A001550 -id:A001550 - cp's OEIS frontend

A357824 Total number A(n,k) of k-tuples of semi-Dyck paths from (0,0) to (n,n-2*j) for j=0..floor(n/2); square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 3, 3, 1, 1, 2, 5, 6, 3, 1, 1, 2, 9, 14, 10, 4, 1, 1, 2, 17, 36, 42, 20, 4, 1, 1, 2, 33, 98, 190, 132, 35, 5, 1, 1, 2, 65, 276, 882, 980, 429, 70, 5, 1, 1, 2, 129, 794, 4150, 7812, 5705, 1430, 126, 6, 1, 1, 2, 257, 2316, 19722, 65300, 78129, 33040, 4862, 252, 6

Offset: 0

Alois P. Heinz, Oct 14 2022

			Square array A(n,k) begins:
  1,  1,   1,    1,     1,       1,        1,         1, ...
  1,  1,   1,    1,     1,       1,        1,         1, ...
  2,  2,   2,    2,     2,       2,        2,         2, ...
  2,  3,   5,    9,    17,      33,       65,       129, ...
  3,  6,  14,   36,    98,     276,      794,      2316, ...
  3, 10,  42,  190,   882,    4150,    19722,     94510, ...
  4, 20, 132,  980,  7812,   65300,   562692,   4939220, ...
  4, 35, 429, 5705, 78129, 1083425, 15105729, 211106945, ...

Alois P. Heinz, Antidiagonals n = 0..121, flattened
Wikipedia, Counting lattice paths

Columns k=0-7 give A008619, A001405, A000108, A003161, A129123, A361887, A382433, A361890.
Rows n=1-5 give: A000012, A007395, A000051, A001550, A074511.
Main diagonal gives A357825.
Cf. A008315, A120730.

b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
     `if`(x=0, 1, add(b(x-1, y+j), j=[-1, 1])))
    end:
A:= (n, k)-> add(b(n, n-2*j)^k, j=0..n/2):
seq(seq(A(n, d-n), n=0..d), d=0..12);

b[x_, y_] := b[x, y] = If[y < 0 || y > x, 0, If[x == 0, 1, Sum[b[x - 1, y + j], {j, {-1, 1}}]]];
A[n_, k_] := Sum[b[n, n - 2*j]^k, { j, 0, n/2}];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Oct 18 2022, after Alois P. Heinz *)

A(n,k) = Sum_{j=0..floor(n/2)} A008315(n,j)^k.

A(n,k) = Sum_{j=0..n} A120730(n,j)^k for k>=1, A(n,0) = A008619(n).

A074507 a(n) = 1^n + 3^n + 5^n.

Original entry on oeis.org

3, 9, 35, 153, 707, 3369, 16355, 80313, 397187, 1972809, 9824675, 49005273, 244672067, 1222297449, 6108298595, 30531927033, 152630937347, 763068593289, 3815084686115, 19074648589593, 95370918425027, 476847618556329

Offset: 0

Robert G. Wilson v, Aug 23 2002

Harvey P. Dale, Table of n, a(n) for n = 0..1000
T. A. Gulliver, Divisibility of sums of powers of odd integers, Int. Math. For. 5 (2010) 3059-3066, eq. (6).
D. Suprijanto, I. W. Suwarno, Observation on Sums of Powers of Integers Divisible by 3k-1, Applied Mathematical Sciences, Vol. 8, 2014, no. 45, 2211 - 2217.
Index entries for linear recurrences with constant coefficients, signature (9,-23,15).

Cf. A001550, A001576, A034513, A001579, A074501-A074580.

Table[1^n + 3^n + 5^n, {n, 0, 22}]
LinearRecurrence[{9,-23,15},{3,9,35},30] (* Harvey P. Dale, Mar 02 2022 *)

a(n) = 1 + 3^n + 5^n; \\ Michel Marcus, Aug 07 2017

a(n) = 8*a(n-1) - 15*a(n-2) + 8.

G.f.: 1/(1-x)+1/(1-3*x)+1/(1-5*x). E.g.f.: e^x+e^(3*x)+e^(5*x). [Mohammad K. Azarian, Dec 26 2008]

A074528 a(n) = 2^n + 3^n + 6^n.

