A218725 -id:A218725

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A218724

a(n) = (21^n - 1)/20.

+10
37

0, 1, 22, 463, 9724, 204205, 4288306, 90054427, 1891142968, 39714002329, 833994048910, 17513875027111, 367791375569332, 7723618886955973, 162195996626075434, 3406115929147584115, 71528434512099266416, 1502097124754084594737, 31544039619835776489478

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Partial sums of powers of 21 (A009965); q-integers for q=21: diagonal k=1 in triangle A022185.

Partial sums are in A014905. Also, the sequence is related to A014938 by A014938(n) = n*a(n) - Sum_{i=0..n-1} a(i) for n > 0. - Bruno Berselli, Nov 06 2012

For n >= 1, 4*a(n) is the total number of holes in a certain box fractal (start with 21 boxes, 4 holes) after n iterations. See illustration in links. - Kival Ngaokrajang, Jan 27 2015

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..700

Kival Ngaokrajang, Illustration of initial terms

Index entries related to partial sums

Index entries for linear recurrences with constant coefficients, signature (22,-21).

FORMULA

a(n) = floor(21^n/20).

G.f.: x/((1-x)*(1-21*x)). - Bruno Berselli, Nov 06 2012

a(n) = 22*a(n-1) - 21*a(n-2). - Vincenzo Librandi, Nov 07 2012

a(n) = 21*a(n-1) + 1. - Kival Ngaokrajang, Jan 27 2015

a(n) = a(n-1) + 21^(n-1), n >= 1, a(0) = 0. - Wolfdieter Lang, Feb 02 2015

E.g.f.: exp(11*x)*sinh(10*x)/10. - Elmo R. Oliveira, Aug 29 2024

MATHEMATICA

LinearRecurrence[{22, -21}, {0, 1}, 40] (* Vincenzo Librandi, Nov 07 2012 *)

PROG

(PARI) A218724(n)=21^n\20

(Maxima) A218724(n):=(21^n-1)/20$ makelist(A218724(n), n, 0, 30); /* Martin Ettl, Nov 05 2012 */

(Magma) [n le 2 select n-1 else 22*Self(n-1) - 21*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

CROSSREFS

Cf. similar sequences of the form (k^n-1)/(k-1): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218725-A218734, A132469, A218736-A218753, A133853, A094028, A218723.

Cf. A009965, A014905, A014938, A022185.

KEYWORD

nonn,easy

AUTHOR

M. F. Hasler, Nov 04 2012

STATUS

approved

A125118

Triangle read by rows: T(n,k) = value of the n-th repunit in base (k+1) representation, 1<=k<=n.

+10
15

1, 3, 4, 7, 13, 21, 15, 40, 85, 156, 31, 121, 341, 781, 1555, 63, 364, 1365, 3906, 9331, 19608, 127, 1093, 5461, 19531, 55987, 137257, 299593, 255, 3280, 21845, 97656, 335923, 960800, 2396745, 5380840, 511, 9841, 87381, 488281, 2015539, 6725601, 19173961, 48427561, 111111111

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Eric Weisstein's World of Mathematics, Repunit

FORMULA

T(n, k) = Sum_{i=0..n-1} (k+1)^i.

T(n+1, k) = (k+1)*T(n, k) + 1.

Sum_{k=1..n} T(n, k) = A125120(n).

T(2*n-1, n) = A125119(n).

T(n, 1) = A000225(n).

T(n, 2) = A003462(n) for n>1.

T(n, 3) = A002450(n) for n>2.

T(n, 4) = A003463(n) for n>3.

T(n, 5) = A003464(n) for n>4.

T(n, 9) = A002275(n) for n>8.

T(n, n) = A060072(n+1).

T(n, n-1) = A023037(n) for n>1.

T(n, n-2) = A031973(n) for n>2.

T(n, k) = A055129(n, k+1) = A104878(n+k, k+1), 1<=k<=n. - Mathew Englander, Dec 19 2020

EXAMPLE

First 4 rows:

1: [1]_2

2: [11]_2 ........ [11]_3

3: [111]_2 ....... [111]_3 ....... [111]_4

4: [1111]_2 ...... [1111]_3 ...... [1111]_4 ...... [1111]_5

1: 1

2: 2+1 ........... 3+1

3: (2+1)*2+1 ..... (3+1)*3+1 ..... (4+1)*4+1

4: ((2+1)*2+1)*2+1 ((3+1)*3+1)*3+1 ((4+1)*4+1)*4+1 ((5+1)*5+1)*5+1.

