A008603 -id:A008603 - cp's OEIS frontend

A008604 Multiples of 22.

0, 22, 44, 66, 88, 110, 132, 154, 176, 198, 220, 242, 264, 286, 308, 330, 352, 374, 396, 418, 440, 462, 484, 506, 528, 550, 572, 594, 616, 638, 660, 682, 704, 726, 748, 770, 792, 814, 836, 858, 880, 902, 924, 946, 968, 990, 1012, 1034, 1056, 1078, 1100, 1122, 1144

Offset: 0

N. J. A. Sloane

Even numbers for which the sum of "digits" base 100 is divisible by 11. - Daniel Forgues, Feb 22 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 334.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008593, A008602, A008603.

Range[0, 1500, 22] (* Vladimir Joseph Stephan Orlovsky, Jun 01 2011 *)
CoefficientList[Series[22 x / (x - 1)^2, {x, 0, 60}], x] (* Vincenzo Librandi, Jun 10 2013 *)
LinearRecurrence[{2,-1},{0,22},50] (* Harvey P. Dale, Aug 06 2018 *)

a(n)=22*n \\ Charles R Greathouse IV, Oct 07 2015

G.f.: 22*x/(x-1)^2. - Vincenzo Librandi, Jun 10 2013
a(n) = A008593(2n). - Daniel Forgues, Feb 22 2016
From Wesley Ivan Hurt, May 19 2024: (Start)
a(n) = 22*n.
a(n) = 2*a(n-1) - a(n-2). (End)
E.g.f.: 22*x*exp(x). - Stefano Spezia, Mar 02 2025

A119457 Triangle read by rows: T(n, 1) = n, T(n, 2) = 2*(n-1) for n>1 and T(n, k) = T(n-1, k-1) + T(n-2, k-2) for 2 < k <= n.

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 4, 6, 6, 5, 5, 8, 9, 10, 8, 6, 10, 12, 15, 16, 13, 7, 12, 15, 20, 24, 26, 21, 8, 14, 18, 25, 32, 39, 42, 34, 9, 16, 21, 30, 40, 52, 63, 68, 55, 10, 18, 24, 35, 48, 65, 84, 102, 110, 89, 11, 20, 27, 40, 56, 78, 105, 136, 165, 178, 144, 12, 22, 30, 45, 64, 91, 126, 170, 220, 267, 288, 233

Offset: 1

Reinhard Zumkeller, May 20 2006

			Triangle begins as:
   1;
   2,  2;
   3,  4,  3;
   4,  6,  6,  5;
   5,  8,  9, 10,  8;
   6, 10, 12, 15, 16, 13;
   7, 12, 15, 20, 24, 26,  21;
   8, 14, 18, 25, 32, 39,  42,  34;
   9, 16, 21, 30, 40, 52,  63,  68,  55;
  10, 18, 24, 35, 48, 65,  84, 102, 110,  89;
  11, 20, 27, 40, 56, 78, 105, 136, 165, 178, 144;
  12, 22, 30, 45, 64, 91, 126, 170, 220, 267, 288, 233;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened
Eric Weisstein's World of Mathematics, Fibonacci Number

Main diagonal: A023607(n).
Sums: A001891 (row), A355020 (signed row).
Columns: A000027(n) (k=1), A005843(n-1) (k=2), A008585(n-2) (k=3), A008587(n-3) (k=4), A008590(n-4) (k=5), A008595(n-5) (k=6), A008603(n-6) (k=7).
Diagonals: A000045(n+1) (k=n), A006355(n+1) (k=n-1), A022086(n-1) (k=n-2), A022087(n-2) (k=n-3), A022088(n-3) (k=n-4), A022089(n-4) (k=n-5), A022090(n-5) (k=n-6).
Cf. A023652, A112469.