Original entry on oeis.org

3, 11, 49, 251, 1393, 8051, 47449, 282251, 1686433, 10097891, 60526249, 362976251, 2177317873, 13062296531, 78368963449, 470199366251, 2821153019713, 16926788715971, 101560344351049, 609360902796251

Offset: 0

Robert G. Wilson v, Aug 23 2002

From Álvar Ibeas, Mar 24 2015: (Start)
Number of isomorphism classes of 3-fold coverings of a connected graph with circuit rank n+1 [Kwak and Lee].
Number of orbits of the conjugacy action of Sym(3) on Sym(3)^(n+1) [Kwak and Lee, 2001].
(End)

J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. [Added by N. J. A. Sloane, Nov 12 2009]

Hakan Icoz, Table of n, a(n) for n = 0..1000 (terms 0..200 from Vincenzo Librandi)
M. W. Hero and J. F. Willenbring, Stable Hilbert series as related to the measurement of quantum entanglement, Discrete Math., 309 (2010), 6508-6514.
J. H. Kwak and J. Lee, Isomorphism classes of graph bundles. Can. J. Math., 42(4), 1990, pp. 747-761.
Index entries for linear recurrences with constant coefficients, signature (11,-36,36).

Cf. A001550, A001576, A034513, A001579, A074501-A074580.
A246985 is essentially identical.
Third row of A160449, shifted.

[2^n + 3^n + 6^n: n in [0..25]]; // Vincenzo Librandi, Jun 11 2011

Table[2^n + 3^n + 6^n, {n, 0, 20}]
LinearRecurrence[{11,-36,36},{3,11,49},30] (* Harvey P. Dale, May 02 2016 *)

a(n)=2^n+3^n+6^n \\ Charles R Greathouse IV, Oct 07 2015

From Mohammad K. Azarian, Dec 26 2008: (Start)
G.f.: 1/(1-2*x)+1/(1-3*x)+1/(1-6*x).
E.g.f.: exp(2*x) + exp(3*x) + exp(6*x). (End)
a(n) = 11*a(n-1) - 36*a(n-2) + 36*a(n-3). - Wesley Ivan Hurt, Aug 21 2020

A341409 a(n) = (Sum_{k=1..3} k^n) mod n.

Original entry on oeis.org

0, 0, 0, 2, 1, 2, 6, 2, 0, 4, 6, 2, 6, 0, 6, 2, 6, 2, 6, 18, 15, 14, 6, 2, 1, 14, 0, 14, 6, 14, 6, 2, 3, 14, 31, 2, 6, 14, 36, 18, 6, 38, 6, 10, 36, 14, 6, 2, 13, 24, 36, 46, 6, 2, 1, 42, 36, 14, 6, 38, 6, 14, 36, 2, 16, 2, 6, 30, 36, 14, 6, 2, 6, 14, 51, 22, 17, 14, 6, 18, 0, 14, 6, 38, 21

Offset: 1

Seiichi Manyama, Feb 11 2021

Seiichi Manyama, Table of n, a(n) for n = 1..10000

(Sum_{k=1..m} k^n) mod n: A096196 (m=2), this sequence (m=3), A341410 (m=4), A341411 (m=5), A341412 (m=6), A341413 (m=7).

Cf. A001550, A045576, A056645, A220235.

a:= n-> add(i&^n, i=1..3) mod n:
seq(a(n), n=1..100);  # Alois P. Heinz, Feb 11 2021

a[n_] := Mod[Sum[k^n, {k, 1, 3}], n]; Array[a, 100] (* Amiram Eldar, Feb 11 2021 *)

PARI
```
a(n) = sum(k=1, 3, k^n)%n;
```

a(n) = A001550(n) mod n.

a(A056645(n)) = 0.

A074506 a(n) = 1^n + 3^n + 4^n.

Original entry on oeis.org

3, 8, 26, 92, 338, 1268, 4826, 18572, 72098, 281828, 1107626, 4371452, 17308658, 68703188, 273218426, 1088090732, 4338014018, 17309009348, 69106897226, 276040168412, 1102998412178, 4408506864308, 17623567104026

Offset: 0

Robert G. Wilson v, Aug 23 2002

Bruno Berselli, Table of n, a(n) for n = 0..1000.
Index entries for linear recurrences with constant coefficients, signature (8,-19,12).

