A000351 -id:A000351 - cp's OEIS frontend

A179606 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + x)/(1 - 3x - 5x^2).

1, 4, 17, 71, 298, 1249, 5237, 21956, 92053, 385939, 1618082, 6783941, 28442233, 119246404, 499950377, 2096083151, 8788001338, 36844419769, 154473265997, 647641896836, 2715292020493, 11384085545659, 47728716739442

Offset: 0

Johannes W. Meijer, Jul 28 2010

a(n) represents the number of n-move routes of a fairy chess piece starting in the central square (m = 5) on a 3 X 3 chessboard. This fairy chess piece behaves like a king on the eight side and corner squares but on the central square the king goes crazy and turns into a red king, see A179596.
The sequence above corresponds to 24 red king vectors, i.e., A[5] vectors, with decimal values 27, 30, 51, 54, 57, 60, 90, 114, 120, 147, 150, 153, 156, 177, 180, 210, 216, 240, 282, 306, 312, 402, 408 and 432. These vectors lead for the corner squares to A015523 and for the side squares to A152187.
This sequence belongs to a family of sequences with g.f. (1 + (k-4)*x)/(1 - 3*x - k*x^2). Red king sequences that are members of this family are A007483 (k= 2), A015521 (k=4), A179606 (k=5; this sequence), A154964 (k=6), A179603 (k=7) and A179599 (k=8). We observe that there is no red king sequence for k=3. Other members of this family are A006190 (k=1), A133494 (k=0) and A168616 (k=-2).
Inverse binomial transform of A052918.
The sequence b(n+1) = 6*a(n), n >= 0 with b(0)=1, is a berserker sequence, see A180147. The b(n) sequence corresponds to 16 A[5] vectors with decimal values between 111 and 492. These vectors lead for the corner squares to sequence c(n+1)=4*A179606(n), n >= 0 with c(0)=1, and for the side squares to A180140. - Johannes W. Meijer, Aug 14 2010
Equals the INVERT transform of A063782: (1, 3, 10, 32, 104, ...). Example: a(3) = 71 = (1, 1, 4, 7) dot (32, 10, 3, 1) = (32 + 10 + 12 + 17). - Gary W. Adamson, Aug 14 2010

Indranil Ghosh, Table of n, a(n) for n = 0..1603
Index entries for linear recurrences with constant coefficients, signature (3,5).

Cf. A179597 (central square).

Cf. A072263, A072264, A152187, A197189.

with(LinearAlgebra): nmax:=22; m:=5; A[1]:= [0,1,0,1,1,0,0,0,0]: A[2]:= [1,0,1,1,1,1,0,0,0]: A[3]:= [0,1,0,0,1,1,0,0,0]: A[4]:= [1,1,0,0,1,0,1,1,0]: A[5]:= [0,0,0,1,1,1,0,0,1]: A[6]:= [0,1,1,0,1,0,0,1,1]: A[7]:= [0,0,0,1,1,0,0,1,0]: A[8]:= [0,0,0,1,1,1,1,0,1]: A[9]:= [0,0,0,0,1,1,0,1,0]: A:=Matrix([A[1],A[2],A[3],A[4],A[5],A[6],A[7],A[8],A[9]]): for n from 0 to nmax do B(n):=A^n: a(n):= add(B(n)[m,k],k=1..9): od: seq(a(n), n=0..nmax);

CoefficientList[Series[(1+x)/(1-3*x-5*x^2), {x, 0, 22}],x] (* or *) LinearRecurrence[{3,5,0},{1,4},23] (* Indranil Ghosh, Mar 05 2017 *)

print(Vec((1 + x)/(1- 3*x - 5*x^2) + O(x^23))); \\ Indranil Ghosh, Mar 05 2017

G.f.: (1+x)/(1 - 3*x - 5*x^2).
a(n) = A015523(n) + A015523(n+1).
a(n) = 3*a(n-1) + 5*a(n-2) with a(0) = 1 and a(1) = 4.
a(n) = ((29 + 7*sqrt(29))*A^(-n-1) + (29-7*sqrt(29))*B^(-n-1))/290 with A = (-3+sqrt(29))/10 and B = (-3-sqrt(29))/10
Limit_{k->oo} a(n+k)/a(k) = (-1)^(n+1)*A000351(n)*A130196(n)/(A015523(n)*sqrt(29) - A072263(n)) for n >= 1.

