A017329 -id:A017329 - cp's OEIS frontend

A017338 a(n) = (10*n + 5)^10.

9765625, 576650390625, 95367431640625, 2758547353515625, 34050628916015625, 253295162119140625, 1346274334462890625, 5631351470947265625, 19687440434072265625, 59873693923837890625, 162889462677744140625, 404555773570791015625, 931322574615478515625

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A016762, A017329.

[(10*n+5)^10: n in [0..10]]; // Vincenzo Librandi, Aug 02 2011

(10 Range[0,20]+5)^10 (* or *) LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{9765625,576650390625,95367431640625,2758547353515625,34050628916015625,253295162119140625,1346274334462890625,5631351470947265625,19687440434072265625,59873693923837890625,162889462677744140625},20] (* Harvey P. Dale, Jul 18 2021 *)

G.f.: -9765625*(x^10 + 59038*x^9 + 9116141*x^8 + 178300904*x^7 + 906923282*x^6 + 1527092468*x^5 + 906923282*x^4 + 178300904*x^3 + 9116141*x^2 + 59038*x + 1)/(x-1)^11. - Colin Barker, Nov 14 2012
From Amiram Eldar, Apr 18 2023: (Start)
a(n) = A017329(n)^10.
a(n) = 5^10 * A016762(n).
Sum_{n>=0} 1/a(n) = 31*Pi^10/28350000000000. (End)

A267755 Expansion of (1 + 2x + x^2 + x^3 + 4x^4 + 2*x^5)/(1 - x - x^5 + x^6).

Original entry on oeis.org

1, 3, 4, 5, 9, 12, 14, 15, 16, 20, 23, 25, 26, 27, 31, 34, 36, 37, 38, 42, 45, 47, 48, 49, 53, 56, 58, 59, 60, 64, 67, 69, 70, 71, 75, 78, 80, 81, 82, 86, 89, 91, 92, 93, 97, 100, 102, 103, 104, 108, 111, 113, 114, 115, 119, 122, 124, 125, 126, 130, 133, 135, 136, 137

Offset: 0

Bruno Berselli, Jan 20 2016

(m^k-1)/11 is a nonnegative integer when
. m is a member of this sequence and k is an odd multiple of 5 (A017329),
. m is a member of A017401 and k is odd but not multiple of 5 (A045572),
. m is a member of A175885 and k is even but not multiple of 5 (A217562),
. m is a member of A160542 and k is a positive multiple of 10 (A008592),
apart from the trivial case in which k=0.
Also, numbers that are congruent to {1, 3, 4, 5, 9} mod 11. Therefore, the product of two terms belongs to the sequence.
Union of this sequence and A267541 is A160542.
a(n) is prime for n = 1, 3, 10, 14, 17, 21, 24, 27, 30, 33, 40, 44, 47, ...

			From the linear recurrence:
(-A267541) ..., -13, -10, -8, -7, -6, -2, 1, 3, 4, 5, 9, 12, ... (A267755)

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Cf. A160542, A017329, A267541.

Related sequences (see the first comment): A017401, A160542, A175885.

m:=70; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+2*x+x^2+x^3+4*x^4+2*x^5)/(1-x-x^5+x^6)));

I:=[1,3,4,5,9,12]; [n le 6 select I[n] else Self(n-1)+Self(n-5)-Self(n-6): n in [1..70]]; // Vincenzo Librandi, Jan 21 2016

gf := (1 + 2*x + x^2 + x^3 + 4*x^4 + 2*x^5)/(1 - x - x^5 + x^6): deg := 64: series(gf, x, deg): seq(coeff(%, x, n), n=0..deg-1); # Peter Luschny, Jan 21 2016

CoefficientList[Series[(1 + 2 x + x^2 + x^3 + 4 x^4 + 2 x^5)/(1 - x - x^5 + x^6), {x, 0, 70}], x]
LinearRecurrence[{1, 0, 0, 0, 1, -1}, {1, 3, 4, 5, 9, 12}, 70]
Select[Range[140], MemberQ[{1, 3, 4, 5, 9}, Mod[#, 11]]&]

Vec((1+2*x+x^2+x^3+4*x^4+2*x^5)/(1-x-x^5+x^6)+O(x^70))

gf = (1 + 2*x + x^2 + x^3 + 4*x^4 + 2*x^5)/(1 - x - x^5 + x^6)
print(taylor(gf, x, 0, 63).list()) # Peter Luschny, Jan 21 2016

G.f.: (1 + 2*x + x^2 + x^3 + 4*x^4 + 2*x^5)/((1 - x)^2*(1 + x + x^2 + x^3 + x^4)).
a(n) = a(n-1) + a(n-5) - a(n-6).
a(-n) = -A267541(n-1).
a(n) = n + 1 + 2*floor(n/5) + 3*floor((n+1)/5) + floor((n+4)/5). - Ridouane Oudra, Sep 06 2023

A326728 A(n, k) = n(k - 1)k/2 - k, square array for n >= 0 and k >= 0 read by ascending antidiagonals.