A119457:= func< n,k | (n-k+1)*Fibonacci(k+1) >;
[A119457(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Apr 16 2025

(* First program *)
T[n_, 1] := n;
T[n_ /; n > 1, 2] := 2 n - 2;
T[n_, k_] /; 2 < k <= n := T[n, k] = T[n - 1, k - 1] + T[n - 2, k - 2];
Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Dec 01 2021 *)
(* Second program *)
A119457[n_,k_]:= (n-k+1)*Fibonacci[k+1];
Table[A119457[n,k], {n,13}, {k,n}]//Flatten (* G. C. Greubel, Apr 16 2025 *)

def A119457(n,k): return (n-k+1)*fibonacci(k+1)
print(flatten([[A119457(n,k) for k in range(1,n+1)] for n in range(1,13)])) # G. C. Greubel, Apr 16 2025

T(n, k) = (n-k+1)*T(k,k) for 1 <= k < n, with T(n, n) = A000045(n+1).
From G. C. Greubel, Apr 15 2025: (Start)
T(n, k) = (n-k+1)*Fibonacci(k+1).
Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = (1/2)*(1-(-1)^n)*A023652(floor((n+1)/2)) + (1+(-1)^n)*A001891(floor(n/2)).
Sum_{k=1..floor((n+1)/2)} (-1)^(k-1)*T(n-k+1, k) = (1/2)*(1-(-1)^n)*A112469(floor((n-1)/2)) + (1+(-1)^n)*A355020(floor((n-2)/2)). (End)

A008605 Multiples of 23.

Original entry on oeis.org

0, 23, 46, 69, 92, 115, 138, 161, 184, 207, 230, 253, 276, 299, 322, 345, 368, 391, 414, 437, 460, 483, 506, 529, 552, 575, 598, 621, 644, 667, 690, 713, 736, 759, 782, 805, 828, 851, 874, 897, 920, 943, 966, 989, 1012, 1035, 1058, 1081, 1104, 1127, 1150

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 335.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008602, A008603, A008604, A008606.

Range[0, 1500, 23] (* Vladimir Joseph Stephan Orlovsky, Jun 01 2011 *)
CoefficientList[Series[23 x / (x - 1)^2, {x, 0, 60}], x] (* Vincenzo Librandi, Jun 10 2013 *)

a(n)=23*n \\ Charles R Greathouse IV, Oct 07 2015

G.f.: 23*x/(x-1)^2. - Vincenzo Librandi, Jun 10 2013
E.g.f.: 23*x*exp(x). - Stefano Spezia, Mar 02 2025
From Elmo R. Oliveira, Apr 10 2025: (Start)
a(n) = 23*n = (A008604(n) + A008606(n))/2.
a(n) = 2*a(n-1) - a(n-2). (End)

A064762 a(n) = 21*n^2.

Original entry on oeis.org

0, 21, 84, 189, 336, 525, 756, 1029, 1344, 1701, 2100, 2541, 3024, 3549, 4116, 4725, 5376, 6069, 6804, 7581, 8400, 9261, 10164, 11109, 12096, 13125, 14196, 15309, 16464, 17661, 18900, 20181, 21504, 22869, 24276, 25725, 27216, 28749

Offset: 0

Roberto E. Martinez II, Oct 18 2001

Number of edges in a complete 7-partite graph of order 7n, K_n,n,n,n,n,n,n.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Similar sequences are listed in A244630.
Cf. A000217, A000290, A033428, A033581, A033582, A033583.
Cf. A008603, A195049.

[21*n^2 : n in [0..50]]; // Wesley Ivan Hurt, Jul 04 2014

21 Range[0, 50]^2 (* Wesley Ivan Hurt, Jul 04 2014 *)
LinearRecurrence[{3,-3,1},{0,21,84},40] (* Harvey P. Dale, Jul 29 2019 *)

a(n)=21*n^2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 42*n + a(n-1) - 21 for n > 0, a(0)=0. - Vincenzo Librandi, Aug 07 2010
a(n) = 21*A000290(n) = 7*A033428(n) = 3*A033582(n). - Omar E. Pol, Jul 03 2014
a(n) = t(7*n) - 7*t(n), where t(i) = i*(i+k)/2 for any k. Special case (k=1): a(n) = A000217(7*n) - 7*A000217(n). - Bruno Berselli, Aug 31 2017
From Elmo R. Oliveira, Nov 30 2024: (Start)
G.f.: 21*x*(1 + x)/(1-x)^3.
E.g.f.: 21*x*(1 + x)*exp(x).
a(n) = n*A008603(n) = A195049(2*n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A317321 Multiples of 21 and odd numbers interleaved.