A193685 5-Stirling numbers of the second kind.

Original entry on oeis.org

1, 5, 1, 25, 11, 1, 125, 91, 18, 1, 625, 671, 217, 26, 1, 3125, 4651, 2190, 425, 35, 1, 15625, 31031, 19981, 5590, 740, 45, 1, 78125, 201811, 170898, 64701, 12250, 1190, 56, 1, 390625, 1288991, 1398097, 688506, 174951, 24150, 1806, 68, 1, 1953125, 8124571, 11075670, 6906145, 2263065, 416451, 44016, 2622, 81, 1, 9765625, 50700551, 85654261, 66324830, 27273730, 6427575, 900627, 75480, 3675, 95, 1

Offset: 0

Wolfdieter Lang, Oct 06 2011

This is the lower triangular Sheffer matrix (exp(5*x),exp(x)-1). For Sheffer matrices see the W. Lang link under A006232 with references, and the rules for the conversion to the umbral notation of S. Roman's book.
The general case is Sheffer (exp(r*x),exp(x)-1), r=0,1,..., corresponding to r-Stirling2 numbers with row and column offsets 0. See the Broder link for r-Stirling2 numbers with offset [r,r].
a(n,m), n >= m >= 0, gives the number of partitions of the set {1.2....,n+5} into m+5 nonempty distinct subsets such that 1,2,3,4 and 5 belong to distinct subsets.
a(n,m) appears in the following normal ordering of Bose operators a and a* satisfying the Lie algebra [a,a*]=1: (a*a)^n (a*)^5 = Sum_{m=0..n} a(n,m)*(a*)^(5+m)*a^m, n >= 0. See the Mikhailov papers, where a(n,m) = S(n+5,m+5,5).
  With a->D=d/dx and a*->x we also have
  (xD)^n x^5 = Sum_{m=0..n} a(n,m)*x^(5+m)*D^m, n >= 0.

			n\m  0    1    2   3  4  5 ...
0    1
1    5    1
2   25   11    1
3  125   91   18   1
4  625  671  217  26  1
5 3125 4651 2190 425 35  1
...
5-restricted S2: a(1,0)=5 from 1,6|2|3|4|5, 2,6|1|3|4|5,
3,6|1|2|4|5, 4,6|1|2|3|5 and 5,6|1|2|3|4.
Recurrence: a(4,2) = (5+2)*a(3,2)+ a(3,1) = 7*18 + 91 = 217.
Normal ordering (n=1): (xD)^1 x^5 = Sum_{m=0..1} a(1,m)*x^(5+m)*D^m = 5*x^5 + 1*x^6*D.
a(2,1) = Sum_{j=0..1} S1(5,5-j)*S2(7-j,6) = 1*21 - 10*1 = 11.

Vincenzo Librandi, Rows n = 0..100, flattened
Peter Bala, Generalized Dobinski formulas
Andrei Z. Broder, The r-Stirling numbers, Discrete Math. 49, 241-259 (1984)
A. Dzhumadildaev and D. Yeliussizov, Path decompositions of digraphs and their applications to Weyl algebra, arXiv preprint arXiv:1408.6764v1 [math.CO], 2014. [Version 1 contained many references to the OEIS, which were removed in Version 2. - _N. J. A. Sloane_, Mar 28 2015]
Askar Dzhumadil’daev and Damir Yeliussizov, Walks, partitions, and normal ordering, Electronic Journal of Combinatorics, 22(4) (2015), #P4.10.
V. V. Mikhailov, Ordering of some boson operator functions, J. Phys A: Math. Gen. 16 (1983) 3817-3827.
V. V. Mikhailov, Normal ordering and generalised Stirling numbers, J. Phys A: Math. Gen. 18 (1985) 231-235.

Cf. A048993, A143494, A143495, A143496.
Cf. A196834 (row sums), A196835 (alternating row sums).
Columns: A000351 (m=0), A005062 (m=1), A019757 (m=2), A028165 (m=3), ...

a[n_, m_] := Sum[ StirlingS1[5, 5-j]*StirlingS2[n+5-j, m+5], {j, 0, Min[5, n-m]}]; Flatten[ Table[ a[n, m], {n, 0, 10}, {m, 0, n}] ] (* Jean-François Alcover, Dec 02 2011, after Wolfdieter Lang *)