Original entry on oeis.org

0, 0, -1, 0, -1, -2, 0, -1, -1, -3, 0, -1, 0, 0, -4, 0, -1, 1, 3, 2, -5, 0, -1, 2, 6, 8, 5, -6, 0, -1, 3, 9, 14, 15, 9, -7, 0, -1, 4, 12, 20, 25, 24, 14, -8, 0, -1, 5, 15, 26, 35, 39, 35, 20, -9, 0, -1, 6, 18, 32, 45, 54, 56, 48, 27, -10

Offset: 0

Peter Luschny, Aug 04 2019

A formal extension of the figurative numbers A139600 to negative n.

			[0] 0, -1, -2, -3, -4, -5, -6,  -7,  -8,  -9, -10, ... A001489
[1] 0, -1, -1,  0,  2,  5,  9,  14,  20,  27,  35, ... A080956
[2] 0, -1,  0,  3,  8, 15, 24,  35,  48,  63,  80, ... A067998
[3] 0, -1,  1,  6, 14, 25, 39,  56,  76,  99, 125, ... A095794
[4] 0, -1,  2,  9, 20, 35, 54,  77, 104, 135, 170, ... A014107
[5] 0, -1,  3, 12, 26, 45, 69,  98, 132, 171, 215, ... A326725
[6] 0, -1,  4, 15, 32, 55, 84, 119, 160, 207, 260, ... A270710
[7] 0, -1,  5, 18, 38, 65, 99, 140, 188, 243, 305, ...

Peter Luschny, Figurate number — a very short introduction. With plots from Stefan Friedrich Birkner.

Cf. A001489 (n=0), A080956 (n=1), A067998 (n=2), A095794 (n=3), A014107 (n=4), A326725 (n=5), A270710 (n=6).
Columns include A008585, A016933, A017329.
Cf. A139600.

A := (n, k) -> n*(k - 1)*k/2 - k:
seq(seq(A(n - k, k), k=0..n), n=0..11);

def A326728Row(n):
    x, y = 1, 1
    yield 0
    while True:
        yield -x
        x, y = x + y - n, y - n
for n in range(8):
    R = A326728Row(n)
print([next(R) for _ in range(11)])

A385623 Array read by ascending antidiagonals: A(n,k) is the number obtained by concatenation of n with k in that order, with k >= 0.

Original entry on oeis.org

0, 10, 1, 20, 11, 2, 30, 21, 12, 3, 40, 31, 22, 13, 4, 50, 41, 32, 23, 14, 5, 60, 51, 42, 33, 24, 15, 6, 70, 61, 52, 43, 34, 25, 16, 7, 80, 71, 62, 53, 44, 35, 26, 17, 8, 90, 81, 72, 63, 54, 45, 36, 27, 18, 9, 100, 91, 82, 73, 64, 55, 46, 37, 28, 19, 10, 110, 101, 92, 83, 74, 65, 56, 47, 38, 29, 110, 11

Offset: 0

Stefano Spezia, Jul 05 2025

			Array begins as:
   0,  1,  2,  3,  4,  5,  6,  7, ...
  10, 11, 12, 13, 14, 15, 16, 17, ...
  20, 21, 22, 23, 24, 25, 26, 27, ...
  30, 31, 32, 33, 34, 35, 36, 37, ...
  40, 41, 42, 43, 44, 45, 46, 47, ...
  50, 51, 52, 53, 54, 55, 56, 57, ...
  60, 61, 62, 63, 64, 65, 66, 67, ...
  ...

Stefano Spezia, Table of n, a(n) for n = 0..11475 (first 151 antidiagonals of the array, flattened)

Columns for k=0..9 give: A008592, A017281, A017293, A017305, A017317, A017329, A017341, A017353, A017365, A017377.

Cf. A001477 (1st row), A020338 (main diagonal), A055642, A385624 (antidiagonal sums).