Original entry on oeis.org

0, 1, 21, 3, 42, 5, 63, 7, 84, 9, 105, 11, 126, 13, 147, 15, 168, 17, 189, 19, 210, 21, 231, 23, 252, 25, 273, 27, 294, 29, 315, 31, 336, 33, 357, 35, 378, 37, 399, 39, 420, 41, 441, 43, 462, 45, 483, 47, 504, 49, 525, 51, 546, 53, 567, 55, 588, 57, 609, 59, 630, 61, 651, 63, 672, 65, 693, 67, 714, 69

Offset: 0

Omar E. Pol, Jul 25 2018

Partial sums give the generalized 25-gonal numbers (A303304).

a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 25-gonal numbers.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A008603 and A005408 interleaved.
Column 21 of A195151.
Sequences whose partial sums give the generalized k-gonal numbers: A026741 (k=5), A001477 (k=6), zero together with A080512 (k=7), A022998 (k=8), A195140 (k=9), zero together with A165998 (k=10), A195159 (k=11), A195161 (k=12), A195312 (k=13), A195817 (k=14).
Cf. A303304.

a[n_] := If[OddQ[n], n, 21*n/2]; Array[a, 70, 0] (* Amiram Eldar, Oct 14 2023 *)

concat(0, Vec(x*(1 + 21*x + x^2) / ((1 - x)^2*(1 + x)^2) + O(x^60))) \\ Colin Barker, Jul 29 2018

a(2n) = 21*n, a(2n+1) = 2*n + 1.
Multiplicative with a(2^e) = 21*2^(e-1), and a(p^e) = p^e for an odd prime p. - Amiram Eldar, Oct 14 2023
Dirichlet g.f.: zeta(s-1) * (1 + 19/2^s). - Amiram Eldar, Oct 26 2023

A139607 a(n) = 21*n + 7.

Original entry on oeis.org

7, 28, 49, 70, 91, 112, 133, 154, 175, 196, 217, 238, 259, 280, 301, 322, 343, 364, 385, 406, 427, 448, 469, 490, 511, 532, 553, 574, 595, 616, 637, 658, 679, 700, 721, 742, 763, 784, 805, 826, 847, 868, 889, 910, 931, 952, 973, 994

Offset: 0

Omar E. Pol, Apr 27 2008

Numbers of the 7th column of positive numbers in the square array of nonnegative and polygonal numbers A139600.

7th transversal numbers (or 7-transversal numbers): (A000217(7)-7)*n + 7.

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189. - From N. J. A. Sloane, Dec 01 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A000217, A008603, A016777, A057145, A139600, A152744.

[21*n+7: n in [0..60]]; // Vincenzo Librandi, Jul 23 2011

Range[7, 1500, 21] (* Vladimir Joseph Stephan Orlovsky, Jun 01 2011 *)
21*Range[0,50]+7 (* or *) LinearRecurrence[{2,-1},{7,28},50] (* Harvey P. Dale, Feb 23 2020 *)

a(n)=21*n+7 \\ Charles R Greathouse IV, Oct 05 2011

a(n) = A057145(n+2,7).
G.f.: 7*(1+2*x)/(x-1)^2. - R. J. Mathar, Jul 28 2016
From Elmo R. Oliveira, Apr 12 2024: (Start)
E.g.f.: 7*exp(x)*(1 + 3*x).
a(n) = 7*A016777(n) = A008603(n) + 7 = A152744(n+1) - A152744(n).
a(n) = 2*a(n-1) - a(n-2) for n >= 2. (End)

A169825 Multiples of 420.