E.g.f. of row polynomials s(n,x):=Sum_{m=0..n} a(n,m)*x^m: exp(5*z + x(exp(z)-1)).
E.g.f. of column no. m (with leading zeros):
  exp(5*x)*((exp(x)-1)^m)/m!, m >= 0 (Sheffer).
O.g.f. of column no. m (without leading zeros):
  1/Product_{j=0..m} (1-(5+j)*x), m >= 0. (Compute the first derivative of the column e.g.f. and compare its Laplace transform with the partial fraction decomposition of the o.g.f. x^(m-1)/Product_{j=0..m} (1-(5+j)*x). This works for every r-restricted Stirling2 triangle.)
Recurrence: a(n,m) = (5+m)*a(n-1,m) + a(n-1,m-1), a(0,0)=1, a(n,m)=0 if n < m, a(n,-1)=0.
a(n,m) = Sum_{j=0..min(5,n-m)} S1(5,5-j)*S2(n+5-j,m+5), n >= m >= 0, with S1 and S2 the Stirling1 and Stirling2 numbers A008275 and A048993, respectively (see the Mikailov papers).
Dobinski-type formula for the row polynomials: R(n,x) = exp(-x)*Sum_{k>=0} k*(4+k)^(n-1)*x^(k-1)/k!. - Peter Bala, Jun 23 2014

A222281 T(n,k) = number of n X k 0..5 arrays with no entry increasing mod 6 by 5 rightwards or downwards, starting with upper left zero.

Original entry on oeis.org

1, 5, 5, 25, 105, 25, 125, 2205, 2205, 125, 625, 46305, 194485, 46305, 625, 3125, 972405, 17153945, 17153945, 972405, 3125, 15625, 20420505, 1513010465, 6354787485, 1513010465, 20420505, 15625, 78125, 428830605, 133450391205

Offset: 1

R. H. Hardin, Feb 14 2013

1/6 the number of 6-colorings of the grid graph P_n X P_k. - Andrew Howroyd, Jun 26 2017

			Table starts
........1................5......................25..........................125
........5..............105....................2205........................46305
.......25.............2205..................194485.....................17153945
......125............46305................17153945...................6354787485
......625...........972405..............1513010465................2354171487645
.....3125.........20420505............133450391205..............872117822449905
....15625........428830605..........11770577485085...........323081602357856985
....78125.......9005442705........1038187247574145........119687637492011211885
...390625.....189114296805.......91570083319317865......44339047670574481807485
..1953125....3971400232905.....8076654937439905005...16425682631297501047982145
..9765625...83399404891005...712376276332499775685.6084998755694142903356375385
.48828125.1751387502711105.62832938018547611186345
...
Some solutions for n=3, k=4:
..0..0..0..0....0..0..0..0....0..0..0..0....0..3..0..0....0..0..0..0
..4..2..0..1....1..2..0..4....0..0..0..1....0..0..3..1....0..2..3..0
..0..4..1..4....1..4..1..2....3..4..4..1....3..0..4..4....4..5..1..3

Andrew Howroyd, Table of n, a(n) for n = 1..325 (terms 1..111 from R. H. Hardin)

Columns 1-7 are A000351(n-1), 5*A009965(n-1), A222276, A222277, A222278, A222279, A222280.
Main diagonal is A068256.
Cf. A078099 (3 colorings), A222444 (4 colorings), A222144 (5 colorings), A198982 (unlabeled 6 colorings), A222340 (7 colorings), A222462 (8 colorings).

T(n, k) = 5 * (24*A198982(n,k) - 12*A198715(n,k) - 8*A207997(n,k) - 3) for n*k > 1. - Andrew Howroyd, Jun 27 2017

A303660 Number of ways to write 2*n+1 as p + 3^k + 5^m, where p is a prime, and k and m are nonnegative integers.

Original entry on oeis.org

0, 1, 2, 3, 3, 4, 4, 4, 4, 5, 3, 4, 4, 3, 6, 7, 5, 6, 8, 5, 5, 9, 6, 5, 8, 3, 6, 8, 4, 4, 7, 6, 4, 8, 6, 5, 9, 4, 4, 8, 3, 6, 8, 7, 4, 9, 6, 4, 9, 5, 5, 9, 6, 6, 11, 7, 7, 9, 5, 3, 8, 5, 3, 9, 7, 7, 11, 8, 8, 12

Offset: 1

Zhi-Wei Sun, Apr 28 2018

nonn

Note that a(21323543) = 0, i.e., the odd number 2*21323543 + 1 = 42647087 cannot be written as the sum of a prime, a power of 3 and a power of 5.

			a(2) = 1 since 2*2+1 = 3 + 3^0 + 5^0 with 3 prime.
a(3) = 2 since 2*3+1 = 3 + 3^1 + 5^0 = 5 + 3^0 + 5^0 with 3 and 5 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Mixed sums of primes and other terms, in: Additive Number Theory (edited by D. Chudnovsky and G. Chudnovsky), pp. 341-353, Springer, New York, 2010.