A[n_,k_]:=FromDigits[Join[IntegerDigits[n],IntegerDigits[k]]]; Table[A[n,k],{n,0,6},{k,0,7}] (* or *)
A[n_,k_]:=If[k==0,10n,n*10^(Floor[Log10[k]]+1)+k]; Table[A[n-k,k],{n,0,11},{k,0,n}]//Flatten

T(n, k) = fromdigits(concat(digits(n), digits(k))); \\ Michel Marcus, Jul 06 2025

A(n,0) = 10*n and A(n,k) = n*10^(floor(log_10(k)) + 1) + k for k > 0.

A017331 a(n) = (10*n + 5)^3.

Original entry on oeis.org

125, 3375, 15625, 42875, 91125, 166375, 274625, 421875, 614125, 857375, 1157625, 1520875, 1953125, 2460375, 3048625, 3723875, 4492125, 5359375, 6331625, 7414875, 8615125, 9938375, 11390625, 12977875, 14706125, 16581375, 18609625, 20796875, 23149125, 25672375

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A002117, A016755, A017329.

[(10*n+5)^3: n in [0..35]]; // Vincenzo Librandi, Aug 02 2011

Table[(10*n + 5)^3, {n, 0, 30}] (* Amiram Eldar, Apr 18 2023 *)
LinearRecurrence[{4,-6,4,-1},{125,3375,15625,42875},30] (* Harvey P. Dale, Aug 23 2024 *)

a(n)=(10*n+5)^3 \\ Charles R Greathouse IV, Aug 02 2011

G.f.: 125*(x+1)*(x^2 + 22*x + 1)/(x-1)^4. - Colin Barker, Nov 14 2012
From Amiram Eldar, Apr 18 2023: (Start)
a(n) = A017329(n)^3.
a(n) = 5^3 * A016755(n).
Sum_{n>=0} 1/a(n) = 7*zeta(3)/1000.
Sum_{n>=0} (-1)^n/a(n) = Pi^3/4000. (End)

A017332 a(n) = (10*n + 5)^4.

Original entry on oeis.org

625, 50625, 390625, 1500625, 4100625, 9150625, 17850625, 31640625, 52200625, 81450625, 121550625, 174900625, 244140625, 332150625, 442050625, 577200625, 741200625, 937890625, 1171350625, 1445900625, 1766100625, 2136750625, 2562890625, 3049800625, 3603000625

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A016756, A017329.

[(10*n+5)^4: n in [0..35]]; // Vincenzo Librandi, Aug 02 2011

Table[(10*n + 5)^4, {n, 0, 30}] (* Amiram Eldar, Apr 18 2023 *)
LinearRecurrence[{5,-10,10,-5,1},{625,50625,390625,1500625,4100625},30] (* Harvey P. Dale, Aug 20 2024 *)

G.f.: -625*(x^4 + 76*x^3 + 230*x^2 + 76*x+1)/(x-1)^5. - Colin Barker, Nov 14 2012
From Amiram Eldar, Apr 18 2023: (Start)
a(n) = A017329(n)^4.
a(n) = 5^4 * A016756(n).
Sum_{n>=0} 1/a(n) = Pi^4/60000. (End)

A017334 a(n) = (10*n + 5)^6.

Original entry on oeis.org

15625, 11390625, 244140625, 1838265625, 8303765625, 27680640625, 75418890625, 177978515625, 377149515625, 735091890625, 1340095640625, 2313060765625, 3814697265625, 6053445140625, 9294114390625, 13867245015625, 20179187015625, 28722900390625, 40089475140625

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A016758, A017329.

[(10*n+5)^6: n in [0..25]]; // Vincenzo Librandi, Aug 02 2011

(10*Range[0,20]+5)^6 (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{15625,11390625,244140625,1838265625,8303765625,27680640625,75418890625},20] (* Harvey P. Dale, Aug 13 2013 *)

G.f.: -15625*(x^6 + 722*x^5 + 10543*x^4 + 23548*x^3 + 10543*x^2 + 722*x + 1)/(x-1)^7. - Colin Barker, Nov 14 2012
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(0)=15625, a(1)=11390625, a(2)=244140625, a(3)=1838265625, a(4)=8303765625, a(5)=27680640625, a(6)=75418890625. - Harvey P. Dale, Aug 13 2013
From Amiram Eldar, Apr 18 2023: (Start)
a(n) = A017329(n)^6.
a(n) = 5^6 * A016758(n).
Sum_{n>=0} 1/a(n) = Pi^6/15000000. (End)

A017335 a(n) = (10*n + 5)^7.