Cf. A001550, A001576, A034513, A001579, A074501..A074580.

Equals A074605(n) + 1.

Table[1^n + 3^n + 4^n, {n, 0, 22}]
LinearRecurrence[{8,-19,12},{3,8,26},30] (* Harvey P. Dale, May 12 2025 *)

a(n) = 7*a(n-1) - 12*a(n-2) + 6 with a(0)=3, a(1)=8. - Vincenzo Librandi, Jul 19 2010
a(n) = 8*a(n-1) - 19*a(n-2) + 12*a(n-3). - R. J. Mathar, Jul 18 2010
From Mohammad K. Azarian, Dec 26 2008: (Start)
G.f.: 1/(1-x) + 1/(1-3*x) + 1/(1-4*x).
E.g.f.: e^x + e^(3*x) + e^(4*x). (End)

A074526 a(n) = 2^n + 3^n + 4^n.

Original entry on oeis.org

3, 9, 29, 99, 353, 1299, 4889, 18699, 72353, 282339, 1108649, 4373499, 17312753, 68711379, 273234809, 1088123499, 4338079553, 17309140419, 69107159369, 276040692699, 1102999460753, 4408508961459, 17623571298329

Offset: 0

Robert G. Wilson v, Aug 23 2002

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (9,-26,24).

Cf. A001550, A001576, A034513, A001579, A074501 - A074580.

[2^n + 3^n + 4^n: n in [0..25]]; // Vincenzo Librandi, Jun 11 2011

Table[2^n + 3^n + 4^n, {n, 0, 23}]
LinearRecurrence[{9,-26,24},{3,9,29},30] (* Harvey P. Dale, Jun 14 2022 *)

From Mohammad K. Azarian, Dec 26 2008: (Start)
G.f.: 1/(1-2*x)+1/(1-3*x)+1/(1-4*x).
E.g.f.: exp(2*x)+exp(3*x)+exp(4*x). (End)

A191488 A companion to Gould’s sequence A001316.

Original entry on oeis.org

4, 6, 8, 10, 8, 12, 16, 18, 8, 12, 16, 20, 16, 24, 32, 34, 8, 12, 16, 20, 16, 24, 32, 36, 16, 24, 32, 40, 32, 48, 64, 66, 8, 12, 16, 20, 16, 24, 32, 36, 16, 24, 32, 40, 32, 48, 64, 68, 16, 24, 32, 40, 32, 48, 64, 72, 32, 48, 64, 80, 64, 96, 128

Offset: 0

Johannes W. Meijer, Jun 05 2011

The row sums of the Sierpinski-Stern triangle A191372 are given by sequence A191487.
The differences diff1(n) = A191487(2*n+3) - A191487(2*n+1) lead to a peculiar number triangle, see the examples. The leading terms of the rows of the diff1(n) triangle clearly stand out from the rest of the terms and are given by A001550(p+1), p>=1; for p=0 this term is 7.
If we ignore the first term of the diff1(n) rows and reverse the order of the remaining terms we get sequence A191488, see the examples; more terms require a higher row number.
Both the diff1(n) and the diff2(n) sequences are related to Gould’s sequence A001316. We ignore the first term and reverse the order of the rest of the terms. The diff2(n) sequence leads directly to A001316, see A191487, while the diff1(n) sequence leads to A001316 in a slightly more complex way. We observe that for Gould’s sequence equation A001316((2*n+1)*2^p-1) = C(p)*A001316(n) with C(p) = 2^p holds, while for its companion A191488 equation A191488((2*n+1)*2^p-1) = C(p)*A001316(n) with C(p) = 2^(p+1)+2 holds; see the Maple program. Furthermore for both sequences a(2^p - 1) = C(p).