Cf. A000040, A000244, A000351, A273812, A302982, A302984, A303233, A303234, A303338, A303363, A303389, A303393, A303399, A303428, A303401, A303432, A303434, A303539, A303540, A303541, A303543, A303601, A303637, A303639, A303656.

tab={};Do[r=0;Do[If[PrimeQ[2n+1-5^x-3^y],r=r+1],{x,0,Log[5,2n]},{y,0,Log[3,2n+1-5^x]}];tab=Append[tab,r],{n,1,70}];Print[tab]

A008566 Digits of powers of 5.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Irregular table with row length sequence A210435. - Jason Kimberley, Nov 26 2012

The constant whose decimal expansion is this sequence is irrational (Mahler, 1981). - Amiram Eldar, Mar 23 2025

			Triangle begins:
  1;
  5;
  2, 5;
  1, 2, 5;
  6, 2, 5;
  3, 1, 2, 5;
  1, 5, 6, 2, 5;
  7, 8, 1, 2, 5,
  3, 9, 0, 6, 2, 5;
  ...

Seiichi Manyama, Rows n = 0..100, flattened
Kurt Mahler, On some irrational decimal fractions, Journal of Number Theory, Vol. 13, No. 2 (1981), pp. 268-269.

Cf. A000351, A210435.

Cf. A000455, A008564, A008565, A008567, A008568, A008569, A008570, A008571, A008572, A008573, A010054.

Flatten[IntegerDigits/@(5^Range[0,20])] (* Harvey P. Dale, Jan 18 2012 *)

A111395 First digit of powers of 5.

Original entry on oeis.org

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

Offset: 0

Almerio A. Castro (almerio.castro(AT)gmail.com), Nov 11 2005

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A008952, A060956.

Cf. A000030, A000351, A008566.

First[IntegerDigits[#]]&/@(5^Range[0,100]) (* Harvey P. Dale, Jan 13 2015 *)

a(n) = digits(5^n)[1]; \\ Michel Marcus, Jan 07 2014

a(n) = A000030(A000351(n)). - Seiichi Manyama, Jul 15 2023

a(0)=1 prepended, and more terms from Michel Marcus, Jan 07 2014

A308566 Number of ways to write n as w^2 + x(x+1) + 4^y5^z with w,x,y,z nonnegative integers.

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 4, 3, 1, 3, 4, 2, 2, 3, 2, 4, 5, 2, 3, 5, 4, 6, 4, 2, 6, 8, 4, 4, 6, 3, 6, 8, 3, 4, 6, 6, 5, 5, 2, 6, 8, 3, 6, 4, 3, 6, 9, 2, 4, 7, 4, 6, 4, 4, 4, 8, 3, 4, 6, 4, 7, 8, 3, 4, 6, 5, 7, 5, 3, 7, 11, 3, 6, 6, 4, 8, 8, 2, 2, 10, 7, 9, 5, 5, 9, 10, 3, 6, 7, 3, 6, 11, 5, 5, 10, 7, 7, 8, 4, 6

Offset: 1

Zhi-Wei Sun, Jun 07 2019

nonn

Recall an observation of Euler: {w^2 + x*(x+1): w,x = 0,1,2,...} = {a*(a+1)/2 + b*(b+1)/2: a,b = 0,1,...}.
Conjecture: a(n) > 0 for all n > 0. Equivalently, each n = 1,2,3,... can be written as a*(a+1)/2 + b*(b+1)/2 + 4^c*5^d with a,b,c,d nonnegative integers.
See also A308584 for a similar conjecture.
We have verified a(n) > 0 for all n = 1..5*10^8.
a(n) > 0 for 0 < n < 10^10. - Giovanni Resta, Jun 08 2019

			a(1) = 1 with 1 = 0^2 + 0*1 + 4^0*5^0.
a(2) = 1 with 2 = 1^2 + 0*1 + 4^0*5^0.
a(3) = 1 with 3 = 0^2 + 1*2 + 4^0*5^0.
a(9) = 1 with 9 = 2^2 + 0*1 + 4^0*5^1.
a(303) = 1 with 303 = 16^2 + 6*7 + 4^0*5^1.
a(585) = 1 with 585 = 5^2 + 15*16 + 4^3*5^1.
a(37863) = 2 with 37863 = 166^2 + 101*102 + 4^0*5^1 = 179^2 + 26*27 + 4^5*5^1.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Mixed sums of squares and triangular numbers, Acta Arith. 127(2007), 103-113.

