A228275 -id:A228275 - cp's OEIS frontend

A104896 a(0) = 0; a(n) = 7*a(n-1) + 7.

0, 7, 56, 399, 2800, 19607, 137256, 960799, 6725600, 47079207, 329554456, 2306881199, 16148168400, 113037178807, 791260251656, 5538821761599, 38771752331200, 271402266318407, 1899815864228856, 13298711049601999, 93090977347214000, 651636841430498007

Offset: 0

Alexandre Wajnberg, Apr 24 2005

Conjecture: this is also the number of integers from 0 to 10^n - 1 that lack 0, 1 and 2 as a digit.

Number of monic irreducible polynomials of degree 1 in GF(7)[x1,...,xn]. - Max Alekseyev, Jan 23 2006

Alois P. Heinz, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (8,-7).

Cf. A000918, A029858, A052379, A052386, A080674.

Row n=7 of A228275.

[(7/6)*(7^n -1): n in [0..30]]; // G. C. Greubel, Jun 09 2021

a:=n->sum (7^j,j=1..n): seq(a(n), n=0..30); # Zerinvary Lajos, Oct 03 2007

RecurrenceTable[{a[n]==7*a[n-1]+7,a[0]==0},a,{n,0,30}] (* Vaclav Kotesovec, Jul 25 2014 *)

concat(0, Vec(7*x/((x-1)*(7*x-1)) + O(x^30))) \\ Colin Barker, Jul 25 2014

[(7/6)*(7^n -1) for n in (0..30)] # G. C. Greubel, Jun 09 2021

a(n) = (7^(n+1) - 7) / 6. - Max Alekseyev, Jan 23 2006
a(n) = a(n-1) + 7^n with a(0)=0. - Vincenzo Librandi, Nov 13 2010
From Colin Barker, Jul 25 2014: (Start)
a(n) = 8*a(n-1) - 7*a(n-2).
G.f.: 7*x / ((x-1)*(7*x-1)). (End)
E.g.f.: (7/6)*(exp(7*x) - exp(x)). - G. C. Greubel, Jun 09 2021

A105281 a(0)=0; a(n) = 6*a(n-1) + 6.

Original entry on oeis.org

0, 6, 42, 258, 1554, 9330, 55986, 335922, 2015538, 12093234, 72559410, 435356466, 2612138802, 15672832818, 94036996914, 564221981490, 3385331888946, 20311991333682, 121871948002098, 731231688012594, 4387390128075570, 26324340768453426, 157946044610720562

Offset: 0

Alexandre Wajnberg, Apr 25 2005

Number of integers from 0 to (10^n) - 1 that lack 0, 1, 2 and 3 as a digit.

a(n) is the expected number of tosses of a single die needed to obtain for the first time a string of n consecutive 6's. - Jean M. Morales, Aug 04 2012

Index entries for linear recurrences with constant coefficients, signature (7,-6).

Cf. A000918, A003464, A029858, A052379, A052386, A080674.

Row n=6 of A228275.

a:=n->add(6^j,j=1..n): seq(a(n),n=0..30); # Zerinvary Lajos, Oct 03 2007

NestList[6#+6&,0,30] (* Harvey P. Dale, Jul 24 2012 *)

PARI
```
a(n)=if(n<0,0, (6^n-1)*6/5)
```

a(n) = 6^n + a(n-1) (with a(0)=0). - Vincenzo Librandi, Nov 13 2010
From Colin Barker, Jan 28 2013: (Start)
a(n) = 7*a(n-1) - 6*a(n-2).
G.f.: 6*x/((x-1)*(6*x-1)). (End)
From Elmo R. Oliveira, Mar 16 2025: (Start)
E.g.f.: 6*exp(x)*(exp(5*x) - 1)/5.
a(n) = 6*(6^n - 1)/5.
a(n) = 6*A003464(n). (End)