Original entry on oeis.org

78125, 170859375, 6103515625, 64339296875, 373669453125, 1522435234375, 4902227890625, 13348388671875, 32057708828125, 69833729609375, 140710042265625, 266001988046875, 476837158203125, 817215093984375, 1347646586640625, 2149422977421875, 3329565857578125

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A013665, A016759, A017329.

[(10*n+5)^7: n in [0..25]]; // Vincenzo Librandi, Aug 02 2011

Table[(10*n + 5)^7, {n, 0, 20}] (* Amiram Eldar, Apr 18 2023 *)

G.f.: 78125*(x+1)*(x^6 + 2178*x^5 + 58479*x^4 + 201244*x^3 + 58479*x^2 + 2178*x + 1)/(x-1)^8. - Colin Barker, Nov 13 2012
From Amiram Eldar, Apr 18 2023: (Start)
a(n) = A017329(n)^7.
a(n) = 5^7 * A016759(n).
Sum_{n>=0} 1/a(n) = 127*zeta(7)/10000000.
Sum_{n>=0} (-1)^n/a(n) = 61*Pi^7/14400000000. (End)

A017336 a(n) = (10*n + 5)^8.

Original entry on oeis.org

390625, 2562890625, 152587890625, 2251875390625, 16815125390625, 83733937890625, 318644812890625, 1001129150390625, 2724905250390625, 6634204312890625, 14774554437890625, 30590228625390625, 59604644775390625, 110324037687890625, 195408755062890625, 333160561500390625

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A016760, A017329.

[(10*n+5)^8: n in [0..20]]; // Vincenzo Librandi, Aug 02 2011

(10Range[0,20]+5)^8 (* Harvey P. Dale, Nov 02 2011 *)

G.f.: -((390625*(x^8 + 6552*x^7 + 331612*x^6 + 2485288*x^5 + 4675014*x^4 + 2485288*x^3 + 331612*x^2 + 6552*x + 1))/(x-1)^9). - Harvey P. Dale, Nov 02 2011
From Amiram Eldar, Apr 18 2023: (Start)
a(n) = A017329(n)^8.
a(n) = 5^8 * A016760(n).
Sum_{n>=0} 1/a(n) = 17*Pi^8/63000000000. (End)

A017337 a(n) = (10*n + 5)^9.

Original entry on oeis.org

1953125, 38443359375, 3814697265625, 78815638671875, 756680642578125, 4605366583984375, 20711912837890625, 75084686279296875, 231616946283203125, 630249409724609375, 1551328215978515625, 3517876291919921875, 7450580596923828125, 14893745087865234375, 28334269484119140625

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A013667, A016761, A017329.

[(10*n+5)^9: n in [0..15]]; // Vincenzo Librandi, Aug 02 2011

(10*Range[0,20]+5)^9 (* or *) LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{1953125,38443359375,3814697265625,78815638671875,756680642578125,4605366583984375,20711912837890625,75084686279296875,231616946283203125,630249409724609375},20] (* Harvey P. Dale, Jul 23 2016 *)

G.f.: 1953125*(x+1)*(x^8 + 19672*x^7 + 1736668*x^6 + 19971304*x^5 + 49441990*x^4 + 19971304*x^3 + 1736668*x^2 + 19672*x + 1)/(x-1)^10. -Colin Barker, Nov 13 2012
From Amiram Eldar, Apr 18 2023: (Start)
a(n) = A017329(n)^9.
a(n) = 5^9 * A016761(n).
Sum_{n>=0} 1/a(n) = 511*zeta(9)/1000000000.
Sum_{n>=0} (-1)^n/a(n) = 277*Pi^9/16128000000000. (End)

cp's OEIS Frontend

A017338 a(n) = (10*n + 5)^10.

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A267755 Expansion of (1 + 2*x + x^2 + x^3 + 4*x^4 + 2*x^5)/(1 - x - x^5 + x^6).

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A326728 A(n, k) = n*(k - 1)*k/2 - k, square array for n >= 0 and k >= 0 read by ascending antidiagonals.

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A385623 Array read by ascending antidiagonals: A(n,k) is the number obtained by concatenation of n with k in that order, with k >= 0.

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A017331 a(n) = (10*n + 5)^3.

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A017332 a(n) = (10*n + 5)^4.

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A017334 a(n) = (10*n + 5)^6.

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A267755 Expansion of (1 + 2x + x^2 + x^3 + 4x^4 + 2*x^5)/(1 - x - x^5 + x^6).

A326728 A(n, k) = n(k - 1)k/2 - k, square array for n >= 0 and k >= 0 read by ascending antidiagonals.