Cf. A000217, A000290, A000302, A000351, A002378, A303656, A303637, A308411, A308547, A308584.

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

A308584 Number of ways to write n as a(a+1)/2 + b(b+1)/2 + 5^c*8^d, where a,b,c,d are nonnegative integers with a <= b.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Jun 08 2019

nonn

Conjecture: a(n) > 0 for all n > 0. Equivalently, each n = 1,2,3,... can be written as w^2 + x*(x+1) + 5^y*8^z with w,x,y,z nonnegative integers.
We have verified a(n) > 0 for all n = 1..4*10^8.
See also A308566 for a similar conjecture.
a(n) > 0 for all 0 < n < 10^10. - Giovanni Resta, Jun 10 2019

			a(13) = 1 with 13 = 3*4/2 + 3*4/2 + 5^0*8^0.
a(48) = 1 with 48 = 5*6/2 + 7*8/2 + 5^1*8^0.
a(87) = 1 with 87 = 1*2/2 + 12*13/2 + 5^0*8^1.
a(90) = 1 with 90 = 4*5/2 + 10*11/2 + 5^2*8^0.
a(423) = 1 with 423 = 9*10/2 + 22*23/2 + 5^3*8^0.
a(517) = 1 with 517 = 17*18/2 + 24*25/2 + 5^0*8^2.
a(985) = 1 with 985 = 19*20/2 + 34*35/2 + 5^2*8^1.
a(2694) = 1 with 2694 = 7*8/2 + 68*69/2 + 5^1*8^2.
a(42507) = 1 with 42507 = 178*179/2 + 223*224/2 + 5^2*8^2.
a(544729) = 1 with 544729 = 551*552/2 + 857*858/2 + 5^5*8^1.
a(913870) = 1 with 913870 = 559*560/2 + 700*701/2 + 5^3*8^4.
a(1843782) = 1 with 1843782 = 808*809/2 + 1668*1669/2 + 5^6*8^1.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Mixed sums of squares and triangular numbers, Acta Arith. 127(2007), 103-113.

Cf. A000217, A000351, A001018, A303656, A303637, A308411, A308547, A308566.

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

A056451 Number of palindromes using a maximum of five different symbols.

Original entry on oeis.org

1, 5, 5, 25, 25, 125, 125, 625, 625, 3125, 3125, 15625, 15625, 78125, 78125, 390625, 390625, 1953125, 1953125, 9765625, 9765625, 48828125, 48828125, 244140625, 244140625, 1220703125, 1220703125, 6103515625, 6103515625, 30517578125, 30517578125, 152587890625, 152587890625

Offset: 0

Marks R. Nester

Number of achiral rows of n colors using up to five colors. For a(3) = 25, the rows are AAA, ABA, ACA, ADA, AEA, BAB, BBB, BCB, BDB, BEB, CAC, CBC, CCC, CDC, CEC, DAD, DBD, DCD, DDD, DED, EAE, EBE, ECE, EDE, and EEE. - Robert A. Russell, Nov 09 2018

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (0,5).

Column k=5 of A321391.
Cf. A000007, A016116, A056391, A056453, A056454, A056455, A056456, A057427,
Cf. A000351 (oriented), A032122 (unoriented), A032088(n>1) (chiral).