			The first few rows of diff1(n) as a triangle, row lengths A000079(p) with p>=0, are:
[7]
[14, 4]
[36, 8, 6, 4]
[98, 16, 12, 8, 10, 8, 6, 4]
[276, 32, 24, 16, 20, 16, 12, 8, 18, 16, 12, 8, 10, 8, 6, 4]
[794, 64, 48, 32, 40, 32, 24, 16, 36, 32, 24, 16, 20, 16, 12, 8, 34, 32, 24, 16, 20, 16, 12, 8, 18, 16, 12, 8, 10, 8, 6, 4]
The first few rows of diff1(n) reversed minus the first term are:
[4]
[4, 6, 8]
[4, 6, 8, 10, 8, 12, 16]
[4, 6, 8, 10, 8, 12, 16, 18, 8, 12, 16, 20, 16, 24, 32]
[4, 6, 8, 10, 8, 12, 16, 18, 8, 12, 16, 20, 16, 24, 32, 34, 8, 12, 16, 20, 16, 24, 32, 36, 16, 24, 32, 40, 32, 48, 64]

Cf. A001316, A191372, A191487.

nmax:=2^6; pmax:=ceil(log(nmax)/log(2)); A001316 := n -> if n<=-1 then 0 else 2^add(i, i=convert(n, base, 2)) fi: C := proc(p): C(p) := 2^(p+1)+2 end: for p from 0 to pmax do for n from 0 to nmax do a((2*n+1)*2^p-1):= C(p)*A001316(n) od: od: seq(a(n), n=0..nmax-2);

a((2*n+1)*2^p - 1) = C(p) * A001316(n) with C(p) = (2^(p+1)+2), p>=0.

a(2^p - 1) = 2^(p+1)+2 = A052548(p+1), p>=0.

A366298 Expansion of e.g.f. 1 / (-2 + Sum_{k=1..3} exp(-k*x)).

Original entry on oeis.org

1, 6, 58, 828, 15766, 375276, 10719118, 357202068, 13603819126, 582854637276, 27747071520478, 1453003753611108, 83005119616449286, 5136947527401250476, 342365553703113120238, 24447711909762202272948, 1862151878019906517540246, 150702660087903415402794876, 12913688931657425188926182398

Offset: 0

Ilya Gutkovskiy, Oct 06 2023

nonn

Cf. A001550, A004701, A005923, A319509, A366299, A366300, A366301, A366302.

nmax = 18; CoefficientList[Series[1/(-2 + Sum[Exp[-k x], {k, 1, 3}]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[(-1)^(k + 1) Binomial[n, k] (1 + 2^k + 3^k) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]

a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k+1) * binomial(n,k) * (1 + 2^k + 3^k) * a(n-k).

A191487 The row sums of the Sierpinski-Stern triangle A191372.

Original entry on oeis.org

0, 1, 3, 8, 9, 22, 24, 26, 27, 62, 66, 70, 72, 76, 78, 80, 81, 178, 186, 194, 198, 206, 210, 214, 216, 224, 228, 232, 234, 238, 240, 242, 243, 518, 534, 550, 558, 574, 582, 590, 594, 610, 618, 626, 630, 638, 642, 646

Offset: 0

Johannes W. Meijer, Jun 05 2011

nonn

The row sums a(n) of the Sierpinski-Stern triangle A191372 equal this sequence.
The differences diff1(n) = a(2*n+3) - a(2*n+1) and diff2(n) = (a(2*n+2) - a(2*n))/3, give rise to patterns that lead to Gould’s sequence A001316, see the examples.
The diff1(n) sequence as a triangle leads to Gould’s sequence in a peculiar way, see A191488. The leading terms of the diff1(n) rows are given by A001550(p+1), p>=1; for p=0 the leading term is 7. The rows sums of diff1(n) as a triangle equal A025192(p+2), p>=1; for p = 0 the row sum is 7. The row sums of diff1(n) as a triangle minus the first term equal 2*A053152(p+1).
The diff2(n) sequence as a triangle leads to Gould’s sequence A001316 in a simple way; just delete the first term and reverse the order of the rest of the terms; more terms require a higher row number. The leading terms of the diff2(n) rows are given by A085281(p), p>=0. The row sums of diff2(n) as a triangle equal A025192(p) and the row sums minus the first term equal A001047(p-1), p>=1; for p=0 the row sum minus the first term is 0.