More terms from Harvey P. Dale, Jul 24 2012

A228250 Total sum A(n,k) of lengths of longest contiguous subsequences with the same value over all s in {1,...,n}^k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 6, 3, 0, 0, 4, 16, 12, 4, 0, 0, 5, 38, 45, 20, 5, 0, 0, 6, 86, 156, 96, 30, 6, 0, 0, 7, 188, 519, 436, 175, 42, 7, 0, 0, 8, 404, 1680, 1916, 980, 288, 56, 8, 0, 0, 9, 856, 5349, 8232, 5345, 1914, 441, 72, 9, 0

Offset: 0

Alois P. Heinz, Aug 18 2013

			A(4,1) = 4 = 1+1+1+1: [1], [2], [3], [4].
A(1,4) = 4: [1,1,1,1].
A(3,2) = 12 = 2+1+1+1+2+1+1+1+2: [1,1], [1,2], [1,3], [2,1], [2,2], [2,3], [3,1], [3,2], [3,3].
A(2,3) = 16 = 3+2+1+2+2+1+2+3: [1,1,1], [1,1,2], [1,2,1], [1,2,2], [2,1,1], [2,1,2], [2,2,1], [2,2,2].
Square array A(n,k) begins:
  0, 0,  0,   0,    0,     0,      0,       0, ...
  0, 1,  2,   3,    4,     5,      6,       7, ...
  0, 2,  6,  16,   38,    86,    188,     404, ...
  0, 3, 12,  45,  156,   519,   1680,    5349, ...
  0, 4, 20,  96,  436,  1916,   8232,   34840, ...
  0, 5, 30, 175,  980,  5345,  28610,  151115, ...
  0, 6, 42, 288, 1914, 12450,  79716,  504492, ...
  0, 7, 56, 441, 3388, 25571, 190428, 1403689, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Project Euler, Problem 427: n-sequences

Columns k=0-3 give: A000004, A001477, A002378, A152618(n+1).
Rows n=0-2 give: A000004, A001477, 2*A102712.
Main diagonal gives: A228194.
Cf. A228275.

b:= proc(n, m, s, i) option remember; `if`(m>i or s>m, 0,
      `if`(i=0, 1, `if`(i=1, n, `if`(s=1, (n-1)*add(
         b(n, m, h, i-1), h=1..m), b(n, m, s-1, i-1)+
      `if`(s=m, b(n, m-1, s-1, i-1), 0)))))
    end:
A:= (n, k)-> add(m*add(b(n, m, s, k), s=1..m), m=1..k):
seq(seq(A(n, d-n), n=0..d), d=0..12);

b[n_, m_, s_, i_] := b[n, m, s, i] = If[m>i || s>m, 0, If[i == 0, 1, If[i == 1, n, If[s == 1, (n-1)*Sum[b[n, m, h, i-1], {h, 1, m}], b[n, m, s-1, i-1] + If[s == m, b[n, m-1, s-1, i-1], 0]]]]]; A[n_, k_] := Sum[m*Sum[b[n, m, s, k], {s, 1, m}], {m, 1, k}]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Jan 19 2015, after Alois P. Heinz *)

A105280 a(0)=0; a(n) = 11*a(n-1) + 11.

Original entry on oeis.org

0, 11, 132, 1463, 16104, 177155, 1948716, 21435887, 235794768, 2593742459, 28531167060, 313842837671, 3452271214392, 37974983358323, 417724816941564, 4594972986357215, 50544702849929376, 555991731349223147, 6115909044841454628, 67274999493256000919, 740024994425816010120

Offset: 0

Alexandre Wajnberg, Apr 25 2005

Harvey P. Dale, Table of n, a(n) for n = 0..960
Index entries for linear recurrences with constant coefficients, signature (12,-11).

Cf. A000918, A016123, A029858, A052379, A052386, A080674.