[5^Floor((n+1)/2): n in [0..40]]; // Vincenzo Librandi, Aug 16 2011

LinearRecurrence[{0,5},{1,5},30] (* or *) Riffle[5^Range[0, 20], 5^Range[20]] (* Harvey P. Dale, Jul 28 2018 *)
Table[5^Ceiling[n/2], {n,0,40}] (* Robert A. Russell, Nov 07 2018 *)

vector(40, n, n--; 5^floor((n+1)/2)) \\ G. C. Greubel, Nov 07 2018

a(n) = 5^floor((n+1)/2).
a(n) = 5*a(n-2). - Colin Barker, May 06 2012
G.f.: (1+5*x) / (1-5*x^2). - Colin Barker, May 06 2012 [Adapted to offset 0 by Robert A. Russell, Nov 07 2018]
a(n) = C(5,0)*A000007(n) + C(5,1)*A057427(n) + C(5,2)*A056453(n) + C(5,3)*A056454(n) + C(5,4)*A056455(n) + C(5,5)*A056456(n). - Robert A. Russell, Nov 08 2018
E.g.f.: cosh(sqrt(5)*x) + sqrt(5)*sinh(sqrt(5)*x). - Stefano Spezia, Jun 06 2023

a(0)=1 prepended by Robert A. Russell, Nov 07 2018

A118738 Number of ones in binary expansion of 5^n.

Original entry on oeis.org

1, 2, 3, 6, 5, 6, 7, 8, 12, 13, 11, 15, 13, 14, 17, 20, 20, 20, 24, 19, 26, 29, 25, 27, 30, 19, 31, 33, 29, 36, 37, 33, 39, 34, 42, 40, 44, 42, 38, 46, 53, 54, 49, 52, 52, 53, 50, 49, 54, 60, 58, 60, 54, 64, 58, 74, 61, 67, 74, 65, 61, 77, 74, 81, 86, 78, 87, 85, 82, 89, 83, 79

Offset: 0

Zak Seidov, May 22 2006

Also binary weight of 10^n, which is verified easily enough: 10^n = 2^n * 5^n; it is obvious that 2^n in binary is a single 1 followed by n 0's, therefore, in the binary expansion of 2^n * 5^n, the 2^n contributes only the trailing zeros. - Alonso del Arte, Oct 28 2012

Conjecture: a(n)/n -> log_4(5) = 1.160964... as n -> oo. - M. F. Hasler, Apr 17 2024

			a(2) = 3 because 5^2 = 25 is 11001, which has 3 on bits.

Robert Israel, Table of n, a(n) for n = 0..10000
Hugo Pfoertner, Plot of a(n) - 1.160964*n, +-4*sqrt(n), n up to 10^6.

Cf. A000120 (Hamming weight), A000351 (5^n), A061785 (floor(log_2(5^n))), A118737 (number of bits 0 in 5^n).

Cf. A011754 (analog for 3^n).

[&+Intseq(5^n, 2): n in [0..100]]; // Vincenzo Librandi, Nov 13 2024

seq(convert(convert(5^n,base,2),`+`),n=0..100); # Robert Israel, Dec 24 2017

Table[DigitCount[5^n, 2, 1], {n, 0, 71}] (* Ray Chandler, Sep 29 2006 *)

a(n) = hammingweight(5^n) \\ Iain Fox, Dec 24 2017

A118738 = lambda n: (5**n).bit_count() # For Python 3.10 and later. - M. F. Hasler, Apr 17 2024

a(n) + A118737(n) = A061785(n) + 1 for n >= 1. - Robert Israel, Dec 24 2017 [corrected by Amiram Eldar, Jul 27 2023]

a(n) = A000120(A000351(n)) = Hammingweight(5^n). - M. F. Hasler, Apr 17 2024

cp's OEIS Frontend

A179606 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + x)/(1 - 3*x - 5*x^2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A193685 5-Stirling numbers of the second kind.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A222281 T(n,k) = number of n X k 0..5 arrays with no entry increasing mod 6 by 5 rightwards or downwards, starting with upper left zero.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A303660 Number of ways to write 2*n+1 as p + 3^k + 5^m, where p is a prime, and k and m are nonnegative integers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A008566 Digits of powers of 5.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A111395 First digit of powers of 5.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A308566 Number of ways to write n as w^2 + x*(x+1) + 4^y*5^z with w,x,y,z nonnegative integers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A308584 Number of ways to write n as a*(a+1)/2 + b*(b+1)/2 + 5^c*8^d, where a,b,c,d are nonnegative integers with a <= b.

Views

Author

Keywords

Comments

Examples

A179606 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + x)/(1 - 3x - 5x^2).

A308566 Number of ways to write n as w^2 + x(x+1) + 4^y5^z with w,x,y,z nonnegative integers.

A308584 Number of ways to write n as a(a+1)/2 + b(b+1)/2 + 5^c*8^d, where a,b,c,d are nonnegative integers with a <= b.