			The first few rows of diff1(n) as a triangle, row lengths A000079(p) with p>=0, are:
[7]
[14, 4]
[36, 8, 6, 4]
[98, 16, 12, 8, 10, 8, 6, 4]
[276, 32, 24, 16, 20, 16, 12, 8, 18, 16, 12, 8, 10, 8, 6, 4]
[794, 64, 48, 32, 40, 32, 24, 16, 36, 32, 24, 16, 20, 16, 12, 8, 34, 32, 24, 16, 20, 16, 12, 8, 18, 16, 12, 8, 10, 8, 6, 4]
The first few rows of diff2(n) as a triangle, row lengths A011782(p) with p>=0, are:
[1]
[2]
[5, 1]
[13, 2, 2, 1]
[35, 4, 4, 2, 4, 2, 2, 1]
[97, 8, 8, 4, 8, 4, 4, 2, 8, 4, 4, 2, 4, 2, 2, 1]
[275, 16, 16, 8, 16, 8, 8, 4, 16, 8, 8, 4, 8, 4, 4, 2, 16, 8, 8, 4, 8, 4, 4, 2, 8, 4, 4, 2, 4, 2, 2, 1]

Cf. A191372, A191488, A001316, A001550, A025192, A053152, A085281, A001047, A007689.

Add the following lines to the Maple program of A191372.
A191487(0):=0: for d from 1 to 2^pmax do A191487(d):= 0: for Tx from 0 to 2^ceil(log(d)/ log(2))-1 do A191487(d):=A191487(d)+S2(Tx,d) od: od: seq(A191487(d),d=0..2^pmax);

a(2*n) = 3*a(n)
diff(n) = a(n+1) - a(n), diff1(n) = a(2*n+3) - a(2*n+1), diff2(n) = (a(2*n+2) - a(2*n))/3
a(2^n+1) - a(2^n) = A085281(n+1) = A007689(n) for n>=0
a(2^(n+1)+1) - a(2^(n+1)-1) = A001550(n+1) for n>=1.

A074502 a(n) = 1^n + 2^n + 6^n.

Original entry on oeis.org

3, 9, 41, 225, 1313, 7809, 46721, 280065, 1679873, 10078209, 60467201, 362799105, 2176786433, 13060702209, 78364180481, 470185017345, 2821109972993, 16926659575809, 101559956930561, 609359740534785, 3656158441111553

Offset: 0

Robert G. Wilson v, Aug 23 2002

From Jonathan Vos Post, Apr 16 2005: (Start)
Primes in this sequence include: a(2) = 41, a(10) = 60467201, a(18) = 101559956930561, a(34) = 286511799958070449017978881, a(58) = 1357602166130257152481187563448636039086735361.
Semiprimes in this sequence include: a(1) = 9 = 3^2, a(4) = 1313 = 13 * 101, a(6) = 46721 = 19 * 2459, a(8) = 1679873 = 13 * 129221, a(12) = 2176786433 = 19 * 114567707, a(13) = 13060702209 = 3 * 4353567403, a(28) = 6140942214465083932673 = 13 * 472380170343467994821, a(29) = 36845653286789429854209 = 3 * 12281884428929809951403, a(72) = 106387358923716524807713475752456398462534338499504504833 = 59670762632990981 * 1782905969847563299479030657520813855693. (End)

Index entries for linear recurrences with constant coefficients, signature (9,-20,12).

Cf. A001550, A001576, A034513, A001579, A074501 - A074580.

Table[1^n + 2^n + 6^n, {n, 0, 20}]
LinearRecurrence[{9,-20,12},{3,9,41},30] (* Harvey P. Dale, Aug 15 2017 *)

G.f.: 1/(1-x)+1/(1-2*x)+1/(1-6*x). E.g.f.: e^x+e^(2*x)+e^(6*x). [Mohammad K. Azarian, Dec 26 2008]

a(n) = 8*a(n-1) - 12*a(n-2) + 5, n> 1. [Gary Detlefs, Jun 21 2010]

cp's OEIS Frontend

A357824 Total number A(n,k) of k-tuples of semi-Dyck paths from (0,0) to (n,n-2*j) for j=0..floor(n/2); square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A074507 a(n) = 1^n + 3^n + 5^n.

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A074528 a(n) = 2^n + 3^n + 6^n.

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A341409 a(n) = (Sum_{k=1..3} k^n) mod n.

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A074506 a(n) = 1^n + 3^n + 4^n.

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A074526 a(n) = 2^n + 3^n + 4^n.

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A191488 A companion to Gould’s sequence A001316.

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A366298 Expansion of e.g.f. 1 / (-2 + Sum_{k=1..3} exp(-k*x)).

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