Original entry on oeis.org

0, 420, 840, 1260, 1680, 2100, 2520, 2940, 3360, 3780, 4200, 4620, 5040, 5460, 5880, 6300, 6720, 7140, 7560, 7980, 8400, 8820, 9240, 9660, 10080, 10500, 10920, 11340, 11760, 12180, 12600, 13020, 13440, 13860, 14280, 14700, 15120, 15540, 15960, 16380, 16800

Offset: 0

N. J. A. Sloane, May 30 2010

Numbers that are divisible by all of 1,2,3,4,5,6,7.

Erich Friedman, What's Special About This Number? (See the entry for "420")
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A001477, A008589, A008596, A008597, A008602, A008603.

Cf. A135628, A169823, A169827, A249674.

Range[0, 100]*420 (* Paolo Xausa, Apr 17 2024 *)

a(n)=420*n \\ Charles R Greathouse IV, Apr 16 2024

a(n) = 420*n. - Wesley Ivan Hurt, Apr 11 2021
From Elmo R. Oliveira, Apr 16 2024: (Start)
G.f.: 420*x/(x-1)^2.
E.g.f.: 420*x*exp(x).
a(n) = 2*a(n-1) - a(n-2) for n >= 2.
a(n) = 7*A169823(n) = 14*A249674(n) = 15*A135628(n) = 20*A008603(n) = 21*A008602(n) = 28*A008597(n) = 30*A008596(n) = 60*A008589(n) = 420*A001477(n) = A169827(n)/2. (End)

A169827 Multiples of 840.

Original entry on oeis.org

0, 840, 1680, 2520, 3360, 4200, 5040, 5880, 6720, 7560, 8400, 9240, 10080, 10920, 11760, 12600, 13440, 14280, 15120, 15960, 16800, 17640, 18480, 19320, 20160, 21000, 21840, 22680, 23520, 24360, 25200, 26040, 26880, 27720, 28560, 29400, 30240, 31080, 31920

Offset: 0

N. J. A. Sloane, May 30 2010

Numbers that are divisible by all of 1,2,3,4,5,6,7,8.

Erich Friedman, What's Special About This Number?
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A001477, A005843, A008596, A008603, A135628, A169823, A169825.

Cf. A249674, A317095.

840*Range[0,40] (* Harvey P. Dale, Nov 03 2015 *)

a(n)=840*n \\ Charles R Greathouse IV, Apr 16 2024

From Elmo R. Oliveira, Apr 16 2024: (Start)
G.f.: 840*x/(x-1)^2.
E.g.f.: 840*x*exp(x).
a(n) = 840*n = 2*a(n-1) - a(n-2) for n >= 2.
a(n) = 2*A169825(n) = 14*A169823(n) = 21*A317095(n) = 28*A249674(n) = 30*A135628(n) = 40*A008603(n) = 60*A008596(n) = 420*A005843(n) = 840*A001477(n). (End)

A260042 Numbers k such that (4^k-1)/3 is not squarefree.

Original entry on oeis.org

9, 10, 18, 20, 21, 27, 30, 36, 40, 42, 45, 50, 54, 55, 60, 63, 68, 70, 72, 78, 80, 81, 84, 90, 99, 100, 105, 108, 110, 117, 120, 126, 130, 135, 136, 140, 144, 147, 150, 153, 155, 156, 160, 162, 165, 168, 170, 171, 180, 182, 189, 190, 198, 200, 204, 207, 210

Offset: 1

N. J. A. Sloane, Jul 22 2015

nonn

Contains all positive multiples of 9 (A008591), because 4^n-1 == 0 (mod 27) for these and (4^n-1)/3 is a multiple of 3^2 then. Contains also all positive multiples of 10 (A008592), because 4^n-1 == 0 (mod 125) for these and (4^n-1)/3 is a multiple of 5^2 then. Contains all positive multiples of 21 (A008603), because 4^n-1 == 0 (mod 147) for these and (4^n-1)/3 is a multiple of 7^2 then. - R. J. Mathar, Aug 02 2015

Complement of A259178. - Omar E. Pol, Aug 03 2015

			(4^9-1)/3 = 3^2*7*19*73 is not squarefree, so 9 is in the sequence. - _R. J. Mathar_, Aug 02 2015

James R. Buddenhagen, Posting to Math Fun Mailing List, Jul 22 2015.

