A104896 -id:A104896 - cp's OEIS frontend

A228275 A(n,k) = Sum_{i=1..k} n^i; square array A(n,k), n>=0, k>=0, read by antidiagonals.

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 6, 3, 0, 0, 4, 14, 12, 4, 0, 0, 5, 30, 39, 20, 5, 0, 0, 6, 62, 120, 84, 30, 6, 0, 0, 7, 126, 363, 340, 155, 42, 7, 0, 0, 8, 254, 1092, 1364, 780, 258, 56, 8, 0, 0, 9, 510, 3279, 5460, 3905, 1554, 399, 72, 9, 0

Offset: 0

Alois P. Heinz, Aug 19 2013

A(n,k) is the total sum of lengths of longest ending contiguous subsequences with the same value over all s in {1,...,n}^k:
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) = 14 = 3+1+1+2+2+1+1+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,  14,   30,    62,    126,    254, ...
  0, 3, 12,  39,  120,   363,   1092,   3279, ...
  0, 4, 20,  84,  340,  1364,   5460,  21844, ...
  0, 5, 30, 155,  780,  3905,  19530,  97655, ...
  0, 6, 42, 258, 1554,  9330,  55986, 335922, ...
  0, 7, 56, 399, 2800, 19607, 137256, 960799, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-10 give: A000004, A001477, A002378, A027444, A027445, A152031, A228290, A228291, A228292, A228293, A228294.
Rows n=0-11 give: A000004, A001477, A000918(k+1), A029858(k+1), A080674, A104891, A105281, A104896, A052379(k-1), A052386, A105279, A105280.
Main diagonal gives A031972.
Lower diagonal gives A226238.
Cf. A228250.

A:= (n, k)-> `if`(n=1, k, (n/(n-1))*(n^k-1)):
seq(seq(A(n, d-n), n=0..d), d=0..12);

a[0, 0] = 0; a[1, k_] := k; a[n_, k_] := n*(n^k-1)/(n-1); Table[a[n-k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 16 2013 *)

A(1,k) = k, else A(n,k) = n/(n-1)*(n^k-1).
A(n,k) = Sum_{i=1..k} n^i.
A(n,k) = Sum_{i=1..k+1} binomial(k+1,i)*A(n-i,k)*(-1)^(i+1) for n>k, given values A(0,k), A(1,k),..., A(k,k). - Yosu Yurramendi, Sep 03 2013

A168572 a(n) = Sum_{k=2..n}(7^k).

Original entry on oeis.org

0, 49, 392, 2793, 19600, 137249, 960792, 6725593, 47079200, 329554449, 2306881192, 16148168393, 113037178800, 791260251649, 5538821761592, 38771752331193, 271402266318400, 1899815864228849, 13298711049601992, 93090977347213993

Offset: 1

Vincenzo Librandi, Nov 30 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (8,-7).

[n le 1 select (n-1) else Self(n-1) + 7^n: n in [1..30] ]; // Vincenzo Librandi, Sep 24 2014

RecurrenceTable[{a[1] == 0, a[n] == a[n-1] + 7^n}, a, {n, 30}] (* Vincenzo Librandi, Sep 24 2014 *)
LinearRecurrence[{8,-7},{0,49}, 25] (* G. C. Greubel, Jul 26 2016 *)
Join[{0},Accumulate[7^Range[2,20]]] (* Harvey P. Dale, Jul 29 2019 *)

a(n)=7*(7^n - 7)/6 \\ Charles R Greathouse IV, Jul 26 2016

a(n) = 7^n + a(n-1), with a(1)=0.
From Robert Israel, Sep 24 2014: (Start)
a(n) = 7*(7^n - 7)/6.
G.f.: 49*x^2/((1-x)*(1-7*x)).
E.g.f.: 7*(exp(7*x) - 7*exp(x)+42)/6. (End)
a(n) = A104896(n) - 7. - Michel Marcus, Sep 25 2014
a(n) = 8*a(n-1) - 7*a(n-2). - G. C. Greubel, Jul 26 2016

Definition and examples simplified by Jon E. Schoenfield, Jun 19 2010

cp's OEIS Frontend

A228275 A(n,k) = Sum_{i=1..k} n^i; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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