Row n=11 of A228275.

a:=n-> add(11^j,j=1..n): seq(a(n),n=0..12); # Zerinvary Lajos, Oct 03 2007

NestList[11#+11&,0,20] (* or *) LinearRecurrence[{12,-11},{0,11},20] (* Harvey P. Dale, Dec 02 2023 *)

a(n) = 11^n + a(n-1) (with a(0)=0). - Vincenzo Librandi, Nov 13 2010
From Elmo R. Oliveira, May 24 2025: (Start)
G.f.: 11*x/((x-1)*(11*x-1)).
E.g.f.: 11*exp(x)*(exp(10*x) - 1)/10.
a(n) = 11*(11^n - 1)/10.
a(n) = 12*a(n-1) - 11*a(n-2).
a(n) = A016123(n) - 1. (End)

Corrected by T. D. Noe, Nov 07 2006

A125120 Sum of values of repunits of length n in base b representation with 1

Original entry on oeis.org

1, 7, 41, 296, 2829, 34637, 519049, 9197344, 188039787, 4356087231, 112754069273, 3224945523736, 100999970565337, 3437517630509497, 126332966608699441, 4986057436997869696, 210331809309776028055, 9443995455881145458715

Offset: 1

Reinhard Zumkeller, Nov 21 2006

			a(4) = [1111]_2 + [1111]_3 + [1111]_4 + [1111]_5 = ((2+1)*2+1)*2+1 + ((3+1)*3+1)*3+1 + ((4+1)*4+1)*4+1 + ((5+1)*5+1)*5+1 = 296.

G. C. Greubel, Table of n, a(n) for n = 1..385
Eric Weisstein's World of Mathematics, Repunit

Row sums of A125118.

Cf. A076015, A228275.

[(&+[(k^n -1)/(k-1): k in [2..n+1]]) : n in [1..30]]; // G. C. Greubel, Aug 14 2022

Table[Sum[(k^n -1)/(k-1), {k, 2, n+1}], {n, 30}] (* G. C. Greubel, Aug 14 2022 *)

a(n) = sum(k=1, n, sum(i=0, n-1, (k+1)^i)); \\ Michel Marcus, Dec 14 2020

[sum(((k+1)^n -1)/k for k in (1..n)) for n in (1..30)] # G. C. Greubel, Aug 14 2022

a(n) = Sum_{k=1..n} Sum_{i=0..n-1} (k+1)^i. [Corrected by Mathew Englander, Dec 14 2020]
a(n) = Sum_{k=1..n} A125118(n,k).
a(n+1) - a(n) = A076015(n+1) + A228275(n+2, n). - Mathew Englander, Dec 14 2020
a(n) = Sum_{j=2..n+1} (j^n - 1)/(j-1)

A228290 a(n) = n^6 + n^5 + n^4 + n^3 + n^2 + n.

Original entry on oeis.org

0, 6, 126, 1092, 5460, 19530, 55986, 137256, 299592, 597870, 1111110, 1948716, 3257436, 5229042, 8108730, 12204240, 17895696, 25646166, 36012942, 49659540, 67368420, 90054426, 118778946, 154764792, 199411800, 254313150, 321272406, 402321276, 499738092

Offset: 0

Alois P. Heinz, Aug 19 2013

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).

Column k=6 of A228275.

a:= n-> (1+(1+(1+(1+(1+n)*n)*n)*n)*n)*n:
seq(a(n), n=0..30);
# second Maple program:
a:= proc(n) option remember; `if`(n<2, 6*n,
      (n+1)*(n^2+n+1)*a(n-1)/((n-1)*(n^2-3*n+3)))
    end:
seq(a(n), n=0..30);
# third Maple program:
a:= n-> `if`(n=1, 6, (n^7-n)/(n-1)):
seq(a(n), n=0..30);

a(n) = n^6 + n^5 + n^4 + n^3 + n^2 + n; \\ Joerg Arndt, Sep 03 2013

a <- c(0, 6, 126, 1092, 5460, 19530, 55986)
for(n in (length(a)+1):30) a[n] <- 7*a[n-1] -21*a[n-2] +35*a[n-3] -35*a[n-4] +21*a[n-5] -7*a[n-6] +a[n-7]
a
[Yosu Yurramendi, Sep 03 2013]