MATHEMATICA

Table[((k+1)^n -1)/k, {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Aug 15 2022 *)

PROG

(Magma) [((k+1)^n -1)/k : k in [1..n], n in [1..12]]; // G. C. Greubel, Aug 15 2022

(SageMath)

def A125118(n, k): return ((k+1)^n -1)/k

flatten([[A125118(n, k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Aug 15 2022

CROSSREFS

This triangle shares some features with triangle A104878.

This triangle is a portion of rectangle A055129.

Each term of A110737 comes from the corresponding row of this triangle.

Diagonals (adjusting offset as necessary): A060072, A023037, A031973, A173468.

Columns (adjusting offset as necessary): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724, A218725, A218726, A218727, A218728, A218729, A218730, A218731, A218732, A218733, A218734, A132469, A218736, A218737, A218738, A218739, A218740, A218741, A218742, A218743, A218744, A218745, A218746, A218747, A218748, A218749, A218750, A218751, A218753, A218752.

Cf. A023037, A031973, A125119, A125120 (row sums).

KEYWORD

nonn,tabl,base

AUTHOR

Reinhard Zumkeller, Nov 21 2006

STATUS

approved

A104878

A sum-of-powers number triangle.

+10
13

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 7, 4, 1, 1, 5, 15, 13, 5, 1, 1, 6, 31, 40, 21, 6, 1, 1, 7, 63, 121, 85, 31, 7, 1, 1, 8, 127, 364, 341, 156, 43, 8, 1, 1, 9, 255, 1093, 1365, 781, 259, 57, 9, 1, 1, 10, 511, 3280, 5461, 3906, 1555, 400, 73, 10, 1, 1, 11, 1023, 9841, 21845

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,5

COMMENTS

Columns are partial sums of the columns of A004248. Row sums are A104879. Diagonal sums are A104880.

The rows of this triangle (apart from the initial "1" in each row) are the antidiagonals of rectangle A055129. The diagonals of this triangle (apart from the initial "1") are the rows of rectangle A055129. The columns of this triangle (apart from the leftmost column) are the same as the columns of rectangle A055129 but shifted downward. - Mathew Englander, Dec 21 2020

LINKS

Table of n, a(n) for n=0..70.

FORMULA

T(n, k) = if(k=1, n, if(k<=n, (k^(n-k+1)-1)/(k-1), 0));

G.f. of column k: x^k/((1-x)(1-k*x)). [corrected by Werner Schulte, Jun 05 2019]

T(n, k) = A069777(n+1,k)/A069777(n,k). [Johannes W. Meijer, Aug 21 2011]

T(n, k) = A055129(n+1-k, k) for n >= k > 0. - Mathew Englander, Dec 19 2020

EXAMPLE

Triangle starts:

1, 1;

1, 2, 1;

1, 3, 3, 1;

1, 4, 7, 4, 1;

1, 5, 15, 13, 5, 1;

1, 6, 31, 40, 21, 6, 1;

...

MAPLE

A104878 :=proc(n, k): if k = 0 then 1 elif k=1 then n elif k>=2 then (k^(n-k+1)-1)/(k-1) fi: end: for n from 0 to 7 do seq(A104878(n, k), k=0..n) od; seq(seq(A104878(n, k), k=0..n), n=0..10); # Johannes W. Meijer, Aug 21 2011

CROSSREFS

Cf. A004248 (first differences by column), A104879 (row sums), A104880 (antidiagonal sums), A125118 (version of this triangle with fewer terms).

This triangle (ignoring the leftmost column) is a rotation of rectangle A055129.

Columns (adjusting offset as necessary): A000012, A000027, A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724, A218725, A218726, A218727, A218728, A218729, A218730, A218731, A218732, A218733, A218734, A132469, A218736, A218737, A218738, A218739, A218740, A218741, A218742, A218743, A218744, A218745, A218746, A218747, A218748, A218749, A218750, A218751, A218753, A218752.

T(2n,n) gives A031973.

KEYWORD

easy,nonn,tabl

AUTHOR

Paul Barry, Mar 28 2005

STATUS

approved

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