Amiram Eldar, Table of n, a(n) for n = 1..169

Cf. A002450, A005117, A013929, A259178.

[n: n in [1..120]| not IsSquarefree((4^n-1) div 3)]; // Vincenzo Librandi, Jul 27 2015

Select[Range[120],!SquareFreeQ[(4^#-1)/3]&] (* Ivan N. Ianakiev, Jul 23 2015 *)

isok(k) = !issquarefree((4^k-1)/3); \\ Michel Marcus, Feb 25 2021

a(24)-a(31) from Ivan N. Ianakiev, Jul 23 2015
a(32)-a(45) from Chai Wah Wu, Jul 26 2015
a(46)-a(57) from Lars Blomberg, Aug 06 2017

A069499 Triangular numbers of the form 21*k.

Original entry on oeis.org

0, 21, 105, 210, 231, 378, 630, 861, 903, 1176, 1596, 1953, 2016, 2415, 3003, 3486, 3570, 4095, 4851, 5460, 5565, 6216, 7140, 7875, 8001, 8778, 9870, 10731, 10878, 11781, 13041, 14028, 14196, 15225, 16653, 17766, 17955, 19110, 20706, 21945, 22155, 23436, 25200

Offset: 1

Amarnath Murthy, Mar 30 2002

Intersection of A000217 and A008603. - Michel Marcus, Sep 17 2013

Let F(r) = Product_{n >= 0} 1 - q^(21*(14*n+r)). The sequence terms occur as the exponents in the expansion of (1 - q^21)*F(5)*F(6)*F(7)*F(8)*F(9)*F(13)*F(14)*F(15) = 1 - q^21 - q^105 + q^210 + q^231 - q^378 - q^630 + + - - ... (by the quintuple product identity). - Peter Bala, Dec 23 2024

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,2,-2,0,0,-1,1).

Cf. A000217, A008603, A069498, A117748.

a[0] := 0:a[1] := 6:a[2] := 14:a[3] := 20:a[4] := 21:a[5] := 27:a[6] := 35:a[7] := 41:seq((42*(floor(i/8))+a[i mod 8])*(42*(floor(i/8))+a[i mod 8]+1)/2,i=0..100);
# alternative program
A := proc (q) local n: for n from 0 to q do if type((1/21)*n*(n+1)/2, integer) then print(n*(n+1)/2) fi; od; end: A(250); # Peter Bala, Dec 24 2024

Select[21Range[1100],OddQ[Sqrt[8#+1]]&] (* Harvey P. Dale, Aug 16 2021 *)
Select[Accumulate[Range[0,300]],IntegerQ[#/21]&] (* Harvey P. Dale, Jun 12 2022 *)

G.f.: -21*x^2*(x^2-x+1)*(x^4+5*x^3+9*x^2+5*x+1) / ((x-1)^3*(x+1)^2*(x^2+1)^2). - Colin Barker, Sep 23 2013
From Peter Bala, Dec 24 2025: (Start)
a(n) is quasi-polynomial in n:
a(4*n) = 21 * n*(21*n - 1)/2; a(4*n+1) = 21 * n*(21*n + 1)/2;
a(4*n+2) = 21 * (3*n + 1)*(7*n + 2)/2; a(4*n+3) = 21 * (3*n + 2)*(7*n + 5)/2. (End)

More terms from Sascha Kurz, Apr 01 2002

a(1)=0 added and edited by Alois P. Heinz, Aug 19 2021

cp's OEIS Frontend

A008604 Multiples of 22.

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A119457 Triangle read by rows: T(n, 1) = n, T(n, 2) = 2*(n-1) for n>1 and T(n, k) = T(n-1, k-1) + T(n-2, k-2) for 2 < k <= n.

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A008605 Multiples of 23.

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A064762 a(n) = 21*n^2.

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A317321 Multiples of 21 and odd numbers interleaved.

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A139607 a(n) = 21*n + 7.

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A169825 Multiples of 420.

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