G.f.: -6*x*(7*x^4+42*x^3+56*x^2+14*x+1)/(x-1)^7.
a(n) = (n+1)*(n^2+n+1)*a(n-1)/((n-1)*(n^2-3*n+3)) for n>1.
a(1) = 6, else a(n) = (n^7-n)/(n-1).
a(n) = 6*A059721(n) = n*(n+1)*(1+n+n^2)*(1-n+n^2). - R. J. Mathar, Aug 21 2013
a(n) = 7*a(n-1) -21*a(n-2) +35*a(n-3) -35*a(n-4) +21*a(n-5) -7*a(n-6) +a(n-7) for n>6, a(0)=0, a(1)=6, a(2)=126, a(3)=1092, a(4)=5460, a(5)=19530, a(6)=55986. - Yosu Yurramendi, Sep 03 2013

A228291 a(n) = Sum_{k=1..7} n^k.

Original entry on oeis.org

0, 7, 254, 3279, 21844, 97655, 335922, 960799, 2396744, 5380839, 11111110, 21435887, 39089244, 67977559, 113522234, 183063615, 286331152, 435984839, 648232974, 943531279, 1347368420, 1891142967, 2613136834, 3559590239, 4785883224, 6357828775, 8353082582

Offset: 0

Alois P. Heinz, Aug 19 2013

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8, -28, 56, -70, 56, -28, 8, -1).

Column k=7 of A228275.

a:= n-> `if`(n=1, 7, (n^8-n)/(n-1)):
seq(a(n), n=0..30);

Table[Total[n^Range[7]],{n,0,30}] (* or *) LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{0,7,254,3279,21844,97655,335922,960799},30] (* Harvey P. Dale, Dec 06 2018 *)

a <- c(0, 7, 254, 3279, 21844, 97655, 335922, 960799)
for(n in (length(a)+1):30) a[n] <- 8*a[n-1] -28*a[n-2] +56*a[n-3] -70*a[n-4] +56*a[n-5] -28*a[n-6] +8*a[n-7] -a[n-8]
a  [Yosu Yurramendi, Sep 03 2013]

G.f.: x*(x^6+78*x^5+981*x^4+2332*x^3+1443*x^2+198*x+7)/(x-1)^8.
a(1) = 7, else a(n) = (n^8-n)/(n-1).
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8) with n > 7, a(0)=0, a(1)=7, a(2)=254, a(3)=3279, a(4)=21844, a(5)=97655, a(6)=335922, a(7)=960799. - Yosu Yurramendi, Sep 03 2013

A228292 a(n) = Sum_{k=1..8} n^k.

Original entry on oeis.org

0, 8, 510, 9840, 87380, 488280, 2015538, 6725600, 19173960, 48427560, 111111110, 235794768, 469070940, 883708280, 1589311290, 2745954240, 4581298448, 7411742280, 11668193550, 17927094320, 26947368420, 39714002328, 57489010370, 81870575520, 114861197400

Offset: 0

Alois P. Heinz, Aug 19 2013

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9, -36, 84, -126, 126, -84, 36, -9, 1).

Column k=8 of A228275.

a:= n-> `if`(n=1, 8, (n^9-n)/(n-1)):
seq(a(n), n=0..30);

Table[Total[n^Range[8]],{n,0,30}] (* or *) LinearRecurrence[ {9,-36,84,-126,126,-84,36,-9,1},{0,8,510,9840,87380,488280,2015538,6725600,19173960},30] (* Harvey P. Dale, Jan 28 2014 *)

G.f.: -2*x*(85*x^6+1695*x^5+7134*x^4+8254*x^3+2769*x^2+219*x+4)/(x-1)^9.
a(1) = 8, else a(n) = (n^9-n)/(n-1).
a(0)=0, a(1)=8, a(2)=510, a(3)=9840, a(4)=87380, a(5)=488280, a(6)=2015538, a(7)=6725600, a(8)=19173960, a(n)=9*a(n-1)-36*a(n-2)+84*a(n-3)- 126*a(n-4)+ 126*a(n-5)- 84*a(n-6)+36*a(n-7)-9*a(n-8)+a(n-9). - Harvey P. Dale, Jan 28 2014

A228293 a(n) = Sum_{k=1..9} n^k.

