469},{3125,6250},{3590,5995},{4750,5125}},
  {{0,15625},{1185,15580},{4375,15000},{5500,14625},{8400,13175},{9375,12500},{10296,11753}}

Cf. A004431, A006496, A024507, A120905.

cp's OEIS Frontend

A045873 a(n) = A006496(n)/2.

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A000351 Powers of 5: a(n) = 5^n.

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A006495 Real part of (1 + 2*i)^n, where i is sqrt(-1).

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A139011 Real part of (2 + i)^n, where i = sqrt(-1).

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A098122 Let (A,B)=(a(2*n),a(2*n+1)), then (A,B) is (even,odd), gcd(A,B)=1 and A^2 + B^2 = 5^n. Note: a(0)=0.

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A098122 Let (A,B)=(a(2n),a(2n+1)), then (A,B) is (even,odd), gcd(A,B)=1 and A^2 + B^2 = 5^n. Note: a(0)=0.

A292495 Triangle read by rows: T(n,k) = (-2)T(n-1,k-1) + T(n,k-1) with T(2m,0) = 0 and T(2*m+1,0) = (-1)^m.