Original entry on oeis.org

0, 9, 1022, 29523, 349524, 2441405, 12093234, 47079207, 153391688, 435848049, 1111111110, 2593742459, 5628851292, 11488207653, 22250358074, 41189313615, 73300775184, 125999618777, 210027483918, 340614792099, 538947368420, 833994048909, 1264758228162

Offset: 0

Alois P. Heinz, Aug 19 2013

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10, -45, 120, -210, 252, -210, 120, -45, 10, -1).

Column k=9 of A228275.

a:= n-> `if`(n=1, 9, (n^10-n)/(n-1)):
seq(a(n), n=0..30);

Table[Total[n^Range[9]],{n,0,30}] (* or *) LinearRecurrence[ {10, -45, 120, -210, 252, -210, 120, -45, 10, -1}, {0,9,1022,29523, 349524, 2441405, 12093234,47079207,153391688,435848049},30] (* Harvey P. Dale, Dec 12 2013 *)

G.f.: x*(x^8 +332*x^7 +11388*x^6 +77356*x^5 +153950*x^4 +99204*x^3 +19708*x^2 +932*x+9) / (x-1)^10.
a(1) = 9, else a(n) = (n^10-n)/(n-1).
a(0)=0, a(1)=9, a(2)=1022, a(3)=29523, a(4)=349524, a(5)=2441405, a(6)=12093234, a(7)=47079207, a(8)=153391688, a(9)=435848049, a(n) = 10*a(n-1) -45*a(n-2) +120*a(n-3) -210*a(n-4) +252*a(n-5) -210*a(n-6) +120*a(n-7) -45*a(n-8) +10*a(n-9) -a(n-10). - Harvey P. Dale, Dec 12 2013

A228294 a(n) = Sum_{k=1..10} n^k.

Original entry on oeis.org

0, 10, 2046, 88572, 1398100, 12207030, 72559410, 329554456, 1227133512, 3922632450, 11111111110, 28531167060, 67546215516, 149346699502, 311505013050, 617839704240, 1172812402960, 2141993519226, 3780494710542, 6471681049900, 10778947368420, 17513875027110

Offset: 0

Alois P. Heinz, Aug 19 2013

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1).

Column k=10 of A228275.

a:= n-> `if`(n=1, 10, (n^11-n)/(n-1)):
seq(a(n), n=0..30);

Table[Total[n^Range[10]],{n,0,30}] (* Harvey P. Dale, May 14 2014 *)

G.f.: -2*x*(341*x^8 +18392*x^7 +194612*x^6 +616880*x^5 +682550*x^4 +267344*x^3 +33308*x^2 +968*x+5) / (x-1)^11.

a(1) = 10, else a(n) = (n^11-n)/(n-1).

cp's OEIS Frontend

A104896 a(0) = 0; a(n) = 7*a(n-1) + 7.

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A105281 a(0)=0; a(n) = 6*a(n-1) + 6.

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A228250 Total sum A(n,k) of lengths of longest contiguous subsequences with the same value over all s in {1,...,n}^k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A105280 a(0)=0; a(n) = 11*a(n-1) + 11.

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Mathematica

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Extensions

A125120 Sum of values of repunits of length n in base b representation with 1

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Magma

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SageMath

Formula

A228290 a(n) = n^6 + n^5 + n^4 + n^3 + n^2 + n.

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A228291 a(n) = Sum_{k=1..7} n^k.

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